test_autograd.py

import contextlib
import gc
import sys
import io
import math
import tempfile
import time
import threading
import unittest
import warnings
from copy import deepcopy
from collections import OrderedDict
from itertools import product
from operator import mul
from functools import reduce
import torch
import json

# TODO: remove this global setting
# Autograd tests use double as the default dtype
torch.set_default_dtype(torch.double)

from torch import nn
from torch._six import inf, nan, istuple
from torch.autograd.gradcheck import gradgradcheck, gradcheck
from torch.autograd.function import once_differentiable
from torch.autograd.profiler import (profile, format_time, EventList,
                                     FunctionEvent, FunctionEventAvg,
                                     record_function, emit_nvtx)
import torch.autograd.functional as autogradF
from torch.utils.checkpoint import checkpoint
from torch.testing._internal.common_utils import (TEST_MKL, TEST_WITH_ROCM, TestCase, run_tests, skipIfNoLapack,
                                                  suppress_warnings, slowTest,
                                                  load_tests, random_symmetric_matrix,
                                                  IS_WINDOWS, IS_MACOS, CudaMemoryLeakCheck)
from torch.autograd import Variable, Function, detect_anomaly
from torch.autograd.function import InplaceFunction
from torch.testing import randn_like
from torch.testing._internal.common_methods_invocations import (method_tests,
                                                                create_input, unpack_variables,
                                                                EXCLUDE_FUNCTIONAL, EXCLUDE_GRADCHECK,
                                                                EXCLUDE_GRADGRADCHECK,
                                                                EXCLUDE_GRADGRADCHECK_BY_TEST_NAME,
                                                                exclude_tensor_method,
                                                                mask_not_all_zeros,
                                                                S)
from torch.testing._internal.common_device_type import (instantiate_device_type_tests, skipCUDAIfRocm,
                                                        onlyCPU, onlyCUDA, dtypes, dtypesIfCUDA,
                                                        deviceCountAtLeast, skipCUDAIfCudnnVersionLessThan,
                                                        skipCUDAIf)

_END_SENTINEL = object()

def getattr_qualified(obj, qname, default=None):
    """ Like getattr but works with qualified names

    e.g. getattr(torch, 'fft.rfft')
    """
    path = qname.split('.')
    if len(path) > 1 and path[0] == 'fft':
        import torch.fft  # noqa: F401

    for name in path:
        obj = getattr(obj, name, _END_SENTINEL)
        if obj is _END_SENTINEL:
            return default
    return obj

# load_tests from common_utils is used to automatically filter tests for
# sharding on sandcastle. This line silences flake warnings
load_tests = load_tests

import pickle

PRECISION = 1e-4


@contextlib.contextmanager
def backward_engine(engine):
    _prev_engine = Variable._execution_engine
    Variable._execution_engine = engine()
    try:
        yield
    finally:
        Variable._execution_engine = _prev_engine


def graph_desc(fn):
    if fn is None:
        return 'None'
    result = type(fn).__name__ + '('
    next_functions = fn.next_functions
    for next_fn, _ in next_functions:
        result += graph_desc(next_fn)
        result += ', '
    if next_functions:
        result = result[:-2]
    return result + ')'


class TestAutograd(TestCase):

    def _function_test(self, cls):
        x = torch.randn(5, 5, requires_grad=True)
        y = torch.randn(5, 5, requires_grad=True)
        result = cls.apply(x, 2, y)
        go = torch.ones((), requires_grad=True)
        result.sum().backward(go, create_graph=True)

        self.assertEqual(x.grad, y + torch.ones(5, 5))
        self.assertEqual(y.grad, x + torch.ones(5, 5) * 2)
        self.assertIsNotNone(x.grad.grad_fn)
        self.assertIsNotNone(y.grad.grad_fn)

        return x, y

    def test_function(self):
        class MyFunction(Function):

            @staticmethod
            def forward(ctx, tensor1, pyscalar, tensor2):
                ctx.pyscalar = pyscalar
                ctx.save_for_backward(tensor1, tensor2)
                return tensor1 + pyscalar * tensor2 + tensor1 * tensor2

            @staticmethod
            def backward(ctx, grad_output):
                var1, var2 = ctx.saved_tensors
                # NOTE: self is the test case here
                self.assertIsInstance(var1, torch.Tensor)
                self.assertIsInstance(var2, torch.Tensor)
                self.assertIsInstance(grad_output, torch.Tensor)
                return (grad_output + grad_output * var2, None,
                        grad_output * ctx.pyscalar + grad_output * var1)

        x, y = self._function_test(MyFunction)

        x_grad_desc = graph_desc(x.grad.grad_fn)
        y_grad_desc = graph_desc(y.grad.grad_fn)
        self.assertExpected(x_grad_desc, "x_grad_desc")
        self.assertExpected(y_grad_desc, "y_grad_desc")

    def test_once_differentiable(self):
        class MyFunction(Function):

            @staticmethod
            def forward(ctx, tensor1, pyscalar, tensor2):
                ctx.pyscalar = pyscalar
                ctx.save_for_backward(tensor1, tensor2)
                return tensor1 + pyscalar * tensor2 + tensor1 * tensor2

            @staticmethod
            @once_differentiable
            def backward(ctx, grad_output):
                self.assertFalse(torch.is_grad_enabled())
                t1, t2 = ctx.saved_tensors
                return (grad_output + grad_output * t2, None,
                        grad_output * ctx.pyscalar + grad_output * t1)

        x, y = self._function_test(MyFunction)
        self.assertEqual(graph_desc(x.grad.grad_fn),
                         'CopyBackwards(None, Error(AccumulateGrad(), None, AccumulateGrad()))')
        self.assertEqual(graph_desc(y.grad.grad_fn),
                         'CopyBackwards(None, Error(AccumulateGrad(), None, AccumulateGrad()))')

    def test_function_returns_input(self):
        class MyFunction(Function):
            @staticmethod
            def forward(ctx, x):
                return x

            @staticmethod
            def backward(ctx, grad):
                return grad * 2

        for shape in [(1,), ()]:
            v = torch.ones(shape, requires_grad=True)
            MyFunction.apply(v).backward()
            self.assertEqual(v.grad, torch.full(shape, 2.))

            with torch.no_grad():
                v.grad.zero_()
            MyFunction.apply(v.clone()).backward()
            self.assertEqual(v.grad, torch.full(shape, 2.))

    def test_function_returns_undefined_tensor(self):
        class MyFunction(Function):
            @staticmethod
            def forward(ctx, x):
                return x * 2

            @staticmethod
            def backward(ctx, grad):
                return None

        # Test that undefined tensors returned from custom backward function
        # are propagated as undefined and not tensor full of zeroes
        x = torch.ones(1, requires_grad=True)

        MyFunction.apply(x).backward()
        self.assertIsNone(x.grad)

        MyFunction.apply(x ** 2).backward()
        self.assertIsNone(x.grad)

        MyFunction.apply(x).sum().backward()
        self.assertIsNone(x.grad)

        self.assertIsNone(torch.autograd.grad(MyFunction.apply(x), x, allow_unused=True)[0])

    def test_materialize_grads(self):
        class MyFunction(Function):
            @staticmethod
            def forward(ctx, x):
                return x

            @staticmethod
            def backward(ctx, grad):
                self.assertEqual(grad, torch.zeros(1))
                return grad

        x = torch.ones(1, requires_grad=True)
        torch._C._functions.UndefinedGrad()(MyFunction.apply(x)).backward()

    def test_dont_materialize_grads(self):
        class MyFunction(Function):
            @staticmethod
            def forward(ctx, x):
                ctx.set_materialize_grads(False)
                return x

            @staticmethod
            def backward(ctx, grad):
                self.assertIsNone(grad)
                return grad

        x = torch.ones(1, requires_grad=True)
        torch._C._functions.UndefinedGrad()(MyFunction.apply(x)).backward()

    def test_legacy_function_deprecation_exception(self):
        # Trigger exception
        class MyFunction(Function):
            def forward(self, x):
                return x

            def backward(self, grad_output):
                return grad_output

        # Check exception occurs
        with self.assertRaisesRegex(
                RuntimeError,
                'Legacy autograd function with non-static forward method is deprecated'):
            MyFunction()(torch.randn(3, 4))

    class SimulateBackwardError(Function):

        @staticmethod
        def forward(ctx, input):
            return input.clone()

        @staticmethod
        @once_differentiable
        def backward(ctx, input):
            raise Exception("Simulate error on backward pass")

    def test_custom_function_exception(self):

        t1 = torch.rand((3, 3), requires_grad=True)
        t2 = torch.rand((3, 3), requires_grad=True)

        tmp = (t1 + t2) * (t1 + t2)
        t3 = TestAutograd.SimulateBackwardError.apply(tmp)
        with self.assertRaisesRegex(Exception, "Simulate error on backward pass"):
            t3.sum().backward()

    def test_invalid_gradients(self):
        class MyFunction(Function):
            @staticmethod
            def forward(ctx, x):
                return x * 2

            @staticmethod
            def backward(ctx, grad_output):
                return torch.randn(10, dtype=torch.float)

        with self.assertRaisesRegex(RuntimeError, 'expected shape'):
            input = torch.randn(5, 5, dtype=torch.float, requires_grad=True)
            MyFunction.apply(input).sum().backward()

    def test_accumulate_grad(self):
        grad_output = torch.ones(5, 5)

        def compute_grad(create_graph):
            x = torch.randn(5, 5, requires_grad=True)
            y = x + 2
            y.backward(grad_output, retain_graph=True)
            x_grad = x.grad
            x_grad_clone = x.grad.clone()
            y.backward(grad_output, create_graph=create_graph)
            return x_grad, x_grad_clone

        # Accumulate in-place when create_graph is False
        x_grad, x_grad_clone = compute_grad(create_graph=False)
        self.assertEqual(x_grad, x_grad_clone * 2)

        # Accumulate out-of-place when create_graph is False
        x_grad, x_grad_clone = compute_grad(create_graph=True)
        self.assertEqual(x_grad, x_grad_clone)

    def test_accumulate_grad_tensor_reference(self):
        def _test_grad_tensor(params_grad_tensor, backward_grad_tensor, should_preserve_reference, create_graph):
            params = torch.tensor([1.5, 1.5]).requires_grad_()
            params.grad = params_grad_tensor
            grad_saved = params.grad
            params.backward(backward_grad_tensor, create_graph=create_graph)
            self.assertEqual(id(grad_saved) == id(params.grad), should_preserve_reference)

        for create_graph in (False, True):
            # Accumulate dense gradient to sparse gradient will change the `params.grad` reference
            _test_grad_tensor(
                torch.sparse_coo_tensor(torch.tensor([[1, 1]]).long(), torch.tensor([1., 1.])),
                torch.tensor([1.5, 1.5]),
                False,  # never accumulates in-place
                create_graph)

            # Accumulate dense gradient to dense gradient will preserve the `params.grad` reference,
            # but only if create_graph=False.
            _test_grad_tensor(
                torch.tensor([1.5, 1.5]),
                torch.tensor([1.5, 1.5]),
                not create_graph,
                create_graph)

            # Accumulate sparse gradient to sparse gradient will preserve the `params.grad` reference,
            # but only if create_graph=False.
            _test_grad_tensor(
                torch.sparse_coo_tensor(torch.tensor([[1, 1]]).long(), torch.tensor([1., 1.])),
                torch.sparse_coo_tensor(torch.tensor([[1, 1]]).long(), torch.tensor([1., 1.])),
                not create_graph,
                create_graph)

    @skipIfNoLapack
    def test_slogdet_sign(self):
        a = torch.randn(3, 3, requires_grad=True)
        s, logdet = a.slogdet()

        # test that sign should not require grad
        self.assertFalse(s.requires_grad)

        # test that backward through computation involving sign works
        def sign_mul_logdet(mat):
            s, logdet = mat.slogdet()
            return s * logdet

        u, s, v = a.detach().svd()
        s.abs_().clamp_(0.0001)
        for sign in (-1, 1):
            s[-1] = sign
            mat = torch.chain_matmul(u, s.diag(), v.t()).requires_grad_()
            gradcheck(sign_mul_logdet, mat)
            gradgradcheck(sign_mul_logdet, mat)

    def test_sum_to_with_empty_dim_grad(self):
        a = torch.rand(4, 0, requires_grad=True)
        b = torch.rand(4, 1, requires_grad=True)
        c = a + b
        assert c.shape == (4, 0)
        c.sum().backward()

        self.assertEqual(b.grad, torch.zeros(4, 1))
        self.assertEqual(a.grad, torch.zeros(4, 0))

    def test_hessian_vector(self):
        x = torch.randn(2, 2, requires_grad=True)
        y = torch.randn(2, 2, requires_grad=True)

        z = x ** 2 + y * x + y ** 2
        z.backward(torch.ones(2, 2), create_graph=True)

        with torch.no_grad():
            x_grad = 2 * x + y
            y_grad = x + 2 * y
        self.assertEqual(x.grad, x_grad)
        self.assertEqual(y.grad, y_grad)

        grad_sum = 2 * x.grad + y.grad
        grad_sum.backward(torch.ones(2, 2))
        x_hv = torch.ones(2, 2) * 5
        y_hv = torch.ones(2, 2) * 4
        self.assertEqual(x.grad, x_grad + x_hv)
        self.assertEqual(y.grad, y_grad + y_hv)

    def test_grad(self):
        x = torch.randn(2, 2, requires_grad=True)
        y = torch.randn(2, 2, requires_grad=True)
        z = x ** 2 + y * x + y ** 2
        z.backward(torch.ones(2, 2), create_graph=True)

        x_grad = 2 * x + y
        y_grad = x + 2 * y
        self.assertEqual(x.grad, x_grad)
        self.assertEqual(y.grad, y_grad)

        grad_sum = 2 * x.grad + y.grad
        x_hv = torch.autograd.grad(
            outputs=[grad_sum], grad_outputs=[torch.ones(2, 2)],
            inputs=[x], create_graph=True)
        expected_x_hv = torch.ones(2, 2) * 5
        expected_y_hv = torch.ones(2, 2) * 4

        self.assertEqual(x_hv[0], expected_x_hv)
        self.assertEqual(x.grad, x_grad)
        self.assertEqual(y.grad, y_grad)

        # Test that grad_outputs and outputs have the same shape
        grad_out = torch.ones(2)
        try:
            torch.autograd.grad(
                outputs=[grad_sum], grad_outputs=[grad_out],
                inputs=[x], create_graph=True)
            self.assertFail()
        except RuntimeError as error:
            self.assertEqual(str(error), "Mismatch in shape: grad_output[0] has a shape of "
                             + str(grad_out.shape) + " and output[0] has a shape of "
                             + str(grad_sum.shape) + ".")

    def test_grad_nonleaf(self):
        x_init = torch.randn(2, 2, requires_grad=True)
        x = x_init
        y = torch.randn(2, 2, requires_grad=True)
        grad_output = torch.ones(2, 2)

        def fn(x):
            return x ** 2 + y * x + y ** 2

        for _ in range(5):
            grad_x, = torch.autograd.grad(
                fn(x), x, grad_outputs=grad_output, create_graph=True)

            grad_x_expected = 2 * x + y
            self.assertIsNone(y.grad)
            self.assertIsNone(x.grad)
            self.assertEqual(grad_x, grad_x_expected)

            x = x + 0.05 * grad_x

        val_init = fn(x_init).sum()
        val_final = fn(x).sum()
        self.assertGreater(val_final, val_init)

        x.backward(grad_output)
        self.assertIsNotNone(y.grad)
        self.assertIsNotNone(x_init.grad)

    def test_grad_nonleaf_many_outputs(self):
        # This checks an edge case for function callbacks
        # We want to capture two grads of a function, but can only
        # register a single callback.
        x = torch.randn(4, 2, requires_grad=True)
        a, b = x.chunk(2)

        def hook(*grads):
            hook_called[0] = True
        hook_called = [False]
        x.register_hook(hook)

        go = torch.randn(2, 2)
        grad_a, grad_b = torch.autograd.grad(
            (a + 2 * b), [a, b], grad_outputs=go, create_graph=True)

        self.assertEqual(grad_a, go)
        self.assertEqual(grad_b, go * 2)
        self.assertFalse(hook_called[0])
        self.assertIsNone(x.grad)

    def test_grad_nonleaf_register_hook(self):
        # This checks an edge case for register_hook.
        # We want to capture grad of a nonleaf tensor,
        # but avoid segfault during backward of other nonleaf tensors
        x = torch.randn(5, requires_grad=True)
        x_list = x.unbind()

        x0 = x_list[0]
        hook_results = [None]

        def hook(grad):
            hook_results[0] = grad
        x0.register_hook(hook)

        x_list[0].backward()
        self.assertEqual(hook_results[0], torch.tensor(1.))
        expected_grad = torch.tensor([1., 0, 0, 0, 0])
        self.assertEqual(x.grad, expected_grad)
        self.assertIsNone(x_list[0].grad)

        for i in range(1, 5, 1):
            x_list[i].backward()
            self.assertEqual(hook_results[0], None)
            expected_grad[i] = 1.0
            self.assertEqual(x.grad, expected_grad)
            self.assertIsNone(x_list[i].grad)

    def test_hook_with_no_name(self):
        # Create a hook that do not have a __name__ attribute
        class MyHookClass:
            def __call__(self, grad):
                return grad.clone()

        x = torch.randn(5, requires_grad=True).clone()
        x.register_hook(MyHookClass())
        x.sum().backward()
        # Should run fine

    def test_sharded_grad(self):
        leaves = [torch.zeros(5, 5, requires_grad=True) for _ in range(10)]
        intermediates = [l * i + l * l for i, l in enumerate(leaves)]
        loss = sum(v * i for i, v in enumerate(intermediates)).sum()

        # define a helper for dividing intermediates into groups
        def group(l, group_size):
            return (l[i:i + group_size] for i in range(0, len(l), group_size))

        # Compute the d loss / d intermediates in chunks of shard_size
        shard_size = 2
        d_intermediates = [d_i for intermediates_batch in group(intermediates, shard_size)
                           for d_i in torch.autograd.grad(loss, intermediates_batch)]
        # Compute rest of backward pass
        torch.autograd.backward(intermediates, d_intermediates)

        for i, l in enumerate(leaves):
            self.assertEqual(l.grad, i * i * (1 + l))

    def test_backward_badcalls(self):
        x = torch.ones(1)
        with self.assertRaisesRegex(RuntimeError, 'does not require grad'):
            x.backward()

    def test_grad_badcalls(self):
        x = torch.ones(1)
        y = x ** 2
        with self.assertRaisesRegex(RuntimeError, 'does not require grad'):
            torch.autograd.grad(x, y)
        with self.assertRaisesRegex(RuntimeError, 'does not require grad'):
            torch.autograd.grad(y, x)

        x = torch.ones(1, requires_grad=True)
        y = x ** 2
        torch.autograd.grad(y, x)  # this should succeed now

    def test_grad_fn_badcalls(self):
        error_regex = 'expected .* arguments, got .* instead'
        x = torch.ones(1, requires_grad=True)
        y = x ** 2
        with self.assertRaisesRegex(TypeError, error_regex):
            y.grad_fn(x.detach(), x.detach())  # too many
        with self.assertRaisesRegex(TypeError, error_regex):
            y.grad_fn()  # too few

        y.grad_fn(x.detach())  # this should succeed

    def test_grad_unreachable(self):
        x = torch.ones(1, requires_grad=True)
        y = torch.ones(1, requires_grad=True)
        # Make sure x and y have grad accumulators allocated
        z = x * 2
        w = y * 2

        grad_x, grad_y = torch.autograd.grad(x * 2, [x, y], allow_unused=True)
        self.assertEqual(grad_x, x * 2)
        self.assertIsNone(grad_y)

        # This is slightly different than the case above, because z doesn't even
        # have a grad accumulator allocated.
        z = torch.ones(1, requires_grad=True)
        grad_x, grad_z = torch.autograd.grad(x * 2, [x, z], allow_unused=True)
        self.assertEqual(grad_x, x * 2)
        self.assertIsNone(grad_z)

        # allow_unused=False, but grads contains None inside, should throw
        with self.assertRaisesRegex(RuntimeError,
                                    "Set allow_unused=True"):
            grad_x, grad_y = torch.autograd.grad(x * 2, [x, y], allow_unused=False)

    def test_hooks(self):
        x = torch.ones(5, 5, requires_grad=True)
        y = torch.ones(5, 5) * 4
        y.requires_grad_(True)

        counter = [0]

        def bw_hook(inc, grad):
            self.assertIsInstance(grad, torch.Tensor)
            counter[0] += inc

        z = x ** 2 + x * 2 + x * y + y
        x.register_hook(lambda *args: bw_hook(0, *args))
        test = z.register_hook(lambda *args: bw_hook(1, *args))
        z.backward(torch.ones(5, 5), retain_graph=True)
        self.assertEqual(counter[0], 1)

        test2 = z.register_hook(lambda *args: bw_hook(2, *args))
        z.backward(torch.ones(5, 5), retain_graph=True)
        self.assertEqual(counter[0], 4)

        test2.remove()
        z.backward(torch.ones(5, 5), retain_graph=True)
        self.assertEqual(counter[0], 5)

        def bw_hook_modify(grad):
            return grad.mul(2)

        test.remove()
        z.register_hook(bw_hook_modify)
        with torch.no_grad():
            y.grad.zero_()
        z.backward(torch.ones(5, 5), retain_graph=True)
        self.assertEqual(y.grad, (x + 1) * 2)

        y.register_hook(bw_hook_modify)
        with torch.no_grad():
            y.grad.zero_()
        z.backward(torch.ones(5, 5))
        self.assertEqual(y.grad, (x + 1) * 4)

    def test_hooks_cpp(self):
        # Tests hooks for autograd function implemented in C++
        bn = torch.nn.BatchNorm1d(5, affine=False)
        bn.eval()

        counter = [0]

        def bw_hook(grad):
            counter[0] += 1
            return grad * 2

        x = torch.ones(5, 5, requires_grad=True)
        z = bn(x)
        z.register_hook(bw_hook)
        z.sum().backward()

        self.assertEqual(counter[0], 1, msg='bw_hook not called')
        self.assertEqual(x.grad, torch.ones(5, 5) * 2, atol=1e-5, rtol=0)

    def test_hook_none(self):
        # WARNING: this is a test for autograd internals.
        # You should never have to use such things in your code.
        class NoneGradientFunction(Function):
            @staticmethod
            def forward(ctx, x, y):
                assert ctx.needs_input_grad[0]
                assert not ctx.needs_input_grad[1]
                return x, y

            @staticmethod
            def backward(ctx, grad_x, grad_y):
                return grad_x, None

        was_called = [False]

        def hook(grad):
            self.assertIsNotNone(grad)
            was_called[0] = True

        x = torch.randn(5, 5, requires_grad=True)
        y = torch.randn(5, 5)
        rx, ry = NoneGradientFunction.apply(x, y)
        rx.register_hook(hook)
        ry.register_hook(hook)
        sum(rx, ry).sum().backward()
        self.assertTrue(was_called[0])

    def test_retain_grad(self):
        input = torch.rand(1, 3, requires_grad=True)
        h1 = input * 3
        out = (h1 * h1).sum()

        # It should be possible to call retain_grad() multiple times
        h1.retain_grad()
        h1.retain_grad()

        # Gradient should be accumulated
        out.backward(retain_graph=True)
        self.assertEqual(h1 * 2, h1.grad)
        out.backward(retain_graph=True)
        self.assertEqual(h1 * 4, h1.grad)

        with torch.no_grad():
            input.grad.zero_()
        # It should be a no-op for leaves
        input.retain_grad()
        input.retain_grad()
        out.backward()
        self.assertEqual(input * 18, input.grad)

    def test_retain_grad_cycle(self):
        import gc
        import weakref
        counter = [0]
        refs = [None]

        x = torch.ones(5, 5, requires_grad=True)

        def run_test():
            y = x * 2
            y.retain_grad()

            def inc(*args):
                counter[0] += 1
            refs[0] = weakref.ref(y, inc)
            return y / 2

        z = run_test()
        gc.collect()
        self.assertIsNone(refs[0]())
        self.assertEqual(counter[0], 1)
        z.sum().backward()

    def test_backward(self):
        v = torch.randn(5, 5, requires_grad=True)
        x = torch.randn(5, 5, requires_grad=True)
        y = (torch.rand(5, 5) + 0.1).requires_grad_(True)
        z = torch.randn(5, 5, requires_grad=True)
        grad_output = torch.randn(5, 5)

        v.backward(grad_output)
        self.assertEqual(v.grad, grad_output)

        a = x + (y * z) + 4 * z ** 2 * x / y
        a.backward(grad_output)
        x_grad = 4 * z.pow(2) / y + 1
        y_grad = z - 4 * x * z.pow(2) / y.pow(2)
        z_grad = 8 * x * z / y + y
        self.assertEqual(x.grad, x_grad * grad_output)
        self.assertEqual(y.grad, y_grad * grad_output)
        self.assertEqual(z.grad, z_grad * grad_output)

    def test_sparse_backward(self):
        class FixedGradientFunction(Function):
            @staticmethod
            def forward(ctx, x, grad_x):
                ctx.save_for_backward(grad_x)
                return x

            @staticmethod
            def backward(ctx, grad_x):
                saved_grad_x, = ctx.saved_tensors
                return saved_grad_x, None

        size = torch.Size([6, 3, 2])
        i1 = torch.LongTensor([
            [0, 3, 4],
            [0, 2, 2],
        ])
        v1 = torch.DoubleTensor([[1, 2], [4, 5], [7, 8]])
        sparse_grad1 = torch.sparse.DoubleTensor(i1, v1, size)
        i2 = torch.LongTensor([
            [0, 1, 3, 4],
            [0, 1, 2, 2],
        ])
        v2 = torch.DoubleTensor([[1, 2], [4, 3], [4, 5], [7, 8]])
        sparse_grad2 = torch.sparse.DoubleTensor(i2, v2, size)
        dense_grad = torch.rand(size).double()
        fn = FixedGradientFunction

        # sparse first
        x = torch.randn(size, requires_grad=True)
        (fn.apply(x, sparse_grad1) + fn.apply(x, dense_grad) + fn.apply(x, sparse_grad2)).sum().backward()
        self.assertEqual(x.grad, dense_grad + sparse_grad1 + sparse_grad2)
        # dense first
        x = torch.randn(size, requires_grad=True)
        (fn.apply(x, dense_grad) + fn.apply(x, sparse_grad1) + fn.apply(x, sparse_grad2)).sum().backward()
        self.assertEqual(x.grad, dense_grad + sparse_grad1 + sparse_grad2)
        # sparse only
        x = torch.randn(size, requires_grad=True)
        (fn.apply(x, sparse_grad1) + fn.apply(x, sparse_grad2)).sum().backward()
        self.assertEqual(x.grad, sparse_grad1 + sparse_grad2)

    def test_sparse_mm_backward(self):
        size = (3, 3)
        sparse = torch.sparse_coo_tensor(size, requires_grad=True)
        dense = torch.randn(size, requires_grad=True)

        with self.assertRaisesRegex(
                RuntimeError,
                "The backward pass for this operation requires the 'mat1' tensor to be strided,"):
            z = dense.addmm(sparse, dense)

        mm_test_cases = [
            # a requires grad, a is sparse, b requires grad, b is sparse, error message
            (False, True, True, False, None),
            (False, False, True, True, "The backward pass for this operation requires the 'mat2'"),
            (False, True, True, True, "The backward pass for this operation requires the 'mat2'"),
            (True, False, True, True, "The backward pass for this operation requires the 'mat2'"),
            (True, True, False, False, "The backward pass for this operation requires the 'self'"),
            (True, True, True, False, "The backward pass for this operation requires the 'self'"),
            (True, True, True, True, "The backward pass for this operation requires the 'mat2'"),
        ]
        for a_req_grad, a_is_sparse, b_req_grad, b_is_sparse, err_msg in mm_test_cases:
            # We should only be testing cases with sparse inputs, and at least one
            # input needs to require grad so we can call a backward pass
            assert a_is_sparse or b_is_sparse
            assert a_req_grad or b_req_grad

            a = torch.randn(size, requires_grad=a_req_grad)
            if a_is_sparse:
                a = a.to_sparse()
            b = torch.randn(size, requires_grad=b_req_grad)
            if b_is_sparse:
                b = b.to_sparse()

            # If no error expected, check that sparse and dense cases match
            if err_msg is None:
                r = a.mm(b)
                r.sum().backward()
                a_grad = None if a.grad is None else a.grad.clone().detach()
                b_grad = None if b.grad is None else b.grad.clone().detach()

                # Redo with only dense tensors
                a = (a.to_dense() if a.is_sparse else a).clone().detach()
                a.requires_grad = a_req_grad
                b = (b.to_dense() if b.is_sparse else b).clone().detach()
                b.requires_grad = b_req_grad
                r = a.mm(b)
                r.sum().backward()

                self.assertEqual(a_grad, a.grad)
                self.assertEqual(b_grad, b.grad)

            else:
                with self.assertRaisesRegex(RuntimeError, err_msg):
                    a.mm(b)

    def test_multi_backward(self):
        x = torch.randn(5, 5, requires_grad=True)
        y = torch.randn(5, 5, requires_grad=True)

        q = torch.randn(5, 5, requires_grad=True)

        a = torch.randn(5, 5, requires_grad=True)
        b = torch.randn(5, 5, requires_grad=True)

        q2 = q * 2
        z = x + y + q2
        c = a * b + q2
        grad_z = torch.randn(5, 5)
        grad_c = torch.randn(5, 5)
        torch.autograd.backward([z, c], [grad_z, grad_c])

        self.assertEqual(x.grad, grad_z)
        self.assertEqual(y.grad, grad_z)
        self.assertEqual(a.grad, grad_c * b)
        self.assertEqual(b.grad, grad_c * a)
        self.assertEqual(q.grad, (grad_c + grad_z) * 2)

    def test_multi_backward_no_grad(self):
        x = torch.randn(5, 5, requires_grad=True)
        y = torch.randn(5, 5, requires_grad=False)

        z = x + y
        q = y * 2

        # NB: we currently raise an exception if any arguments to backwards
        # have requires_grad=False and don't have a grad_fn. We may want to
        # relax that check to a warning.
        def call_backwards():
            torch.autograd.backward([z, q], [torch.ones(5, 5), torch.ones(5, 5)])
        self.assertRaises(RuntimeError, call_backwards)

    def test_backward_with_inputs(self):
        x = torch.randn(2, 2, requires_grad=True)
        y = torch.randn(2, 2, requires_grad=True)

        def fn():
            return x ** 2 + y * x + y ** 2

        gradient = torch.ones(2, 2)
        x_grad_expected = 2 * x + y
        y_grad_expected = x + 2 * y

        @torch.no_grad()
        def reset_grad():
            x.grad.zero_()
            y.grad.zero_()

        torch.autograd.backward(fn(), gradient, inputs=[x, y])
        self.assertEqual(x.grad, x_grad_expected)
        self.assertEqual(y.grad, y_grad_expected)

        reset_grad()
        torch.autograd.backward(fn(), gradient, inputs=[x])
        self.assertEqual(x.grad, x_grad_expected)
        self.assertEqual(y.grad, torch.zeros(2, 2))

        reset_grad()
        torch.autograd.backward(fn(), gradient, inputs=[y])
        self.assertEqual(y.grad, y_grad_expected)
        self.assertEqual(x.grad, torch.zeros(2, 2))

        reset_grad()
        self.assertRaisesRegex(RuntimeError, 'cannot be empty',
                               lambda: torch.autograd.backward(fn(), gradient, inputs=[]))

    def test_backward_with_nonleaf_inputs(self):
        x = torch.randn(2, 2, requires_grad=True)
        x_nonleaf = x * 1
        y = torch.randn(2, 2, requires_grad=True)
        z = torch.randn(2, 2, requires_grad=True)

        out = x_nonleaf ** 2 + y * x_nonleaf + y ** 2

        out.backward(torch.ones(2, 2), create_graph=True, inputs=[x, y])
        x_grad_expected = 2 * x + y
        y_grad_expected = x + 2 * y

        self.assertEqual(y.grad, y_grad_expected)
        self.assertEqual(x.grad, x_grad_expected)

        self.assertRaisesRegex(RuntimeError, 'not a leaf Tensor',
                               lambda: out.backward(torch.ones(2, 2), create_graph=True, inputs=[x, y, x_nonleaf]))

        # backward doesn't have an allow_unused flag, so the behavior of backward
        # when variable is not part of the graph is as if allow_used were true
        # x.grad will simply be None.
        out.backward(torch.ones(2, 2), create_graph=True, inputs=[z])
        self.assertIsNone(z.grad)

    def test_dependent_backward(self):
        x = torch.randn(10, requires_grad=True)
        y = x ** 2
        z = y ** 3

        go_y = torch.randn(10)
        go_z = torch.randn(10)
        torch.autograd.backward([y, z], [go_y, go_z])

        xd = x
        self.assertEqual(x.grad, 2 * xd * go_y + 6 * xd.pow(5) * go_z)

    def test_save_output_nr(self):
        x = torch.randn(10, requires_grad=True)

        class MultiOutputFn(Function):
            @staticmethod
            def forward(ctx, x):
                return x[:5], x[5:]

            @staticmethod
            def backward(ctx, *grad):
                return torch.cat(grad)

        a, b = MultiOutputFn.apply(x)
        self.assertEqual(b.output_nr, 1)

        class TestFn(Function):
            @staticmethod
            def forward(ctx, b):
                ctx.save_for_backward(b)
                return b * 2

            @staticmethod
            def backward(ctx, grad_b):
                b, = ctx.saved_tensors
                self.assertEqual(b.output_nr, 1)

        TestFn.apply(b).sum().backward()

    def test_free_deep_graph(self):
        def scope():
            depth = 150000
            x = torch.randn(1, requires_grad=True)
            y = x.clone()

            # build a "chain" computation graph
            for _ in range(depth):
                y = y + y * 0.000001

            # graph deletion occurs when the above locals go out of scope.
            # In this case `del y` will trigger it but it's easier to leave
            # it to Python to delete the locals.

        # Should not stack overflow
        scope()

    def test_free_deep_graph_complicated(self):
        def scope():
            depth = 100000
            randchoice = torch.randint(2, [depth, 2])
            x = torch.randn(1, requires_grad=True)
            y = x.clone()

            # Hold the two previous values
            prev_values = [None, None]

            # Build a "chain with skip connections" graph
            for _ in range(depth):
                prev_tensors = [tensor for tensor in prev_values[:-1]
                                if tensor is not None]
                prev_values.append(y)
                prev_values.pop(0)

                # Definitely pick one tensor to add
                y += y * 0.000001

                # Possibly add other tensors
                nprev = len(prev_tensors)
                if nprev == 2:
                    y += randchoice[depth].mul(torch.cat(prev_tensors)).sum()

            # graph deletion occurs when the above locals go out of scope.

        # Should not stack overflow
        scope()

    def test_free_deep_graph_pyfunction(self):
        class MyOp(Function):
            @staticmethod
            def forward(ctx, tensor1, tensor2):
                return tensor1 + tensor2

            @staticmethod
            def backward(ctx, grad_output):
                return grad_output, grad_output

        def scope():
            depth = 150000
            x = torch.randn(1, requires_grad=True)
            y = x.clone()

            # build deeply nested computation graph
            for _ in range(depth):
                y = MyOp.apply(y, y)

            # graph deletion occurs when the above locals go out of scope.

        # Should not stack overflow
        scope()

    def test_no_unnecessary_save(self):
        # If we kept x in the derivative Function of x * 2 we would
        # get an error in the backward that would complain that we've
        # modified x, which was needed for gradient computation.
        # Since we should elide unnecessary saves, this test should pass.
        mu = torch.ones(1, requires_grad=True)
        x = torch.empty(1)
        loss = 0
        for i in range(3):
            x.detach_()
            x.copy_(mu + i)
            ft = torch.tensor([float(i)])
            multiplied = x * ft
            s = multiplied.sum()
            loss += s
        loss.backward()

    def test_no_grad(self):
        x = torch.ones(5, 5, requires_grad=True)
        y = torch.ones(5, 5) * 4
        with torch.no_grad():
            w = x + y

        @torch.no_grad()
        def adder(x, y):
            return x + y

        z = adder(x, y)

        self.assertFalse(w.requires_grad)
        self.assertRaises(RuntimeError, lambda: w.backward(torch.ones(5, 5)))
        self.assertIsNone(w.grad_fn)
        self.assertFalse(z.requires_grad)
        self.assertRaises(RuntimeError, lambda: z.backward(torch.ones(5, 5)))
        self.assertIsNone(z.grad_fn)

        # test nested decorator and with-statement on no_grad
        with torch.no_grad():
            self.assertFalse(torch.is_grad_enabled())
            w = adder(x, y)
            self.assertFalse(torch.is_grad_enabled())

    def test_set_grad_generator_functions(self):
        @torch.no_grad()
        def gen_no_grad():
            for i in range(10):
                self.assertEqual(torch.is_grad_enabled(), False)
                yield i

        with torch.enable_grad():
            for _ in gen_no_grad():
                self.assertEqual(torch.is_grad_enabled(), True)

        @torch.enable_grad()
        def gen_enable_grad():
            for i in range(10):
                self.assertEqual(torch.is_grad_enabled(), True)
                yield i

        with torch.no_grad():
            for _ in gen_enable_grad():
                self.assertEqual(torch.is_grad_enabled(), False)

    def test_set_grad_generator_functions_recursive(self):
        # enable_grad_decorator_recursive and no_grad_decorator_recursive call each other
        # recursively, to ensure that the decorators preserve the caller's setting
        @torch.enable_grad()
        def enable_grad_decorator_recursive(depth):
            self.assertTrue(torch.is_grad_enabled())
            if depth > 0:
                no_grad_decorator_recursive(depth - 1)
                self.assertTrue(torch.is_grad_enabled())

        @torch.no_grad()
        def no_grad_decorator_recursive(depth):
            self.assertFalse(torch.is_grad_enabled())
            if depth > 0:
                enable_grad_decorator_recursive(depth - 1)
                self.assertFalse(torch.is_grad_enabled())

        # enable_grad_context_manager_recursive and no_grad_context_manager_recursive call
        # each other recursively, to ensure that the decorators preserve the caller's setting
        def enable_grad_context_manager_recursive(depth):
            with torch.enable_grad():
                self.assertTrue(torch.is_grad_enabled())
                if depth > 0:
                    no_grad_context_manager_recursive(depth - 1)
                    self.assertTrue(torch.is_grad_enabled())

        def no_grad_context_manager_recursive(depth):
            with torch.no_grad():
                self.assertFalse(torch.is_grad_enabled())
                if depth > 0:
                    enable_grad_context_manager_recursive(depth - 1)
                    self.assertFalse(torch.is_grad_enabled())

        with torch.enable_grad():
            self.assertTrue(torch.is_grad_enabled())
            enable_grad_decorator_recursive(10)
            self.assertTrue(torch.is_grad_enabled())
            enable_grad_context_manager_recursive(10)
            self.assertTrue(torch.is_grad_enabled())

        with torch.no_grad():
            self.assertFalse(torch.is_grad_enabled())
            enable_grad_decorator_recursive(10)
            self.assertFalse(torch.is_grad_enabled())
            enable_grad_context_manager_recursive(10)
            self.assertFalse(torch.is_grad_enabled())

    def test_no_grad_python_function(self):
        """Python Functions should respect grad mode."""
        x = torch.ones(5, 5, requires_grad=True)

        class MyOp(Function):
            @staticmethod
            def forward(self, x):
                return x + 1

            @staticmethod
            def backward(self, dy):
                return dy

        with torch.no_grad():
            y = MyOp.apply(x)
        self.assertFalse(y.requires_grad)

    def test_indexing(self):
        x = torch.arange(1., 17).view(4, 4)
        y = Variable(x, requires_grad=True)

        def compare(x, y, idx, indexed_tensor, indexed_var):
            indexed_var_t = indexed_var.data
            if not isinstance(indexed_tensor, torch.Tensor):
                indexed_var_t = indexed_var_t[0]
            self.assertEqual(indexed_tensor, indexed_var_t)

            indexed_var.sum().backward()
            expected_grad = torch.Tensor(x.size()).fill_(0)
            expected_grad[idx] = 1
            self.assertEqual(y.grad, expected_grad)

        def check_index(x, y, idx):
            if y.grad is not None:
                with torch.no_grad():
                    y.grad.zero_()
            indexed_tensor = x[idx]
            indexed_var = y[idx]
            compare(x, y, idx, indexed_tensor, indexed_var)

        check_index(x, y, 1)
        check_index(x, y, (1, 1))
        check_index(x, y, slice(1, None))
        check_index(x, y, slice(None, 2))
        check_index(x, y, (slice(None, 2), 2))
        check_index(x, y, (slice(1, 2), 2))
        check_index(x, y, (1, slice(2, None)))
        check_index(x, y, (slice(None, None), slice(2, None)))
        check_index(x, y, torch.LongTensor([0, 2]))
        check_index(x, y, torch.rand(4, 4).bernoulli().bool())
        check_index(x, y, (Ellipsis, slice(2, None)))
        check_index(x, y, ([0], [0]))
        check_index(x, y, ([1, 2, 3], [0]))
        check_index(x, y, ([1, 2], [2, 1]))
        check_index(x, y, ([[1, 2], [3, 0]], [[0, 1], [2, 3]]))
        check_index(x, y, ([slice(None), [2, 3]]))
        check_index(x, y, ([[2, 3], slice(None)]))

        # advanced indexing, with less dim, or ellipsis
        check_index(x, y, ([0]))
        check_index(x, y, ([0], ))

        x = torch.arange(1., 49).view(4, 3, 4)
        y = Variable(x, requires_grad=True)

        check_index(x, y, (slice(None), [0], [0]))
        check_index(x, y, ([0], [0], slice(None)))
        check_index(x, y, (slice(None), [0, 1, 2], [0]))
        check_index(x, y, ([0, 1, 2], [0], slice(None)))
        check_index(x, y, (slice(None), [1, 2], [2, 1]))
        check_index(x, y, ([1, 2], [2, 1], slice(None)))
        check_index(x, y, (slice(None), [[1, 2], [2, 0]], [[0, 1], [2, 3]]))
        check_index(x, y, ([[1, 2], [3, 0]], [[0, 1], [2, 2]], slice(None)))
        check_index(x, y, (slice(None), slice(None), [2, 1]))
        check_index(x, y, (slice(None), [2, 1], slice(None)))
        check_index(x, y, ([2, 1], slice(None), slice(None)))

        # advanced indexing, with less dim, or ellipsis
        check_index(x, y, ([0], ))
        check_index(x, y, ([0], slice(None)))
        check_index(x, y, ([0], Ellipsis))
        check_index(x, y, ([1, 2], [0, 1]))
        check_index(x, y, ([1, 2], [0, 1], Ellipsis))
        check_index(x, y, (Ellipsis, [1, 2], [0, 1]))

        # advanced indexing, with a tensor wrapped in a variable
        z = torch.LongTensor([0, 1])
        zv = Variable(z, requires_grad=False)
        seq = [z, Ellipsis]
        seqv = [zv, Ellipsis]

        if y.grad is not None:
            with torch.no_grad():
                y.grad.zero_()
        indexed_tensor = x[seq]
        indexed_var = y[seqv]
        compare(x, y, seq, indexed_tensor, indexed_var)

    def test_indexing_duplicates(self):
        x = torch.arange(1., 17).view(4, 4)
        y = Variable(x, requires_grad=True)

        idx = torch.LongTensor([1, 1, 3, 2, 1, 2])
        y[idx].sum().backward()
        expected_grad = torch.zeros(4, 4)
        for i in idx:
            expected_grad[i] += 1
        self.assertEqual(y.grad, expected_grad)

        # with advanced indexing
        x = torch.arange(1., 17).view(4, 4)
        y = Variable(x, requires_grad=True)

        idx = [[1, 1, 3, 2, 1, 2], [0]]
        y[idx].sum().backward()
        expected_grad = torch.zeros(4, 4)
        for i in idx[0]:
            for j in idx[1]:
                expected_grad[i][j] += 1

        self.assertEqual(y.grad, expected_grad)

        x = torch.arange(1., 17).view(4, 4)
        y = Variable(x, requires_grad=True)
        idx = [[[1, 2], [0, 0]], [[0, 1], [1, 1]]]
        y[idx].sum().backward()
        expected_grad = torch.Tensor([[0, 2, 0, 0],
                                      [1, 0, 0, 0],
                                      [0, 1, 0, 0],
                                      [0, 0, 0, 0]])
        self.assertEqual(y.grad, expected_grad)

        x = torch.arange(1., 65).view(4, 4, 4)
        y = Variable(x, requires_grad=True)

        idx = [[1, 1, 1], slice(None), slice(None)]
        y[idx].sum().backward()
        expected_grad = torch.Tensor(4, 4, 4).zero_()
        expected_grad[1].fill_(3)
        self.assertEqual(y.grad, expected_grad)

    def test_index_backward_does_not_save_tensor(self):
        # Example from https://github.com/pytorch/pytorch/issues/24853.
        # if `index(tensor, indices)` saves `tensor` for backwards, then it will
        # trigger a version check on `tensor` during the backward pass, which
        # will cause the following code to error because `tensor` gets modified
        # by the indexing line.
        a = torch.tensor([1., 0, 0])
        b = torch.zeros(3, requires_grad=True)
        tensor = b + 0
        tensor[a != 0] = tensor[a != 0]
        tensor.backward(torch.zeros_like(tensor))

    def test_volatile_deprecated(self):
        v = torch.autograd.torch.randn(3, 3)
        with warnings.catch_warnings(record=True) as w:
            self.assertFalse(v.volatile)
        self.assertIn('volatile', str(w[0].message))

    def test_saved_variables_deprecated(self):
        class MyFunction(Function):
            @staticmethod
            def forward(ctx, tensor1, tensor2):
                ctx.save_for_backward(tensor1, tensor2)
                return tensor1 + tensor2

            @staticmethod
            def backward(ctx, grad_output):
                var1, var2 = ctx.saved_variables
                return (grad_output, grad_output)

        with warnings.catch_warnings(record=True) as warns:
            warnings.simplefilter("always")
            x = torch.randn((3, 3), requires_grad=True)
            y = torch.randn((3, 3), requires_grad=True)
            model = MyFunction()
            model.apply(x, y).sum().backward()

            has_deprecated = map(lambda warn:
                                 'deprecated' in str(warn) and
                                 'saved_variables' in str(warn),
                                 warns)
            has_deprecated = reduce(lambda x, y: x or y, has_deprecated)
            self.assertTrue(has_deprecated)

    def test_requires_grad(self):
        x = torch.randn(5, 5)
        y = torch.randn(5, 5)
        z = torch.randn(5, 5, requires_grad=True)
        a = x + y
        self.assertFalse(a.requires_grad)
        b = a + z
        self.assertTrue(b.requires_grad)

        def error():
            raise RuntimeError
        # Make sure backward isn't called on these
        a._backward_hooks = OrderedDict()
        x._backward_hooks = OrderedDict()
        y._backward_hooks = OrderedDict()
        a._backward_hooks['test'] = error
        x._backward_hooks['test'] = error
        y._backward_hooks['test'] = error
        b.backward(torch.ones(5, 5))

    def test_requires_grad_(self):
        x = torch.randn(5, 5)
        y = torch.randn(5, 5, requires_grad=True)
        self.assertIs(x, x.requires_grad_())
        self.assertTrue(x.requires_grad)
        self.assertIs(y, y.requires_grad_())
        self.assertTrue(y.requires_grad)
        self.assertIs(x, x.requires_grad_(True))
        self.assertTrue(x.requires_grad)
        self.assertIs(y, y.requires_grad_(True))
        self.assertTrue(y.requires_grad)
        z = x * y
        self.assertRaises(RuntimeError, lambda: z.requires_grad_(False))
        self.assertIs(z, z.requires_grad_())
        self.assertTrue(z.requires_grad)
        self.assertIs(z, z.requires_grad_(True))
        self.assertTrue(z.requires_grad)

        self.assertIs(x, x.requires_grad_(False))
        self.assertFalse(x.requires_grad)
        self.assertIs(y, y.requires_grad_(False))
        self.assertFalse(y.requires_grad)

    def test_requires_grad_inplace(self):
        a = torch.randn(5, 5)
        b = torch.randn(5, 5, requires_grad=True)
        a += b
        self.assertTrue(a.requires_grad)

        # non-leaf
        a = torch.randn(5, 5) + 0
        b = torch.randn(5, 5, requires_grad=True)
        a += b
        self.assertTrue(a.requires_grad)

    def test_no_requires_grad_inplace(self):
        # basic case, should be able to modify inplace while requires_grad is False
        a = torch.randn(2, 3)
        a.add_(5)
        a.requires_grad = True
        a.sum().backward()
        self.assertEqual(a.grad, torch.ones(2, 3))

        # same but with a view
        a = torch.randn(2, 3)
        b = a[:]
        b.add_(5)
        a.requires_grad = True
        a.sum().backward()
        self.assertEqual(a.grad, torch.ones(2, 3))

        # should fail if requires_grad = True when we modify inplace
        a = torch.randn(2, 3)
        b = a[:]
        a.requires_grad = True
        with self.assertRaises(RuntimeError):
            a.add_(5)
        with self.assertRaises(RuntimeError):
            b.add_(5)

    def test_attribute_deletion(self):
        x = torch.randn((5, 5), requires_grad=True)
        del x.grad
        self.assertIsNone(x.grad)
        with self.assertRaises(RuntimeError):
            del x.data
        with self.assertRaises(TypeError):
            x.data = None
        with self.assertRaises(RuntimeError):
            del x.requires_grad
        with self.assertRaises(RuntimeError):
            del x._grad_fn
        with self.assertRaises(RuntimeError):
            del x._backward_hooks

    def test_duplicate_backward_root(self):
        a = torch.randn(5, 5, requires_grad=True)
        b = torch.randn(5, 5, requires_grad=True)

        x = a * b
        grad_output = torch.randn_like(x)
        torch.autograd.backward([x, x], [grad_output, grad_output])

        self.assertEqual(a.grad, b * grad_output * 2)
        self.assertEqual(b.grad, a * grad_output * 2)

    def test_backward_no_grad(self):
        a = torch.randn(5, 5, requires_grad=True)
        b = a + 2
        with self.assertRaises(RuntimeError):
            torch.autograd.backward([b], [None])

    def test_backward_twice_with_saved_values(self):
        b = torch.randn(3, requires_grad=True, dtype=torch.double)
        c = torch.zeros(3, dtype=torch.double)
        c[[1, 2]] = b[[1, 1]]
        c.backward(torch.tensor([1, 1, 1], dtype=torch.double))
        self.assertRaisesRegex(RuntimeError, 'Specify retain_graph=True',
                               lambda: c.backward(torch.tensor([1, 1, 1], dtype=torch.double)))

    def test_backward_twice_retained_graph_with_saved_values(self):
        b = torch.randn(3, requires_grad=True, dtype=torch.double)
        c = torch.zeros(3, dtype=torch.double)
        c[[1, 2]] = b[[1, 1]]
        c.backward(torch.tensor([1, 1, 1], dtype=torch.double), retain_graph=True)
        c.backward(torch.tensor([1, 1, 1], dtype=torch.double))

    def test_backward_twice_without_saved_values(self):
        b = torch.randn(3, requires_grad=True, dtype=torch.double)
        c = b + 1
        c.backward(torch.tensor([1, 1, 1], dtype=torch.double))
        c.backward(torch.tensor([1, 1, 1], dtype=torch.double))

    def test_backward_twice_retained_graph_without_saved_values(self):
        b = torch.randn(3, requires_grad=True, dtype=torch.double)
        c = torch.zeros(3, dtype=torch.double)
        c[[1, 2]] = b[[1, 1]]
        c.backward(torch.tensor([1, 1, 1], dtype=torch.double), retain_graph=True)
        c.backward(torch.tensor([1, 1, 1], dtype=torch.double))

    def test_next_functions(self):
        x = torch.randn(5, 5, requires_grad=True)
        y = torch.randn(5, 5, requires_grad=True)

        a = x + y
        self.assertIsNotNone(a.grad_fn)
        next_functions = a.grad_fn.next_functions
        self.assertEqual(len(next_functions), 2)
        self.assertIsInstance(next_functions[0][0], torch._C._functions.AccumulateGrad)
        self.assertEqual(next_functions[0][1], 0)
        self.assertIsInstance(next_functions[1][0], torch._C._functions.AccumulateGrad)
        self.assertEqual(next_functions[1][1], 0)

        b = a + 5
        next_functions = b.grad_fn.next_functions
        self.assertEqual(len(next_functions), 2)
        self.assertIs(next_functions[0][0], a.grad_fn)
        self.assertIs(next_functions[1][0], None)

    def test_inplace(self):
        x = torch.ones(5, 5, requires_grad=True)
        y = Variable(torch.ones(5, 5) * 4, requires_grad=True)

        z = x * y
        q = z + y
        w = z * y
        z.add_(2)
        # Add doesn't need it's inputs to do backward, so it shouldn't raise
        q.backward(torch.ones(5, 5), retain_graph=True)
        # Mul saves both inputs in forward, so it should raise
        self.assertRaises(RuntimeError, lambda: w.backward(torch.ones(5, 5)))

        z = x * y
        q = z * y
        r = z + y
        w = z.add_(y)
        # w is a the last expression, so this should succeed
        w.backward(torch.ones(5, 5), retain_graph=True)
        # r doesn't use the modified value in backward, so it should succeed
        r.backward(torch.ones(5, 5), retain_graph=True)
        # q uses dirty z, so it should raise
        self.assertRaises(RuntimeError, lambda: q.backward(torch.ones(5, 5)))

        with torch.no_grad():
            x.grad.zero_()
        m = x / 2
        z = m + y / 8
        q = z * y
        r = z + y
        prev_version = z._version
        w = z.exp_()
        self.assertNotEqual(z._version, prev_version)
        r.backward(torch.ones(5, 5), retain_graph=True)
        self.assertEqual(x.grad, torch.ones(5, 5) / 2)
        w.backward(torch.ones(5, 5), retain_graph=True)
        self.assertEqual(x.grad, torch.Tensor(5, 5).fill_((1 + math.e) / 2))
        self.assertRaises(RuntimeError, lambda: q.backward(torch.ones(5, 5)))

        leaf = torch.ones(5, 5, requires_grad=True)
        x = leaf.clone()
        x.add_(10)
        self.assertEqual(x, torch.ones(5, 5) * 11)
        # x should be still usable
        y = x + 2
        y.backward(torch.ones(5, 5))
        self.assertEqual(leaf.grad, torch.ones(5, 5))
        z = x * y
        x.add_(2)
        self.assertRaises(RuntimeError, lambda: z.backward(torch.ones(5, 5)))

    def test_mark_non_differentiable(self):
        class MyFunction(Function):
            @staticmethod
            def forward(ctx, input):
                output = input > 0
                ctx.mark_non_differentiable(output)
                return output

            @staticmethod
            def backward(ctx, grad_output):
                return (grad_output * 0).to(torch.double)

        x = torch.randn(5, 5, requires_grad=True)
        mask = MyFunction.apply(x)
        self.assertFalse(mask.requires_grad)
        y = x.masked_fill(mask, 0)
        y.sum().backward()

    def test_mark_non_differentiable_mixed(self):
        class MyFunction(Function):
            @staticmethod
            def forward(ctx, input):
                a = input + 1
                b = input + 2
                ctx.mark_non_differentiable(a)
                return a, b

            @staticmethod
            def backward(ctx, grad_a, grad_b):
                self.assertTrue((grad_a == 0).all())
                self.assertTrue((grad_b == 1).all())
                return grad_b

        x = torch.randn(5, 5, requires_grad=True)
        a, b = MyFunction.apply(x)
        self.assertFalse(a.requires_grad)
        self.assertTrue(b.requires_grad)
        b.sum().backward()
        self.assertEqual(x.grad, torch.ones(5, 5))

    def test_mark_non_differentiable_none(self):
        # This used to segfault because MyFunction would send back null
        # gradients to MulBackward, which is implemented in C++. C++
        # implemented functions expect incoming  grad_ouptuts to be non-null.
        class MyFunction(Function):
            @staticmethod
            def forward(ctx, input):
                output = input.clone()
                ctx.mark_non_differentiable(output)
                return output

            @staticmethod
            def backward(ctx, grad_output):
                return None

        x = torch.randn(5, 5, requires_grad=True)
        r = MyFunction.apply(x * x)
        (r * x).sum().backward()

    def test_return_duplicate(self):
        class DoubleDuplicate(Function):
            @staticmethod
            def forward(ctx, x):
                output = x * 2
                return output, output

            @staticmethod
            def backward(ctx, grad1, grad2):
                return grad1 * 2 + grad2 * 2

        def fn(x):
            a, b = DoubleDuplicate.apply(x)
            self.assertIs(a, b)
            return a + b

        x = torch.randn(5, 5, requires_grad=True)
        gradcheck(fn, [x])
        gradgradcheck(fn, [x])

    def test_return_duplicate_inplace(self):
        class DoubleInplace(Function):
            @staticmethod
            def forward(ctx, x):
                x.mul_(2)
                ctx.mark_dirty(x)
                return x, x

            @staticmethod
            def backward(ctx, grad1, grad2):
                return grad1 * 2 + grad2 * 2

        def inplace_fn(x):
            a, b = DoubleInplace.apply(x.clone())
            self.assertIs(a, b)
            return a + b

        x = torch.randn(5, 5, requires_grad=True)
        gradcheck(inplace_fn, [x])
        gradgradcheck(inplace_fn, [x])

        # Can't modify leaf variables in-place
        self.assertRaises(RuntimeError, lambda: InplaceFunction.apply(x))
        # Functions which modify views in-place must return only one output
        self.assertRaises(RuntimeError, lambda: InplaceFunction.apply(x.clone()[0]))

    @suppress_warnings
    def test_resize(self):
        x = torch.ones(2, 3)
        self.assertTrue(x.resize(3, 2).size() == (3, 2))

    def _test_setitem(self, size, index):
        x = torch.ones(*size, requires_grad=True)
        y = x + 2
        y_version = y._version
        y[index] = 2
        self.assertNotEqual(y._version, y_version)
        y.backward(torch.ones(*size))
        expected_grad = torch.ones(*size)
        expected_grad[index] = 0
        self.assertEqual(x.grad, expected_grad)

    def _test_setitem_tensor(self, size, index):
        x = torch.ones(*size, requires_grad=True)
        y = x + 2
        y_version = y._version
        value = x.new(x[index].size()).fill_(7)
        value.requires_grad = True
        y[index] = value
        self.assertNotEqual(y._version, y_version)
        y.backward(torch.ones(*size))
        expected_grad_input = torch.ones(*size)
        expected_grad_input[index] = 0
        self.assertEqual(x.grad, expected_grad_input)
        self.assertEqual(value.grad, torch.ones_like(value))

        # case when x broadcasts to as y[1]
        x = torch.randn(4, requires_grad=True)
        y = torch.zeros(2, 3, 4)
        y[1] = x
        y.backward(torch.randn(2, 3, 4))
        self.assertEqual(x.size(), x.grad.size())

    def test_setitem(self):
        self._test_setitem((5, 5), 1)
        self._test_setitem((5,), 1)
        self._test_setitem((1,), 0)
        self._test_setitem((10,), [[0, 4, 2]])
        self._test_setitem((5, 5), [[0, 4], [2, 2]])
        self._test_setitem((5, 5, 5), [slice(None), slice(None), [1, 3]])
        self._test_setitem((5, 5, 5), [slice(None), [1, 3], slice(None)])
        self._test_setitem((5, 5, 5), [[1, 3], slice(None), slice(None)])
        self._test_setitem((5, 5, 5), [slice(None), [2, 4], [1, 3]])
        self._test_setitem((5, 5, 5), [[1, 3], [2, 4], slice(None)])
        self._test_setitem_tensor((5, 5), 3)
        self._test_setitem_tensor((5, 5), [[0, 1], [1, 0]])
        self._test_setitem_tensor((5,), 3)
        self._test_setitem_tensor((5,), Variable(torch.LongTensor([3]), requires_grad=False).sum())
        self._test_setitem_tensor((5,), [[0, 1, 2, 3]])
        self._test_setitem_tensor((5, 5, 5), [slice(None), slice(None), [1, 3]])
        self._test_setitem_tensor((5, 5, 5), [slice(None), [1, 3], slice(None)])
        self._test_setitem_tensor((5, 5, 5), [[1, 3], slice(None), slice(None)])
        self._test_setitem_tensor((5, 5, 5), [slice(None), [2, 4], [1, 3]])
        self._test_setitem_tensor((5, 5, 5), [[1, 3], [2, 4], slice(None)])
        self._test_setitem_tensor((5, 5, 5), [Variable(torch.LongTensor([1,
                                              3]), requires_grad=False), [2, 4], slice(None)])

    def test_setitem_mask(self):
        mask = torch.BoolTensor(5, 5).bernoulli_()
        self._test_setitem((5, 5), Variable(mask))
        self._test_setitem((5,), Variable(mask[0]))
        self._test_setitem((1,), Variable(mask[0, 0:1]))
        self._test_setitem_tensor((5, 5), Variable(mask))
        self._test_setitem_tensor((5,), Variable(mask[0]))

    def test_select_sum(self):
        # both select and sum return Scalars in ATen; ensure they work together.
        x = torch.randn(10, requires_grad=True)

        def func(x):
            return x.select(0, 1).sum()

        gradcheck(func, [x])
        gradgradcheck(func, [x])

    def test_diagonal_expanded_v(self):
        value = torch.rand([])
        v_expanded = torch.tensor(value).expand(10)
        a = torch.rand(10, 10, requires_grad=True)
        result, = torch.autograd.grad(a.diagonal(), a, v_expanded)
        self.assertEqual(result, torch.eye(10) * value)

    def test_select_expanded_v(self):
        v_expanded = torch.rand(10).expand(10, 10)
        a = torch.rand(10, 10, 10, requires_grad=True)
        result, = torch.autograd.grad(a[0], a, v_expanded)
        expected = torch.zeros(10, 10, 10)
        expected[0] = v_expanded
        self.assertEqual(result, expected)

    def test_slice_expanded_v(self):
        v_expanded = torch.rand(10, 1).expand(2, 10, 10)
        a = torch.rand(10, 10, 10, requires_grad=True)
        result, = torch.autograd.grad(a[3:5], a, v_expanded)
        expected = torch.zeros(10, 10, 10)
        expected[3:5] = v_expanded
        self.assertEqual(result, expected)

    def test_stack(self):
        x = torch.randn(10, 10, requires_grad=True)
        y = torch.randn(10, 10, requires_grad=True)
        z = torch.randn(10, 10, requires_grad=True)
        stacked = torch.stack([x, y, z], 0)
        grad = torch.randn(3, 10, 10)
        stacked.backward(grad)
        self.assertEqual(x.grad, grad[0])
        self.assertEqual(y.grad, grad[1])
        self.assertEqual(z.grad, grad[2])

    def test_hstack(self):
        x = torch.randn(10, 10, requires_grad=True)
        y = torch.randn(10, 10, requires_grad=True)
        z = torch.randn(10, 10, requires_grad=True)
        stacked = torch.hstack([x, y, z])
        grad = torch.randn(10, 30)
        stacked.backward(grad)
        self.assertEqual(x.grad, grad[:, 0:10])
        self.assertEqual(y.grad, grad[:, 10:20])
        self.assertEqual(z.grad, grad[:, 20:30])

        x = torch.randn(10, requires_grad=True)
        y = torch.randn(10, requires_grad=True)
        z = torch.randn(10, requires_grad=True)
        stacked = torch.hstack([x, y, z])
        grad = torch.randn(30)
        stacked.backward(grad)
        self.assertEqual(x.grad, grad[0:10])
        self.assertEqual(y.grad, grad[10:20])
        self.assertEqual(z.grad, grad[20:30])

    def test_vstack(self):
        x = torch.randn(10, 10, requires_grad=True)
        y = torch.randn(10, 10, requires_grad=True)
        z = torch.randn(10, 10, requires_grad=True)
        stacked = torch.vstack([x, y, z])
        grad = torch.randn(30, 10)
        stacked.backward(grad)
        self.assertEqual(x.grad, grad[0:10])
        self.assertEqual(y.grad, grad[10:20])
        self.assertEqual(z.grad, grad[20:30])

    def test_dstack(self):
        x = torch.randn(10, 10, requires_grad=True)
        y = torch.randn(10, 10, requires_grad=True)
        z = torch.randn(10, 10, requires_grad=True)
        stacked = torch.dstack([x, y, z])
        grad = torch.randn(10, 10, 3)
        stacked.backward(grad)
        self.assertEqual(x.grad, grad[:, :, 0])
        self.assertEqual(y.grad, grad[:, :, 1])
        self.assertEqual(z.grad, grad[:, :, 2])

    def test_unbind(self):
        stacked = torch.randn(3, 10, 10, requires_grad=True)
        x, y, z = stacked.unbind()
        grad = torch.randn(3, 10, 10)
        torch.autograd.backward([x, y, z], grad.unbind())
        self.assertEqual(stacked.grad, grad)
        # check that it works with only one gradient provided (#9977)
        for i in range(3):
            stacked = torch.randn(3, 10, 10, requires_grad=True)
            outs = stacked.unbind()
            gi = grad.unbind()[i]
            g, = torch.autograd.grad(outs[i], stacked, gi)
            g_expected = torch.stack([gi if j == i else torch.zeros_like(gi)
                                      for j in range(3)], dim=0)
            self.assertEqual(g, g_expected)

    def test_put(self):
        root = torch.randn(4, 5, requires_grad=True)
        values = torch.randn(6, requires_grad=True)
        idx = Variable(torch.LongTensor([1, 2, 3, -1, -2, -3]))

        def func(root, values):
            x = root.clone()
            x.put_(idx, values)
            return x

        gradcheck(func, [root, values])
        gradgradcheck(func, [root, values])

    def test_put_accumulate(self):
        root = torch.randn(4, 5, requires_grad=True)
        values = torch.randn(6, requires_grad=True)
        idx = Variable(torch.LongTensor([1, 2, 3, 1, 2, 3]))

        def func(root, values):
            x = root.clone()
            x.put_(idx, values, accumulate=True)
            return x

        gradcheck(func, [root, values])
        gradgradcheck(func, [root, values])

    def test_fill(self):
        root = torch.randn(4, 5, requires_grad=True)

        def func(root):
            x = root.clone()
            x.fill_(2)
            return x

        gradcheck(func, [root])
        gradgradcheck(func, [root])

    def test_unused_output(self):
        x = torch.randn(10, 10, requires_grad=True)
        outputs = x.chunk(5)
        o = outputs[2]
        o = o * 4 + 2
        o.sum().backward()
        expected_grad = torch.zeros(10, 10)
        expected_grad[4:6] = 4
        self.assertEqual(x.grad, expected_grad)

        with torch.no_grad():
            x.grad.zero_()
        grad_output = torch.randn(2, 10)
        outputs = x.chunk(5)
        outputs[0].backward(grad_output)
        expected_grad = torch.zeros(10, 10)
        expected_grad[:2] = grad_output
        self.assertEqual(x.grad, expected_grad)

    def _test_sparse_gather(self, size_x, size_ind, dim):
        x = torch.randn(size_x, requires_grad=True)
        if len(size_ind) > 0 and len(size_x) > 0:
            ind = torch.randint(x.size(dim), size_ind)
        else:
            ind = torch.zeros(size_ind, dtype=torch.int64)
        out = torch.gather(x, dim, ind, sparse_grad=False)
        grad = torch.rand_like(out)
        out.backward(grad)
        grad_dense = x.grad.clone()
        x.grad = None
        out = torch.gather(x, dim, ind, sparse_grad=True)
        out.backward(grad)
        self.assertEqual(grad_dense, x.grad.to_dense())

    def test_sparse_gather_dim0(self):
        self._test_sparse_gather((10, 10), (5, 10), 0)

    def test_sparse_gather_dim1(self):
        self._test_sparse_gather((10, 10, 5), (10, 5, 5), 1)

    def test_sparse_gather_dim_neg(self):
        self._test_sparse_gather((10, 10, 5), (10, 10, 2), -1)

    def test_sparse_gather_ind_scalar(self):
        self._test_sparse_gather((10,), (), 0)

    def test_sparse_gather_x_scalar(self):
        self._test_sparse_gather((), (2,), 0)

    def test_sparse_gather_both_scalar(self):
        self._test_sparse_gather((), (), 0)

    def test_gc_in_destructor(self):
        """
        Previously, if a Function destructor triggered a garbage collection,
        the Variable's tp_dealloc handler would get called twice leading to a
        segfault.
        """
        class CollectOnDelete(Function):
            def forward(self, x):
                return x

            def backward(self, grad_output):
                return grad_output

            def __del__(self):
                gc.collect()

        for _ in range(10):
            CollectOnDelete().forward(torch.randn(1, requires_grad=True)).backward()

    # Delete this test when legacy custom autograd functions are deleted.
    def test_naughty_legacy_variable_grad_fn(self):
        class Id(Function):
            def forward(self, x):
                return x

            def backward(self, grad_x):
                return grad_x

        self.assertRaises(RuntimeError, lambda: Variable(torch.zeros(1), _grad_fn=Id()))

    # Delete this test when legacy custom autograd functions are deleted.
    def test_naughty_legacy_function_backward_before_forward(self):
        class Id(Function):
            def forward(self, x):
                return x

            def backward(self, grad_x):
                return grad_x

        f = Id()
        self.assertRaises(RuntimeError, lambda: f._do_backward((torch.zeros(0), ), False))

    # Delete this test when legacy custom autograd functions are deleted.
    def test_naughty_legacy_function_early_access(self):
        class Id(Function):
            def forward(self, x):
                return x

            def backward(self, grad_x):
                return grad_x

        f = Id()
        # A legacy autograd function is not fully initialized until you actually
        # apply it.  That means a lot of accessors on them don't actually work.
        # Test that we properly error in this case.
        self.assertRaises(RuntimeError, lambda: f.register_hook(lambda x, y: None))
        self.assertRaises(RuntimeError, lambda: f.next_functions)
        self.assertRaises(RuntimeError, lambda: f.metadata)

    @unittest.expectedFailure
    def test_naughty_anomaly_access(self):
        class MyFunction(Function):
            @staticmethod
            def forward(ctx, x):
                return x

            @staticmethod
            def backward(ctx, g):
                return g

        x = torch.zeros(1, requires_grad=True)
        y = MyFunction.apply(x)
        y.backward()
        y.grad_fn.metadata
        g = y.grad_fn
        del y
        g.metadata  # this currently fails, but shouldn't

    def test_naughty_autograd_function_stashing_ctx(self):
        saved_ctx = []

        class Id(Function):
            @staticmethod
            def forward(ctx, x):
                ctx.save_for_backward(x)
                return x

            @staticmethod
            def backward(ctx, grad_x):
                saved_ctx.append(ctx)
                return ctx.saved_tensors

        p = torch.zeros(1, requires_grad=True)
        loss = Id.apply(p)
        loss.backward(retain_graph=True)
        del loss
        # At this point in time, it complains that the graph has been freed
        # (which indeed true, although a somewhat indirect way of stating the
        # problem).
        self.assertRaises(RuntimeError, lambda: saved_ctx[0].saved_tensors)

    def test_custom_autograd_repeated_grad_grad(self):
        # This test failed the equality check in PR #22983; it's an interesting
        # and different test case worth enshrining.  mult1 is not testing
        # anything that interesting, but mult2 is the interesting case.

        def mult1(x):
            return x.prod(dim=-1).prod(dim=-1)

        class Mult(torch.autograd.Function):
            @staticmethod
            def forward(ctx, x):
                y = mult1(x)
                ctx.save_for_backward(x, y)
                return y

            @staticmethod
            def backward(ctx, grad_output):
                x, y = ctx.saved_tensors
                return (grad_output * y)[:, None, None] / x

        mult2 = Mult.apply

        def check_gradgrad_repeated(x, y):
            gy, = torch.autograd.grad(y[0], x, create_graph=True)
            ggy_1, = torch.autograd.grad(gy[0, 0, 0], x, retain_graph=True)
            gy, = torch.autograd.grad(y[0], x, create_graph=True)
            ggy_2, = torch.autograd.grad(gy[0, 0, 0], x, retain_graph=True)
            self.assertEqual(ggy_1[0, 0, 1], ggy_2[0, 0, 1])

        x = torch.ones(2, 4, 4).requires_grad_()
        check_gradgrad_repeated(x, mult1(x))
        check_gradgrad_repeated(x, mult2(x))

    def test_custom_autograd_no_early_free(self):
        # This test failed complaining that buffers had already been freed
        # prior to #22983.  Also pretty interesting test case.
        class Double(torch.autograd.Function):
            @staticmethod
            def forward(ctx, x):
                y = x ** 2
                ctx.save_for_backward(x, y)
                return y

            @staticmethod
            def backward(ctx, grad_output):
                x, _ = ctx.saved_tensors
                return grad_output * 2 * x

        # this is equivalent, but uses the output of .forward() in .backward()
        class Double2(Double):
            @staticmethod
            def backward(ctx, grad_output):
                x, y = ctx.saved_tensors
                return grad_output * 2 * y / x

        double = Double.apply
        double2 = Double2.apply

        x = torch.tensor(2).double().requires_grad_()

        self.assertTrue(torch.autograd.gradcheck(double, x))
        self.assertTrue(torch.autograd.gradgradcheck(double, x))
        self.assertTrue(torch.autograd.gradcheck(double2, x))
        self.assertTrue(torch.autograd.gradgradcheck(double2, x))

        y = double(x)
        torch.autograd.grad(y, x, create_graph=True)
        torch.autograd.grad(y, x)

        y = double2(x)
        torch.autograd.grad(y, x, create_graph=True)
        torch.autograd.grad(y, x)  # should not error!

    def test_detach(self):
        x = torch.randn(10, 10, requires_grad=True)
        y = x + 2
        y = y.detach()
        z = y * 4 + 2
        self.assertFalse(y.requires_grad)
        self.assertFalse(z.requires_grad)

        x = torch.randn(10, 10, requires_grad=True)
        y = x * 2
        y = y.detach()
        self.assertFalse(y.requires_grad)
        self.assertIsNone(y.grad_fn)
        z = x + y
        z.sum().backward()
        # This is an incorrect gradient, but we assume that's what the user
        # wanted. detach() is an advanced option.
        self.assertEqual(x.grad, torch.ones(10, 10))

        # in-place detach
        x = torch.randn(10, 10, requires_grad=True)
        y = torch.randn(10, 10, requires_grad=True)
        a = x * 2
        (y + a).sum().backward(retain_graph=True)
        a.detach_()
        self.assertFalse(a.requires_grad)
        (y + a).sum().backward()  # this won't backprop to x
        self.assertEqual(x.grad, torch.ones(10, 10) * 2)
        self.assertEqual(y.grad, torch.ones(10, 10) * 2)

        # in-place deatch on a view raises an exception
        view = x.narrow(0, 1, 4)
        self.assertRaisesRegex(RuntimeError, 'view', lambda: view.detach_())

    def test_detach_base(self):
        "detaching base does not detach view"
        x = torch.randn(10, 10, requires_grad=True)
        view = x.narrow(0, 1, 4)
        x.detach_()
        self.assertFalse(x.requires_grad)
        self.assertTrue(view.requires_grad)
        self.assertIsNotNone(view.grad_fn)
        self.assertIs(view._base, x)

    def _test_type_conversion_backward(self, t, ):
        fvar = Variable(t(torch.randn(5, 5).float()), requires_grad=True)
        fvar.double().sum().backward()
        self.assertEqual(fvar.grad, torch.ones_like(fvar))
        self.assertEqual(type(fvar.grad), type(fvar))
        dvar = Variable(t(torch.randn(5, 5).double()), requires_grad=True)
        dvar.float().sum().backward()
        self.assertEqual(dvar.grad, torch.ones_like(dvar))
        self.assertEqual(type(dvar.grad), type(dvar))

    def test_type_conversions(self):
        x = torch.randn(5, 5)
        self.assertIsInstance(x.float(), torch.FloatTensor)
        self.assertIsInstance(x.int(), torch.IntTensor)
        if torch.cuda.is_available():
            self.assertIsInstance(x.float().cuda(), torch.cuda.FloatTensor)
            self.assertIsInstance(x.int().cuda(), torch.cuda.IntTensor)
            self.assertIsInstance(x.int().cuda().cpu(), torch.IntTensor)
            if torch.cuda.device_count() >= 2:
                x2 = x.float().cuda(1)
                self.assertIsInstance(x2, torch.cuda.FloatTensor)
                self.assertIs(x2.get_device(), 1)
                x2 = x.float().cuda()
                self.assertIsInstance(x2, torch.cuda.FloatTensor)
                self.assertIs(x2.get_device(), 0)
                x2 = x2.cuda(1)
                self.assertIsInstance(x2, torch.cuda.FloatTensor)
                self.assertIs(x2.get_device(), 1)
                y = Variable(torch.randn(5).cuda(1), requires_grad=True)
                y.cpu().sum().backward()
                self.assertIs(y.grad.get_device(), 1)
                self.assertIs(y.long().get_device(), 1)

        for t in [torch.DoubleTensor, torch.FloatTensor, torch.IntTensor, torch.ByteTensor]:
            for y_var in (True, False):
                y = torch.randint(5, (5, 5), dtype=t.dtype)
                y = Variable(y) if y_var else y
                self.assertIsInstance(x.type(t), t)
                self.assertIsInstance(x.type_as(y), t)
                # TODO: t.dtype should work
                t_dtype = t().dtype
                self.assertIsInstance(x.type(t_dtype), t)
                self.assertIs(t_dtype, x.type(t_dtype).dtype)
                self.assertEqual(y.data_ptr(), y.type(t).data_ptr())
                if torch.cuda.is_available():
                    for x_cuda in (True, False):
                        for y_cuda in (True, False):
                            x_c = x.cuda() if x_cuda else x
                            y_c = y.cuda() if y_cuda else y
                            _, y_type = y_c.type().rsplit('.', 1)
                            y_typestr = ('torch.cuda.' if y_cuda else 'torch.') + y_type
                            self.assertEqual(y_c.type(), x_c.type(y_typestr).type())
                            self.assertIs(y_c.dtype, x_c.type(y_c.dtype).dtype)
                            self.assertEqual(y_c.data_ptr(), y_c.cuda().data_ptr() if y_cuda else y_c.data_ptr())

        self._test_type_conversion_backward(lambda x: x)
        if torch.cuda.is_available():
            self._test_type_conversion_backward(lambda x: x.cuda())
            if torch.cuda.device_count() >= 2:
                # one of these has to be the non-default device
                self._test_type_conversion_backward(lambda x: x.cuda(0))
                self._test_type_conversion_backward(lambda x: x.cuda(1))

    def test_isolated_node(self):
        x = torch.randn(5, 5, requires_grad=True)
        y = torch.randn(5, 5, requires_grad=True)

        a = x + y
        b = torch.max(a, 1, True)[1].repeat(1, 5).double()
        o = (b + a).sum()
        o.backward()

    def test_shape(self):
        x = torch.randn(3, 4)
        self.assertEqual(2, len(x.shape))
        self.assertEqual(x.shape[0], 3)
        self.assertEqual(x.shape[1], 4)

    def test_numpy_requires_grad(self):
        x = torch.randn(2, 2, requires_grad=True)
        err_msg_outputs = r"Can't call numpy\(\) on Tensor that requires grad. Use tensor.detach\(\).numpy\(\) instead."
        with self.assertRaisesRegex(RuntimeError, err_msg_outputs):
            x.numpy()

        with torch.no_grad():
            x.numpy()

        x = torch.randn(2, 2)
        x.numpy()

        with torch.no_grad():
            x.numpy()

    def test_return_leaf(self):
        class Identity(Function):
            @staticmethod
            def forward(ctx, a, b):
                return a, a + b

            @staticmethod
            def backward(ctx, grad_a, grad_b):
                return grad_a + grad_b, grad_b

        hook_called = [False]
        x = torch.randn(5, 5, requires_grad=True)
        y = torch.randn(5, 5, requires_grad=True)

        q, p = Identity.apply(x, y)

        # Make sure hooks only receive grad from usage of q, not x.
        def hook(grad):
            hook_called[0] = True
            self.assertEqual(grad, torch.ones(5, 5))

        q.register_hook(hook)
        (q + p + x).sum().backward()
        self.assertEqual(x.grad, torch.ones(5, 5) * 3)
        self.assertEqual(y.grad, torch.ones(5, 5))
        self.assertTrue(hook_called[0])

    def test_return_leaf_inplace(self):
        class Inplace(InplaceFunction):
            @staticmethod
            def forward(ctx, a, b):
                ctx.mark_dirty(a)
                return a.add_(b), b + 2

            @staticmethod
            def backward(ctx, grad_a, grad_b):
                return grad_a, grad_a + grad_b

        x = torch.randn(5, 5)
        y = torch.randn(5, 5, requires_grad=True)

        fn = Inplace(True)
        q, p = fn.apply(x, y)
        self.assertIs(q, x)
        self.assertIs(q.grad_fn.__class__, fn._backward_cls)
        self.assertTrue(q.requires_grad)
        q.sum().backward()
        self.assertEqual(y.grad, torch.ones(5, 5))

    def test_leaf_assignment(self):
        x = torch.randn(5, 5)
        y = torch.randn(5, requires_grad=True)
        z = torch.randn(5, requires_grad=True)

        x[0] = y
        x[1] = 2 * z
        self.assertTrue(x.requires_grad)
        self.assertIsNot(x.grad_fn, None)
        x.sum().backward()
        self.assertEqual(y.grad, torch.ones(5))
        self.assertEqual(z.grad, torch.ones(5) * 2)

    def test_no_grad_assignment(self):
        x = torch.randn(5, 5, requires_grad=True)
        y = torch.randn(5)
        with torch.no_grad():
            x[0] = y

        self.assertTrue(x.requires_grad)
        self.assertIsNone(x.grad_fn)

    def test_no_grad_modifies_version(self):
        x = torch.randn(5, requires_grad=True)
        y = torch.randn(5, requires_grad=True)
        z = (x * y).sum()
        with torch.no_grad():
            x *= 2
        self.assertRaisesRegex(RuntimeError, 'modified by an inplace operation',
                               lambda: z.backward())

    def test_no_grad_input(self):
        class MyFunction(Function):
            @staticmethod
            def forward(self, x):
                return x

            @staticmethod
            def backward(self, grad_output):
                return grad_output

        x = torch.randn(5, requires_grad=True)
        with torch.no_grad():
            y = MyFunction.apply(x)

        self.assertTrue(x.requires_grad)
        self.assertIsNone(y.grad_fn)

    def test_backward_copy(self):
        # This tests checks backward engine for a very subtle bug that appreared
        # in one of the initial versions of autograd. Gradients tensors were
        # simply stored in lists while the function waited for all its gradients
        # to be computed. However, sometimes an output was used multiple times,
        # so the gradients needed to be summed. Engine used to keep a need_copy
        # set of tensors that will need a clone upon next addition and removed
        # them from the set as soon as the clone was performed. However, this
        # could lead to incorrect results if the same gradient tensor was
        # buffered in three places in the graph:
        # 1. When accumulating gradients in one of these places it was cloned
        #    and removed from need_copy set.
        # 2. When accumulating in second place, it wasn't in the need_copy set,
        #    so the gradients were simply accumulated in-place (which already
        #    modified the grad in 3rd place)
        # 3. When accumulating in the third place, it wasn't in the need_copy set
        #    as well, so the incoming gradient was summed in-place, yielding
        #    incorrect results in all functions, except the first one.
        x = torch.ones(5, 5, requires_grad=True)
        y = torch.ones(5, 5, requires_grad=True)
        # Simulate that we're in the middle of the graph
        a = x + 2
        b = y + 2
        c = x + 2
        # This op will just return grad_output two times in backward
        add1 = a + b
        add2 = add1 + c
        # Simulate a long branch, so grad_output will get buffered.
        for _ in range(4):
            a = a * 2
            b = b * 2
            c = c * 2
        branch = a + b + c
        out = add2 + branch
        # expected gradients are:
        # for x: 34 (16 from final a, 16 from final c, 2 from add2)
        # for y: 17 (16 from final b, 1 from add2)
        grad_output = torch.ones(5, 5)
        out.backward(grad_output)
        self.assertEqual(x.grad, torch.ones(5, 5) * 34)
        self.assertEqual(y.grad, torch.ones(5, 5) * 17)

    def test_save_none_for_backward(self):
        test_case = self

        class MyFn(Function):
            @staticmethod
            def forward(ctx, input):
                ctx.save_for_backward(None, input, None)
                return input * input

            @staticmethod
            def backward(ctx, grad_output):
                n1, input, n2 = ctx.saved_tensors
                test_case.assertIsNone(n1)
                test_case.assertIsNone(n2)
                return 2 * input * grad_output

        x = torch.randn(5, 5, requires_grad=True)
        y = MyFn.apply(x)
        y.sum().backward()
        self.assertEqual(x.grad, 2 * x)

    def test_too_many_grads(self):
        class MyFn(Function):
            @staticmethod
            def forward(ctx, input):
                return input

            @staticmethod
            def backward(ctx, grad_output):
                return grad_output, None, None

        x = torch.randn(5, 5, requires_grad=True)
        y = MyFn.apply(x)
        y.sum().backward()
        self.assertEqual(x.grad, torch.ones_like(x))

    def test_pickle(self):
        x = torch.randn(10, 10, requires_grad=True)
        y = torch.randn(10, 10, requires_grad=False)

        def assert_strict_equal(var1, var2):
            self.assertEqual(var1, var2)
            self.assertEqual(var1.requires_grad, var2.requires_grad)

        serialized = [pickle.dumps([x, y], protocol=p) for p in range(3)]
        for dump in serialized:
            xc, yc = pickle.loads(dump)
            assert_strict_equal(xc, x)
            assert_strict_equal(yc, y)

    def test_dep_nograd(self):
        class F1(Function):
            @staticmethod
            def forward(ctx, input):
                out = torch.randn(input.size())
                ctx.mark_non_differentiable(out)
                return input, out

            @staticmethod
            def backward(ctx, grad_output, ignored):
                return grad_output

        class F2(Function):
            @staticmethod
            def forward(ctx, input, ignored):
                return input

            @staticmethod
            def backward(ctx, grad_output):
                return grad_output, None

        x = torch.randn(5, requires_grad=True)
        a, b = F1.apply(x)
        b = b + 1  # separate F1 from F2 by another op
        self.assertTrue(a.requires_grad)
        self.assertFalse(b.requires_grad)
        c = F2.apply(a, b)
        c.backward(torch.ones(c.size()))
        self.assertEqual(x.grad, torch.ones(x.size()))

    def test_set_grad_enabled(self):
        x = torch.tensor([1.], requires_grad=True)
        with torch.set_grad_enabled(False):
            y = x * 2
        self.assertFalse(y.requires_grad)
        with torch.set_grad_enabled(True):
            y = x * 2
        self.assertTrue(y.requires_grad)
        with torch.set_grad_enabled(False):
            torch.set_grad_enabled(True)
            y = x * 2
        self.assertTrue(y.requires_grad)

    def test_simple_reentrant(self):
        y_data = torch.randn(2, 2)

        class Reenter(Function):
            @staticmethod
            def forward(ctx, x):
                with torch.enable_grad():
                    ctx.x = Variable(x, requires_grad=True)
                    ctx.y = Variable(y_data, requires_grad=True)
                    ctx.output_var = ctx.x * ctx.y
                return ctx.output_var.detach()

            @staticmethod
            def backward(ctx, grad_output):
                with torch.enable_grad():
                    ctx.output_var.sum().backward()
                return ctx.x.grad * grad_output

        # Reentrant starts on CPU thread, finishs on GPU thread
        x = torch.randn(2, 2, requires_grad=True)
        out = Reenter.apply(x)
        out.sum().backward()
        self.assertEqual(x.grad, y_data)

    def test_reentrant_child_error(self):
        # Parent graph.
        a = torch.rand(3, 3, requires_grad=True)
        c = a * a

        # Reentrant child graph.
        b = torch.rand(3, 3, requires_grad=True)
        e = b * b
        f = TestAutograd.SimulateBackwardError.apply(e)
        reentrant_root = f.sum()

        class ReentrantFunc(Function):

            @staticmethod
            def forward(ctx, inp):
                return inp.clone()

            @staticmethod
            def backward(ctx, grad):
                # Reentrant backward in child will throw an error.
                reentrant_root.backward()
                return grad

        d = ReentrantFunc.apply(c)
        with self.assertRaisesRegex(Exception, 'Simulate error'):
            d.sum().backward()

    def test_broadcast_tensors(self):
        f_args_variable = (torch.randn(3, requires_grad=True),
                           torch.randn(1, 2, 1, requires_grad=True),
                           torch.randn(1, 1, requires_grad=True),
                           torch.randn(5, 1, 1, requires_grad=True))
        f_args_tensor = deepcopy(unpack_variables(f_args_variable))
        run_functional_checks(self, "test_broadcast_tensors", "broadcast",
                              lambda a, b, c, d: torch.broadcast_tensors(a, b, c, d),
                              True, f_args_variable, f_args_tensor)

    def test_block_diag(self):
        f_args_variable = (torch.randn(1, S, requires_grad=True),
                           torch.randn(2, S, requires_grad=True),
                           torch.randn(3, S, requires_grad=True))
        f_args_tensor = deepcopy(unpack_variables(f_args_variable))
        run_functional_checks(self, "test_block_diag", "block_diag",
                              lambda a, b, c: torch.block_diag(a, b, c),
                              True, f_args_variable, f_args_tensor)

    def test_cat(self):
        f_args_variable = (torch.randn(1, S, S, requires_grad=True),
                           torch.randn(2, S, S, requires_grad=True),
                           torch.randn(3, S, S, requires_grad=True),
                           0)
        f_args_tensor = deepcopy(unpack_variables(f_args_variable))
        run_functional_checks(self, "test_cat", "cat",
                              lambda a, b, c, dim: torch.cat((a, b, c), dim),
                              True, f_args_variable, f_args_tensor)

    def test_cat_negdim_1(self):
        f_args_variable = (torch.randn(S, S, 1, requires_grad=True),
                           torch.randn(S, S, 2, requires_grad=True),
                           torch.randn(S, S, 3, requires_grad=True),
                           -1)
        f_args_tensor = deepcopy(unpack_variables(f_args_variable))
        run_functional_checks(self, "test_cat_negdim_1", "cat",
                              lambda a, b, c, dim: torch.cat((a, b, c), dim),
                              True, f_args_variable, f_args_tensor)

    def test_cat_negdim_2(self):
        f_args_variable = (torch.randn(S, 1, S, requires_grad=True),
                           torch.randn(S, 2, S, requires_grad=True),
                           torch.randn(S, 3, S, requires_grad=True),
                           -2)
        f_args_tensor = deepcopy(unpack_variables(f_args_variable))
        run_functional_checks(self, "test_cat_negdim_2", "cat",
                              lambda a, b, c, dim: torch.cat((a, b, c), dim),
                              True, f_args_variable, f_args_tensor)

    def test_cat_empty_legacy(self):
        f_args_variable = (torch.randn(0, requires_grad=True),
                           torch.randn(S, S, requires_grad=True))
        # gradgradcheck doesn't work, probably because legacy size tracking is wrong somewhere,
        # hence False passed below, but gradcheck checked explicitly.
        f_args_tensor = deepcopy(unpack_variables(f_args_variable))
        run_functional_checks(self, "test_cat_empty_legacy", "cat",
                              lambda a, b: torch.cat((a, b)),
                              False, f_args_variable, f_args_tensor)
        self.assertTrue(gradcheck(lambda a, b: torch.cat((a, b)), f_args_variable, eps=1e-6, atol=PRECISION))

    def test_cat_empty(self):
        f_args_variable = (torch.randn(0, S, requires_grad=True),
                           torch.randn(S, S, requires_grad=True))
        f_args_tensor = deepcopy(unpack_variables(f_args_variable))
        run_functional_checks(self, "test_cat_empty", "cat",
                              lambda a, b: torch.cat((a, b)),
                              True, f_args_variable, f_args_tensor)

    def test_trapz(self):
        f_args_variable = (torch.randn(2, 3, requires_grad=True),
                           torch.tensor([[1.0, 2.0, 5.5], [2.3, 0.5, 6.2]], requires_grad=True))
        f_args_tensor = deepcopy(unpack_variables(f_args_variable))
        run_functional_checks(self, "test_trapz", "trapz",
                              lambda y, x: torch.trapz(y, x),
                              True, f_args_variable, f_args_tensor)


    def test_var_mean_differentiable(self):
        dim = [2, 4]
        keepdim = False
        input1 = torch.randn(3, 4, 5, 6, 2, 3, requires_grad=True)
        input2 = deepcopy(input1)
        var1, mean1 = torch.var_mean(input1, dim=dim, keepdim=keepdim)
        var2 = input2.var(dim=dim, keepdim=keepdim)
        mean2 = input2.mean(dim=dim, keepdim=keepdim)
        grad = torch.randn(3, 4, 6, 3, requires_grad=True)

        r1 = var1 * var1 * mean1 * mean1
        r2 = var2 * var2 * mean2 * mean2
        self.assertTrue(torch.allclose(r1, r2, rtol=0.01, atol=0.0))

        torch.autograd.backward(r1, grad)
        torch.autograd.backward(r2, grad)
        self.assertTrue(torch.allclose(input1.grad, input2.grad, rtol=0.01, atol=0.0))

    @skipIfNoLapack
    def test_cholesky_solve(self):
        def _test_with_size(A_dims, B_dims, upper):
            root = torch.rand(*A_dims).requires_grad_()
            b = torch.rand(*B_dims).requires_grad_()

            def func(root, b, upper):
                if upper:
                    A = root.triu()
                else:
                    A = root.tril()
                return torch.cholesky_solve(b, A, upper)

            gradcheck(func, [root, b, upper])
            gradgradcheck(func, [root, b, upper])

        for (a_size, b_size), upper in product([((3, 3), (3, 4)), ((3, 3), (3, 2)),
                                                ((2, 3, 3), (2, 3, 4)), ((2, 3, 3), (2, 3, 2))],
                                               [True, False]):
            _test_with_size(a_size, b_size, upper)

    @skipIfNoLapack
    def test_eig(self):
        def func(B):
            return torch.eig(B, eigenvectors=True)

        def func_eigvals(B):
            return torch.eig(B, eigenvectors=True)[0]

        def func_eigvecs(B):
            return torch.eig(B, eigenvectors=True)[1]

        def run_test(dims):
            # The backward operation for eig only works for real eigenvalues,
            # so the matrix should be B = U^{-1}*A*U where A is a random
            # symmetric matrix and U is a random full-rank matrix.
            # Slight change to the matrix should not make the eigenvalues
            # complex, so we apply requires_grad_ to B, not A and U

            A = random_symmetric_matrix(dims[-1], *dims[:-2])
            U = torch.rand(*dims)
            Uinv = torch.inverse(U)
            B = torch.matmul(Uinv, torch.matmul(A, U)).requires_grad_()

            gradcheck(func, [B])
            gradgradcheck(func, [B])
            gradcheck(func_eigvals, [B])
            gradgradcheck(func_eigvals, [B])
            gradcheck(func_eigvecs, [B])
            gradgradcheck(func_eigvecs, [B])

        for dims in [(3, 3), (5, 5)]:
            run_test(dims)

    @skipIfNoLapack
    def test_symeig(self):
        def func(root, upper):
            x = 0.5 * (root + root.transpose(-2, -1))
            return torch.symeig(x, eigenvectors=True, upper=upper)

        def run_test(upper, dims):
            root = torch.rand(*dims, requires_grad=True)

            gradcheck(func, [root, upper])
            gradgradcheck(func, [root, upper])

            root = random_symmetric_matrix(dims[-1], *dims[:-2]).requires_grad_()
            w, v = root.symeig(eigenvectors=True)
            (w.sum() + v.sum()).backward()
            self.assertEqual(root.grad, root.grad.transpose(-1, -2))  # Check the gradient is symmetric

        for upper, dims in product([True, False], [(3, 3), (5, 3, 3), (4, 3, 2, 2)]):
            run_test(upper, dims)

    @slowTest
    @skipIfNoLapack
    def test_lobpcg(self):

        def func(k, A, largest=True, B=None):
            X_shape = list(A.shape)
            X_shape[-1] = k
            X = torch.eye(A.size(-2), k, dtype=A.dtype, device=A.device)
            if A.dim() > 2:
                X = X.expand(X_shape)

            D, U = torch.lobpcg(A=A, k=k, B=B, X=X)

            # LOBPCG uses a random initial eigenspace approximation
            # if parameter `X` is not provided.
            # This may cause a non-deterministic behavior
            # when it comes to the sign of an eigenvector
            # (note if v is an eigenvector, so is -v),
            # hence we eliminate this non-determinism
            # by making sure that each column of U
            # gets multiplied by the sign of its max (in absolute value) element.
            # Also, gradcheck changes the content of the input by +/- eps (default to 1e-06)
            # to compute the numerical gradient which can also cause the signs to flip.
            _, idx = U.abs().max(-2, keepdim=True)
            sign = U.gather(-2, idx).sign()
            U = U * sign
            return D, U

        def run_symeig_test(k, sizes, largest=True):
            A = torch.rand(*sizes).double()
            A = A.matmul(A.transpose(-1, -2)) / 10
            A.requires_grad_(True)

            gradcheck(lambda A: func(k, A, largest), A)

            # Custom gradient vectors for better stability due to some
            # non-determinism in the lobpcg's forward.
            # Note it is not required if symeig is in forward instead (tested).
            D_grad = torch.rand(*A.shape[:-2], k) / 100
            U_grad = torch.rand(*A.shape[:-1], k) / 100
            gradgradcheck(lambda A: func(k, A, largest), A, [D_grad, U_grad], atol=1e-4)

            # check whether A.grad is symmetric
            A = A.detach().requires_grad_(True)
            D, U = func(k, A, largest)
            (D.sum() + U.sum()).backward()
            self.assertEqual(A.grad, A.grad.transpose(-1, -2))

        # the tests below take about 1-2 minutes to finish,
        # but we want to be extra sure that the backward is correct.
        for largest in [True, False]:
            run_symeig_test(1, (6, 6), largest=largest)
            run_symeig_test(1, (2, 6, 6), largest=largest)
            run_symeig_test(1, (2, 2, 6, 6), largest=largest)
            run_symeig_test(2, (6, 6), largest=largest)
            run_symeig_test(2, (2, 6, 6), largest=largest)
            run_symeig_test(2, (2, 2, 6, 6), largest=largest)
            run_symeig_test(3, (9, 9), largest=largest)
            run_symeig_test(3, (2, 9, 9), largest=largest)
            run_symeig_test(3, (2, 2, 9, 9), largest=largest)

    @skipIfNoLapack
    def test_cholesky_inverse(self):
        def _test_with_size(upper, dims):
            # We require to create a Cholesky factor which requires that the diagonal elements are positive.
            # Initializing too small values for the diagonal elements could cause issues when being perturbed
            # to obtain the numerical Jacobian, thereby leading to inconsistent gradcheck
            A = torch.randn(*dims)
            A.diagonal().uniform_(0.1, 5.0)
            A.requires_grad_()

            def func(A, upper):
                if upper:
                    root = A.triu()
                else:
                    root = A.tril()
                return torch.cholesky_inverse(root, upper)

            gradcheck(func, [A, upper])
            gradgradcheck(func, [A, upper])

        for upper, dims in product([True, False], [(3, 3), (5, 5)]):
            _test_with_size(upper, dims)

    @skipIfNoLapack
    def test_triangular_solve(self):
        def run_test(A_dims, B_dims, dtype):
            A = torch.rand(*A_dims, dtype=dtype).requires_grad_()
            b = torch.rand(*B_dims, dtype=dtype).requires_grad_()

            for upper, transpose, unitriangular in product((True, False), repeat=3):
                def func(A, b):
                    return torch.triangular_solve(b, A, upper, transpose, unitriangular)

                gradcheck(func, [A, b])
                gradgradcheck(func, [A, b])

        for dtype in (torch.double, torch.cdouble):
            run_test((3, 3), (3, 4), dtype)
            run_test((3, 3), (3, 2), dtype)
            run_test((2, 3, 3), (2, 3, 4), dtype)
            run_test((2, 3, 3), (2, 3, 2), dtype)

    @unittest.skipIf(not TEST_MKL, "PyTorch is built without MKL support")
    def test_fft_ifft_rfft_irfft(self):
        def _test_complex(sizes, signal_ndim):
            x = torch.randn(sizes, requires_grad=True, dtype=torch.double)

            for normalized in (True, False):
                def fft(x):
                    return x.fft(signal_ndim, normalized=normalized)

                gradcheck(fft, [x])
                gradgradcheck(fft, [x], gen_non_contig_grad_outputs=True)

                def ifft(fx):
                    return fx.ifft(signal_ndim, normalized=normalized)

                # Use output of fft(x) for inverse fft, due to symmetry requirements
                fx = fft(x).detach()
                fx.requires_grad = True
                gradcheck(ifft, [fx])
                gradgradcheck(ifft, [fx], gen_non_contig_grad_outputs=True)

        def _test_real(sizes, signal_ndim):
            x = torch.randn(sizes, requires_grad=True, dtype=torch.double)
            if x.dim() == signal_ndim:
                start_dim = 0
            else:
                start_dim = 1
            signal_sizes = x.size()[start_dim:start_dim + signal_ndim]

            for normalized, onesided in product((True, False), repeat=2):
                def rfft(x):
                    return x.rfft(signal_ndim, normalized=normalized, onesided=onesided)

                gradcheck(rfft, [x])
                gradgradcheck(rfft, [x], gen_non_contig_grad_outputs=True)

                # Generally speaking, irfft itself won't and can't pass the
                # current gradcheck as it assumes the input follows conjugate
                # symmetry, an requirement that is never true with our point
                # numerical Jacobian estimate. Without input symmtry, irfft's
                # behavior is undefined.
                #
                # Even onesided results can't remove all redundancy. For
                # example, consider the .select(last_signal_dim, 0) slice.
                # It is entirely represented in the onesided results (except
                # for 1D), and will be reflected onto itself!
                #
                # So only 1D onesided irfft should pass grad check as it is
                # guaranteed that the input has no symmetrical values.
                #
                # In other cases, we test a function that first uses rfft to
                # generate a tensor that follows the conjugate symmetry irfft
                # expects, and then feeds it into irfft. Since rfft is already
                # tested above, we thereby verify the correctness of irfft.
                if signal_ndim == 1 and onesided:
                    def irfft(fx):
                        return fx.irfft(signal_ndim, normalized=normalized,
                                        onesided=onesided, signal_sizes=signal_sizes)

                    # Use output of rfft(x) for inverse rfft, due to symmetry requirements
                    fx = rfft(x).detach()
                    fx.requires_grad = True
                    gradcheck(irfft, [fx])
                    gradgradcheck(irfft, [fx], gen_non_contig_grad_outputs=True)
                else:
                    # Test this function: f(x) = ifft(rfft(x) + rfft(z)), where
                    # z is some fixed tensor of same size as x. rfft(z) term is
                    # needed because otherwise f becomes identity.
                    z = torch.randn(sizes, dtype=torch.double)
                    fz = z.rfft(signal_ndim, normalized=normalized, onesided=onesided)

                    def rfft_irfft(x):
                        fx = x.rfft(signal_ndim, normalized=normalized, onesided=onesided)
                        y = fx + fz
                        return y.irfft(signal_ndim, normalized=normalized,
                                       onesided=onesided, signal_sizes=signal_sizes)

                    gradcheck(rfft_irfft, [x])
                    gradgradcheck(rfft_irfft, [x], gen_non_contig_grad_outputs=True)

        _test_real((2, 10), 1)
        _test_real((2, 3, 4), 2)
        _test_real((2, 3, 4, 3), 3)

        _test_complex((2, 2, 10, 2), 1)
        _test_complex((1, 2, 3, 4, 2), 2)
        _test_complex((2, 1, 3, 4, 3, 2), 3)

    def test_gradcheck_fail_when_no_differentiable_outputs_and_num_grad_not_zero(self):
        def autograd_fn(input):
            output = torch.detach(input)
            self.assertFalse(output.requires_grad)
            return output

        f_args_variable = torch.ones(S, S, requires_grad=True)
        self.assertRaisesRegex(RuntimeError, 'Numerical gradient for function expected to be zero',
                               lambda: gradcheck(autograd_fn, f_args_variable, eps=1e-6, atol=PRECISION))

    def test_variable_traverse(self):
        def get_out_and_unrefed_cycle():
            inp = torch.randn(10, requires_grad=True)
            tmp = inp.view(10, 1)
            out = tmp.view(10)

            # Create a reference cycle that contains an
            # intermediary Variable in the graph
            my_list = []
            my_list.append(tmp)
            my_list.append(my_list)

            return out

        out = get_out_and_unrefed_cycle()
        gc.collect()
        # This will segfault if things have been erroneously released
        out.backward(torch.randn(out.size()))

    def test_norm_subgradient(self):
        def run_test(input_size, norm_deg):
            input = torch.zeros(*input_size, requires_grad=True)
            input.norm(norm_deg).backward()
            self.assertEqual(input.grad.abs().sum(), 0)

        run_test((10,), 2)
        run_test((10, 10), 2)
        run_test((10,), 3)
        run_test((10,), 1)
        run_test((10,), 1.5)

    def test_pow_zero_tensor_gradient(self):
        def run_test(input_size, exponent):
            input = torch.zeros(*input_size, requires_grad=True)
            input.pow(exponent).sum().backward()
            self.assertEqual(input.grad.abs().sum(), 0)

        run_test((10,), torch.zeros(10))
        run_test((10, 10), torch.zeros(10, 10))
        run_test((10,), 0)

    def test_pow_scalar_base(self):
        a = torch.arange(1, 13, dtype=torch.double).view(3, 4).requires_grad_()
        gradcheck(lambda a: torch.pow(2, a), (a,))

    def test_igamma(self):
        # 1e-3 offset to avoid zeros
        # NOTE: derivative for s is not implemented
        s = (torch.rand(100, dtype=torch.double) + 1e-3)
        x = (torch.rand(100, dtype=torch.double) + 1e-3).requires_grad_()
        gradcheck(torch.igamma, (s, x))
        gradgradcheck(torch.igamma, (s, x))

    @skipIfNoLapack
    def test_pinverse(self):
        # Why is pinverse tested this way, and not ordinarily as other linear algebra methods?
        # 1. Pseudo-inverses are not generally continuous, which means that they are not differentiable
        # 2. Derivatives for pseudo-inverses exist typically for constant rank (Golub et al, 1973)
        # 3. This method creates two orthogonal matrices, and a constructs a test case with large
        #    singular values (given by x to the function).
        # 4. This will ensure that small perturbations don't affect the rank of matrix, in which case
        #    a derivative exists.
        # 5. This test exists since pinverse is implemented using SVD, and is hence a backpropable method
        m, n = 5, 10
        U = torch.randn(n, m).qr()[0].t()  # Orthogonal with dimensions m x n
        V = torch.randn(n, m).qr()[0].t()  # Orthogonal with dimensions m x n

        def func(x):
            S = torch.cat([x, torch.zeros(n - m)], 0)
            M = U.mm(torch.diag(S)).mm(V.t())
            return M.pinverse()

        gradcheck(func, [torch.rand(m).add_(1).requires_grad_()])
        gradcheck(func, [torch.rand(m).add_(10).requires_grad_()])
        gradgradcheck(func, [torch.rand(m).add_(1).requires_grad_()])
        gradgradcheck(func, [torch.rand(m).add_(10).requires_grad_()])

    def test_chain_matmul(self):
        def gen_matrices(p):
            matrices = []
            for (pi, pi_1) in zip(p[:-1], p[1:]):
                matrices.append(torch.randn(pi, pi_1).requires_grad_())
            return matrices

        gradcheck(torch.chain_matmul, gen_matrices([5, 10, 15, 5]))
        gradcheck(torch.chain_matmul, gen_matrices([3, 5, 2, 6]))
        gradcheck(torch.chain_matmul, gen_matrices([6, 2, 4, 8, 10]))
        gradgradcheck(torch.chain_matmul, gen_matrices([5, 10, 15, 5]))
        gradgradcheck(torch.chain_matmul, gen_matrices([3, 5, 2, 6]))
        gradgradcheck(torch.chain_matmul, gen_matrices([6, 2, 4, 8, 10]))

    @unittest.skipIf(IS_WINDOWS, """File open permission error on Windows,
            https://github.com/pytorch/pytorch/issues/34086""")
    def test_profiler_tracing(self):
        t1, t2 = torch.ones(1), torch.ones(1)
        with torch.autograd.profiler.profile() as prof:
            torch.add(t1, t2)

        with tempfile.NamedTemporaryFile(mode="w+") as f:
            prof.export_chrome_trace(f.name)
            # read the trace and expect valid json
            # if the JSON generated by export_chrome_trace is not valid, this will throw and fail the test.
            json.load(f)

        # Same test but for cuda.
        if not torch.cuda.is_available():
            return

        device = torch.device("cuda:0")
        t1, t2 = torch.ones(1, device=device), torch.ones(1, device=device)
        with torch.autograd.profiler.profile(use_cuda=True) as prof:
            torch.add(t1, t2)

        with tempfile.NamedTemporaryFile(mode="w+") as f:
            prof.export_chrome_trace(f.name)
            # Now validate the json
            json.load(f)

    def test_profiler(self):
        x = torch.randn(10, 10)

        with profile() as p:
            self.assertTrue(torch.autograd._profiler_enabled())
            y = x * 2 + 4

        self.assertFalse(torch.autograd._profiler_enabled())

        last_end = 0
        names = ['aten::mul', 'aten::to', 'aten::empty_strided', 'aten::copy_',
                 'aten::empty', 'aten::add', 'aten::to', 'aten::empty_strided',
                 'aten::copy_', 'aten::empty']
        top_level_names = ['aten::mul', 'aten::add']
        top_level_iter = iter(top_level_names)
        self.assertEqual(len(p.function_events), len(names))
        for info, expected_name in zip(p.function_events, names):
            if info.cpu_interval.start > last_end:
                top_level_name_expected = next(top_level_iter)
                self.assertEqual(info.name, top_level_name_expected)
                last_end = info.cpu_interval.end
            self.assertEqual(info.name, expected_name)

    def test_profiler_seq_nr(self):
        with profile() as p:
            x = torch.randn(10, 10, requires_grad=True)
            y = torch.randn(10, 10, requires_grad=True)
            z = x + y
            s = z.sum()
            s.backward()
        # expecting aten::add, aten::sum to have the sequence numbers,
        # expecting the corresponding backward nodes to have the same numbers
        # as the forward ops
        add_seq_nr = -1
        sum_seq_nr = -1
        found_add = found_sum = False
        found_bwd_add = found_bwd_sum = False
        found_empty = False
        for e in p.function_events:
            if e.name == "aten::add":
                add_seq_nr = e.sequence_nr
                self.assertFalse(found_add)
                found_add = True
            elif e.name == "aten::sum":
                sum_seq_nr = e.sequence_nr
                self.assertFalse(found_sum)
                found_sum = True
            elif "Add" in e.name and "Backward" in e.name:
                self.assertEqual(e.sequence_nr, add_seq_nr)
                self.assertFalse(found_bwd_add)
                found_bwd_add = True
            elif "Sum" in e.name and "Backward" in e.name:
                self.assertEqual(e.sequence_nr, sum_seq_nr)
                self.assertFalse(found_bwd_sum)
                found_bwd_sum = True
            # check that nested ops (e.g. empty) don't have
            # sequence number
            if e.name == "aten::empty":
                self.assertEqual(e.sequence_nr, -1)
                found_empty = True
        self.assertGreaterEqual(add_seq_nr, 0)
        self.assertGreaterEqual(sum_seq_nr, 0)
        self.assertNotEqual(add_seq_nr, sum_seq_nr)
        self.assertTrue(found_add)
        self.assertTrue(found_sum)
        self.assertTrue(found_bwd_add)
        self.assertTrue(found_bwd_sum)
        self.assertTrue(found_empty)

    def test_profiler_unboxed_only(self):
        x = torch.rand(3, 4)

        with torch.autograd.profiler.profile() as prof:
            x.resize_([3, 2])

    def test_profiler_propagation(self):
        def foo(x):
            with record_function("in_foo") as rf:
                return x * 2

        x = torch.rand(3, 4)
        traced_foo = torch.jit.trace(foo, x)

        def bar(x):
            with record_function("in_bar") as rf:
                # we expect that profiler will be able
                # propagate across fork
                fut = torch.jit._fork(traced_foo, x)
                y = torch.jit._wait(fut)
                # note: continuation (and rf's end) can
                # be executed in a different thread
                with record_function("in_bar_after_wait") as rf2:
                    y = y * 2
                return y

        traced_bar = torch.jit.trace(bar, x)

        with profile() as p:
            traced_bar(x)

        found_foo = False
        found_bar = False
        found_bar_after_wait = False
        for info in p.function_events:
            if info.name == "in_foo":
                self.assertFalse(found_foo)
                found_foo = True
            elif info.name == "in_bar":
                self.assertFalse(found_bar)
                found_bar = True
            elif info.name == "in_bar_after_wait":
                self.assertFalse(found_bar_after_wait)
                found_bar_after_wait = True
        self.assertTrue(found_foo)
        self.assertTrue(found_bar)
        self.assertTrue(found_bar_after_wait)

    def test_record_function_callbacks(self):
        x = torch.randn(10, 10)
        with profile() as p:
            with record_function("foo"):
                y = x * 2 + 4

        function_events = p.function_events
        foo_event = [event for event in function_events if "foo" in event.name][0]
        self.assertEqual(foo_event.count, 1)

    def test_profiler_aggregation_fake(self):
        events = EventList()
        id = [0]

        def get_id():
            id[0] = id[0] + 1
            return id[0]

        # [[thread_id, [(start, end, id), ....]], ...]
        # Using list instead of a dict so order is guaranteed for any Python
        # version
        threads = [
            [1, [(0, 1, get_id()), (1, 2, get_id())]],
            [0, [(0, 2, get_id()), (1, 2, get_id()), (1, 3, get_id())]],
        ]
        for thread, ranges in threads:
            for range in ranges:
                assert(len(range) == 3)
                events.append(
                    FunctionEvent(
                        id=range[2],
                        node_id=0,
                        name="",
                        thread=thread,
                        cpu_start=range[0],
                        cpu_end=range[1],
                    )
                )

        events.populate_cpu_children()

        # Note that [1, 3] pushes out [0, 2] first. Then we record [1, 2]
        # as a child of [1, 3]
        res = [[], [], [], [], [4]]

        def get_children_ids(event):
            return [child.id for child in event.cpu_children]

        assert([get_children_ids(event) for event in events] == res)

    def test_profiler_aggregation_table(self):
        """
        Test if the profiling result is aggregated for `str(prof)`

        See: https://github.com/pytorch/pytorch/issues/37500
        """

        x = torch.randn(1024)
        with torch.autograd.profiler.profile() as prof:
            torch.einsum("i->", x)

        prof_str = str(prof)
        prof_table = prof.table()

        self.assertEqual(prof_table, prof_str)

    def test_profiler_function_event_avg(self):
        avg = FunctionEventAvg()
        avg.add(FunctionEvent(id=0, node_id=0, name="foo", thread=0, cpu_start=10, cpu_end=15))
        avg.add(FunctionEvent(id=1, node_id=0, name="foo", thread=0, cpu_start=20, cpu_end=30))
        avg.add(avg)
        self.assertEqual(avg.key, "foo")

        # aggregate stats
        self.assertEqual(avg.count, 4)
        self.assertEqual(avg.cpu_time_total, 30)
        self.assertEqual(avg.self_cpu_time_total, 30)
        self.assertEqual(avg.cuda_time_total, 0)

        # average stats
        self.assertEqual(avg.cpu_time, 7.5)
        self.assertEqual(avg.cuda_time_total, 0)

    def test_profiler_shapes(self):
        print("")
        layer1 = torch.nn.Linear(20, 30)
        layer2 = torch.nn.Linear(30, 40)
        input = torch.randn(128, 20)
        with profile(record_shapes=True) as prof:
            layer2(layer1(input))

        print(prof.function_events)

        top_level_expected_events_and_shapes = [
            (None, [[30, 20]]),
            ('aten::addmm', [[30], [128, 20], [20, 30], [], []]),
            (None, [[40, 30]]),
            ('aten::addmm', [[40], [128, 30], [30, 40], [], []])
        ]

        expected_iter = iter(top_level_expected_events_and_shapes)
        last_end = 0

        for event in prof.function_events:
            if event.cpu_interval.start > last_end:
                name_expected, input_shape_expected = next(expected_iter)
                if name_expected is not None:
                    self.assertEqual(event.name, name_expected)
                self.assertEqual(event.input_shapes, input_shape_expected)
                last_end = event.cpu_interval.end

    def test_profiler_no_cuda(self):
        print("")
        layer = torch.nn.Linear(20, 30)
        x = torch.randn(128, 20)
        with profile(use_cuda=False) as prof:
            layer(x)

        prof_str = str(prof)
        print(prof_str)
        self.assertTrue('cpu' in prof_str.lower())
        self.assertTrue('cuda' not in prof_str.lower())

    def test_profiler_aggregation_lstm(self):
        print("")
        rnn = torch.nn.LSTM(10, 20, 2)
        total_time_s = 0
        with profile(record_shapes=True) as prof:
            for i in range(20):
                input = torch.randn(5, 3, 10)
                h = torch.randn(2, 3, 20)
                c = torch.randn(2, 3, 20)
                start = time.time()
                rnn(input, (h, c))
                end = time.time()
                total_time_s += end - start

        print(prof.table(
            sort_by="self_cpu_time_total", row_limit=10, header="TEST"))
        print(prof.key_averages(group_by_input_shape=True).table(
            sort_by="self_cpu_time_total", row_limit=10))
        print(prof.table(
            sort_by="self_cpu_time_total", row_limit=10, header="TEST", top_level_events_only=True))
        print(prof.key_averages(group_by_input_shape=True).table(
            sort_by="self_cpu_time_total", row_limit=10, top_level_events_only=True))

        total_time_us = total_time_s * 1000.0 * 1000.0  # make it us which is profiler default
        print(
            "Total time based on python measurements: ",
            format_time(total_time_us)
        )
        print(
            "CPU time measurement python side overhead: {:.2f}%".format(
                (total_time_us / prof.self_cpu_time_total - 1.0) * 100.0
            )
        )

        if sys.platform != "win32":
            with tempfile.NamedTemporaryFile() as trace_file:
                prof.export_chrome_trace(trace_file.name)

    def test_memory_profiler(self):
        def run_profiler(tensor_creation_fn, metric):
            # collecting allocs / deallocs
            with profile(profile_memory=True, record_shapes=True) as prof:
                x = None
                with record_function("test_user_scope_alloc"):
                    x = tensor_creation_fn()
                with record_function("test_user_scope_dealloc"):
                    del x
            stats = prof.key_averages(group_by_input_shape=True)
            print(stats.table(sort_by=metric))
            return stats

        def check_metrics(stats, metric, allocs=None, deallocs=None):
            stat_metrics = {}
            for stat in stats:
                stat_metrics[stat.key] = getattr(stat, metric)
            if allocs is not None:
                for alloc_fn in allocs:
                    self.assertTrue(alloc_fn in stat_metrics)
                    self.assertTrue(stat_metrics[alloc_fn] > 0)
            if deallocs is not None:
                for dealloc_fn in deallocs:
                    self.assertTrue(dealloc_fn in stat_metrics)
                    self.assertTrue(stat_metrics[dealloc_fn] < 0)

        def create_cpu_tensor():
            return torch.rand(10, 10)

        def create_cuda_tensor():
            return torch.rand(10, 10).cuda()

        def create_mkldnn_tensor():
            return torch.rand(10, 10, dtype=torch.float32).to_mkldnn()

        print("Running CPU test")
        stats = run_profiler(create_cpu_tensor, "cpu_memory_usage")
        check_metrics(
            stats,
            "cpu_memory_usage",
            allocs=[
                "aten::empty",
                "aten::rand",
                "test_user_scope_alloc",
            ],
            deallocs=[
                "test_user_scope_dealloc",
            ]
        )

        if torch.cuda.is_available():
            create_cuda_tensor()
            print("Running CUDA test")
            stats = run_profiler(create_cuda_tensor, "cuda_memory_usage")
            check_metrics(
                stats,
                "cuda_memory_usage",
                allocs=[
                    "test_user_scope_alloc",
                    "aten::to",
                    "aten::empty_strided",
                ],
                deallocs=[
                    "test_user_scope_dealloc",
                ]
            )
            check_metrics(
                stats,
                "cpu_memory_usage",
                allocs=[
                    "aten::rand",
                    "aten::empty",
                ]
            )

        if torch._C.has_mkldnn:
            create_mkldnn_tensor()
            print("Running MKLDNN test")
            stats = run_profiler(create_mkldnn_tensor, "cpu_memory_usage")
            check_metrics(
                stats,
                "cpu_memory_usage",
                allocs=[
                    "test_user_scope_alloc",
                    "aten::rand",
                    "aten::empty",
                    "aten::to_mkldnn",
                ],
                deallocs=[
                    "test_user_scope_dealloc",
                ]
            )

        # check partial overlap of tensor allocation with memory profiler
        x = torch.rand(10, 10)
        with profile(profile_memory=True, record_shapes=True) as prof:
            del x
            x = torch.rand(10, 10)
        del x
        stats = prof.key_averages(group_by_input_shape=True)
        check_metrics(
            stats,
            "cpu_memory_usage",
            allocs=[
                "aten::rand",
                "aten::empty",
            ]
        )

    def test_record_function(self):
        x = torch.randn(10, 10)

        def forward(x):
            with record_function("outer"):
                y = x * 2 + 4
                with record_function("inner"):
                    y = y - 1
            y = y / 1

        forward(x)

        with profile() as p:
            forward(x)

        events = p.function_events
        important_events = [
            'outer',
            'aten::mul',
            'aten::add',
            'inner',
            'aten::sub',
            'aten::div'
        ]
        idx = 0
        for info in events:
            if info.name == important_events[idx]:
                idx = idx + 1
            if idx == len(important_events):
                break
        self.assertEqual(idx, len(important_events))

        # We can also use record_function to decorate arbitrary function
        @record_function('my_func')
        def f(x, y):
            return x + y

        with profile() as p:
            f(1, 2)

        self.assertTrue('my_func' in str(p))

    def test_record_function_multithreaded(self):
        rf = record_function("outer")
        rf.__enter__()
        with record_function("inner"):
            # test that exiting the record function after starting another one
            # doesn't throw.
            rf.__exit__(None, None, None)

        with record_function("inner"):
            rf.__enter__()
        # test that exiting the record function after ending another one
        # doesn't throw.
        rf.__exit__(None, None, None)


    def test_dir(self):
        x = torch.randn(10, 10)
        keys = dir(x)
        self.assertIn('shape', keys)

        # real and imag are only implemented for complex tensors.
        y = torch.randn(10, 10, dtype=torch.cfloat)
        for key in ['real', 'imag']:
            self.assertRaises(RuntimeError, lambda: hasattr(x, key))
            self.assertTrue(hasattr(y, key))
            keys.remove(key)

        for key in keys:
            self.assertTrue(hasattr(x, key))

    def test_as_strided(self):

        def test(x, prepro_fn, size, strides, offset=None):
            x = x.to(torch.double).detach().requires_grad_()

            # Check that forward will **not** resize storage because it may
            # cause NaN in output and fail numerical Jacobian check consequently
            with torch.no_grad():
                y = prepro_fn(x) if prepro_fn is not None else x
                max_offset = sum((si - 1) * st for si, st in zip(size, strides))
                max_offset += offset if offset is not None else y.storage_offset()
                assert max_offset < len(y.storage()), "test case resizes storage"

            def closure(x):
                if prepro_fn is not None:
                    x = prepro_fn(x)
                return x.as_strided(size, strides, offset)

            gradcheck(closure, [x])
            gradgradcheck(closure, [x])

        # test
        test(torch.arange(0, 25), lambda x: x.view(5, 5), [3, 3], [6, 2], 2)

        # test crazy stride at dim with size 1 case
        test(torch.randn(12), None, [1, 2, 1, 5], [0, 5, 100, 1], 2)

        # test expand case
        test(torch.randn(5), None, [3, 3, 3], [0, 1, 0], 2)
        test(torch.randn(5), None, [3, 3, 3], [0, 0, 0], 4)
        test(torch.randn(5), lambda x: x.expand(5, 5), [5, 5], [0, 1], 0)

        # test non-expand overlapping case
        test(torch.randn(35), None, [6, 6], [5, 1], 2)
        test(torch.randn(15), None, [3, 2], [3, 6], 2)

        # test transpose case
        test(torch.randn(3, 4), None, [4, 3], [1, 4])

        # test "getting things outside the input" case
        x = torch.randn(6, 2)
        test(x[3:], None, [3, 2], [2, 1], 0)  # should be all zeros
        self.assertEqual(x[3:].as_strided([3, 2], [2, 1], 0), x[:3])

        # test select on expanded input case
        test(torch.randn(2, 3), lambda x: x.expand(10, 2, 3), [2, 3], [3, 1], 0)

    def _test_lerp_tensor_weights(self, cast):
        def construct_inputs(*shapes):
            start = cast(torch.randn(shapes[0])).requires_grad_()
            end = cast(torch.randn(shapes[1])).requires_grad_()
            weight = cast(torch.randn(shapes[2])).requires_grad_()
            return [start, end, weight]

        all_test_shapes = [((3, 3, 3), (3, 3, 3), (3, 3, 3)),  # no broadcasting
                           ((3,), (3, 3, 3), (3, 3, 3)),  # start broadcasting - 1
                           ((3, 3, 3), (3,), (3, 3, 3)),  # end broadcasting - 1
                           ((3, 3, 3), (3, 3, 3), (3,)),  # weight broadcasting - 1
                           ((), (3, 3, 3), (3, 3, 3)),  # start broadcasting - 2
                           ((3, 3, 3), (), (3, 3, 3)),  # end broadcasting - 2
                           ((3, 3, 3), (3, 3, 3), ()),  # weight broadcasting - 2
                           ((3, 3), (3, 3, 3), (3,))]  # all broadcasting

        for shapes in all_test_shapes:
            cur_inputs = construct_inputs(*shapes)
            gradcheck(torch.lerp, cur_inputs)
            gradgradcheck(torch.lerp, cur_inputs)

    def test_lerp_tensor_weights(self):
        self._test_lerp_tensor_weights(lambda t: t)

    def test_reduce_dtype(self):
        def test_reduction(op, has_no_dim, takes_dtype=True):
            x = torch.randn(3, 3, dtype=torch.float, requires_grad=True)

            if has_no_dim:
                grad1, = torch.autograd.grad([op(x)], [x])
                grad2, = torch.autograd.grad([op(x, dtype=torch.double)], [x])
                self.assertEqual(grad1, grad2)
                self.assertEqual(grad2.dtype, torch.float)

            gi = torch.randn(op(x, dim=0).shape, dtype=torch.float)
            grad1, = torch.autograd.grad([op(x, dim=0)], [x], gi)
            if takes_dtype:
                grad2, = torch.autograd.grad([op(x, dim=0, dtype=torch.double)], [x], gi.double())
            else:
                grad2, = torch.autograd.grad([op(x.double(), dim=0)], [x], gi.double())
            self.assertEqual(grad1, grad2)
            self.assertEqual(grad2.dtype, torch.float)

        test_reduction(torch.sum, True)
        test_reduction(torch.prod, True)
        test_reduction(torch.cumsum, False)
        test_reduction(torch.cumprod, False)
        test_reduction(torch.logcumsumexp, False, takes_dtype=False)

    def test_inplace_view_saved_output(self):
        # Test an in-place operation on a view in which the in-place op saves
        # its output. Previously, this created a reference cycle.
        dealloc = [0]

        class IncrementOnDelete(object):
            def __del__(self):
                dealloc[0] += 1

        def test():
            root = torch.randn(3, 3, requires_grad=True)
            copy = root.clone()
            copy.grad_fn.register_hook(IncrementOnDelete())
            view = copy.view(9)
            torch.nn.functional.relu(view, inplace=True)

        test()
        self.assertEqual(dealloc[0], 1)

    def test_inplace_view_leaf_errors(self):
        # Issue #21875: Fail faster (when we try to modify the view vs. in backward())
        x = torch.zeros(1, requires_grad=True)
        y = x.view_as(x)
        with self.assertRaisesRegex(RuntimeError,
                                    "a view of a leaf Variable that "
                                    "requires grad is being used in "
                                    "an in-place operation."):
            y.add_(1)

    def test_inplace_view_backward(self):
        # Issue #10532: Make sure that this does not raise RuntimeError.
        net = nn.Sequential(
            nn.InstanceNorm2d(2),
            nn.ReLU(True)
        )

        x = torch.tensor([[[[1.0, 1.0]]]], requires_grad=True)
        g, = torch.autograd.grad(net(x).pow(2), [x], grad_outputs=x.new_ones(x.shape) , create_graph=True)
        torch.autograd.grad(g.sum(), [x])
        self.assertEqual(x, torch.tensor([[[[1.0, 1.0]]]]))

        # https://discuss.pytorch.org/t/freeing-buffer-strange-behavior/31955/8
        inputs = torch.ones((1, 3, 256, 256), requires_grad=True)

        tmp1 = (inputs + 1).view_as(inputs)
        tmp2 = torch.nn.functional.threshold(tmp1, 0., 0., True)
        prob_interpolated = torch.sigmoid(tmp2)

        gradients = torch.autograd.grad(outputs=prob_interpolated, inputs=inputs,
                                        grad_outputs=torch.ones(prob_interpolated.size()),
                                        create_graph=True, retain_graph=True)[0]

        gradient_penalty = gradients.sum()
        gradient_penalty.backward()

        fn = gradient_penalty.grad_fn.next_functions[0][0].next_functions[1][0]
        self.assertEqual(fn.name(), "ThresholdBackwardBackward")

    def test_inplace_view_weak_grad_fn(self):
        # Issue 23502: Test that b's grad_fn is preserved.
        a = torch.arange(10.0, requires_grad=True)

        b = a.narrow(0, 0, 2).clone().view(-1)
        b.relu_()

        c = b.clone()
        del b
        gc.collect()

        s = c.sum()
        s.backward()
        self.assertEqual(s, torch.tensor(1.0))

        # Issue #21875: Fail faster (when we try to modify the view vs. in backward())
        a = torch.rand(10, requires_grad=True).narrow(0, 0, 10)
        with self.assertRaises(RuntimeError):
            b = a.relu_()

    def test_mul_out(self):
        a = torch.randn(2, 2, requires_grad=True)
        b = torch.randn(2, 2, requires_grad=True)
        x = torch.zeros_like(a)

        # out=... functions don't support automatic differentiation currently
        self.assertRaisesRegex(RuntimeError, 'out=', lambda: torch.mul(a, b, out=x))

        # the inputs can require grad if we're in no_grad() mode
        with torch.no_grad():
            torch.mul(a, b, out=x)
            self.assertEqual(x, a * b)

    def test_mul_out_result_requires_grad(self):
        a = torch.randn(2, 2)
        b = torch.randn(2, 2)
        x = torch.zeros(2, 2, requires_grad=True)
        # we should throw an exception if the output requires grad
        self.assertRaisesRegex(RuntimeError, 'out=', lambda: torch.mul(a, b, out=x))

    def test_diagonal_derivative_requires_grad(self):
        # test that the backward requires grad
        # we do this is because diagonal_backward uses inplace
        # operations and gradgradcheck does not catch whether
        # they works as expected (it will succeed even if
        # the gradient has requires_grad == False
        a = torch.randn(5, 6, requires_grad=True)
        b = torch.diagonal(a)**2
        c = b.sum()
        d, = torch.autograd.grad(c, a, retain_graph=True, create_graph=True)
        self.assertTrue(d.requires_grad)

    def test_anomaly_detect_nan(self):
        size = 10

        class MyFunc(Function):
            @staticmethod
            def forward(ctx, inp1, inp2, fail_0th):
                ctx.fail_0th = fail_0th
                return inp1.sum(0, keepdim=True)

            @staticmethod
            def backward(ctx, gO):
                gI = gO.clone().expand(size)
                gI[0] = 0
                gI[0] /= 0  # Generate a nan
                if ctx.fail_0th:
                    return gI, None, None
                else:
                    return None, gI, None

        inp = torch.rand(size, requires_grad=True)
        out = MyFunc.apply(inp, inp, True)
        out.backward()  # Should not fail

        inp = torch.rand(size, requires_grad=True)
        out = MyFunc.apply(inp, inp, True)
        with self.assertRaisesRegex(RuntimeError, "Function 'MyFuncBackward' returned nan values in its 0th output."):
            with warnings.catch_warnings(record=True) as w:
                with detect_anomaly():
                    out.backward()
            self.assertIn('No forward pass information', str(w[0].message))

        inp = torch.rand(size, requires_grad=True)
        with self.assertRaisesRegex(RuntimeError, "Function 'MyFuncBackward' returned nan values in its 1th output."):
            with warnings.catch_warnings(record=True) as w:
                with detect_anomaly():
                    out = MyFunc.apply(inp, inp, False)
                    out.backward()
            self.assertIn('MyFunc.apply', str(w[0].message))

    def test_nested_anomaly_detect_nan(self):
        size = 10

        class MyFunc(Function):
            @staticmethod
            def forward(ctx, inp1, fail_0th):
                ctx.fail_0th = fail_0th
                ctx.save_for_backward(inp1)
                return inp1.sum(0, keepdim=True)

            @staticmethod
            def backward(ctx, gO):
                inp, = ctx.saved_tensors
                fail_0th = ctx.fail_0th
                g = gO.clone().expand(size)
                gI = MyFunc2.apply(g * inp, g + inp, fail_0th)
                return gI, None

        class MyFunc2(Function):
            @staticmethod
            def forward(ctx, inp1, inp2, fail_0th):
                ctx.fail_0th = fail_0th
                return inp1 * 2.0 + inp2

            @staticmethod
            def backward(ctx, gO):
                fail_0th = ctx.fail_0th
                g1 = gO.clone()
                g2 = gO.clone()
                g1[0] = 0
                g2[0] = 0
                # generate a nan
                if fail_0th:
                    g1[0] /= 0
                else:
                    g2[0] /= 0
                return g1, g2, None

        inp = torch.rand(size, requires_grad=True)
        out = MyFunc.apply(inp, True)
        ginp, = torch.autograd.grad(out, (inp,), create_graph=True)
        gsum = ginp.sum()
        gsum.backward()  # should not fail

        inp = torch.rand(size, requires_grad=True)
        out = MyFunc.apply(inp, True)
        ginp, = torch.autograd.grad(out, (inp,), create_graph=True)
        gsum = ginp.sum()
        with warnings.catch_warnings(record=True) as w:
            with self.assertRaisesRegex(RuntimeError, "Function 'MyFunc2Backward' returned nan values in its 0th output."):
                with detect_anomaly():
                    gsum.backward()
        self.assertIn('No forward pass information', str(w[1].message))

        inp = torch.rand(size, requires_grad=True)
        with warnings.catch_warnings(record=True) as w:
            with self.assertRaisesRegex(RuntimeError, "Function 'MyFunc2Backward' returned nan values in its 1th output."):
                with detect_anomaly():
                    out = MyFunc.apply(inp, False)
                    ginp, = torch.autograd.grad(out, (inp,), create_graph=True)
                    gsum = ginp.sum()
                    gsum.backward()
        self.assertIn('MyFunc2.apply', str(w[1].message))
        self.assertIn('MyFunc.apply', str(w[2].message))

    def test_anomaly_grad_warnings(self):
        # PyTorch won't throw warnings if there is an error
        # but we'd want to at least see them in stderr

        class StdErrDiverter:
            def __enter__(self):
                self.stderr_orig = sys.stderr
                self.stderr_new = io.StringIO()
                sys.stderr = self.stderr_new
                return self

            def __exit__(self, *args):
                self.captured = self.stderr_new.getvalue()
                sys.stderr = self.stderr_orig


        # if the warnings don't throw, they will be handled as regular warnings
        with self.assertRaisesRegex(RuntimeError,
                                    "one of the variables needed for gradient computation has been "
                                    "modified by an inplace operation"):
            with warnings.catch_warnings(record=True) as w:
                with detect_anomaly():
                    a = torch.randn(5, requires_grad=True)
                    d1 = a + 1
                    d2 = d1 ** 2
                    d1 += 1
                    torch.autograd.grad(d2.sum(), a)

        self.assertEqual(len(w), 2)
        self.assertIn('Anomaly Detection has been enabled', str(w[0].message))
        self.assertIn('Error detected in PowBackward0', str(w[1].message))

        # if the warning throws, it will be printed to sys.stderr
        with self.assertRaisesRegex(RuntimeError,
                                    "one of the variables needed for gradient computation has been "
                                    "modified by an inplace operation"):
            with warnings.catch_warnings(record=True) as w:
                with detect_anomaly():
                    warnings.simplefilter("error")
                    with StdErrDiverter() as s:
                        a = torch.randn(5, requires_grad=True)
                        d1 = a + 1
                        d2 = d1 ** 2
                        d1 += 1
                        torch.autograd.grad(d2.sum(), a)

        self.assertEqual(len(w), 1)
        self.assertIn('Anomaly Detection has been enabled', str(w[0].message))
        self.assertIn('Error detected in PowBackward0', s.captured)

    @skipIfNoLapack
    def test_eig_no_eigenvectors(self):
        A = torch.tensor([[1., 2.], [2., 4.]], dtype=torch.float32, requires_grad=True)
        w, v = torch.eig(A, eigenvectors=False)
        with self.assertRaisesRegex(RuntimeError, 'cannot compute backward'):
            torch.autograd.backward([w, v], [torch.ones_like(w), torch.ones_like(v)])

    @skipIfNoLapack
    def test_eig_complex_eigenvalues(self):
        A = torch.tensor([[0., -1.], [1., 0.]], dtype=torch.float32, requires_grad=True)
        w, v = torch.eig(A, eigenvectors=True)
        with self.assertRaisesRegex(RuntimeError, 'does not support complex eigenvalues'):
            torch.autograd.backward([w, v], [torch.ones_like(w), torch.ones_like(v)])

    @skipIfNoLapack
    def test_symeig_no_eigenvectors(self):
        A = torch.tensor([[1., 2.], [2., 4.]], dtype=torch.float32, requires_grad=True)
        w, v = torch.symeig(A, eigenvectors=False)
        with self.assertRaisesRegex(RuntimeError, 'cannot compute backward'):
            torch.autograd.backward([w, v], [torch.ones_like(w), torch.ones_like(v)])

    @skipIfNoLapack
    def test_svd_no_singularvectors(self):
        A = torch.randn(2, 2, dtype=torch.float32, requires_grad=True)
        u, s, v = torch.svd(A, compute_uv=False)
        with self.assertRaisesRegex(RuntimeError, 'cannot compute backward'):
            torch.autograd.backward([u, s, v], [torch.ones_like(u), torch.ones_like(s), torch.ones_like(v)])

    def test_no_grad_copy(self):
        # create autograd function that saves grad pointer as class static
        class MyFunc(Function):
            static_grad_ptr = None

            @staticmethod
            def forward(ctx, inp1, inp2):
                return inp1 + inp2

            @staticmethod
            def backward(ctx, grad):
                MyFunc.static_grad_ptr = grad.data_ptr()
                return grad, grad

        class NonContGradFunc(Function):
            @staticmethod
            def forward(ctx, inp1):
                ctx.size = inp1.size()
                return torch.tensor([1.])

            @staticmethod
            def backward(ctx, grad):
                return torch.ones(1).expand(ctx.size)

        a = torch.randn(5, 6, requires_grad=True)
        b = torch.randn(5, 6, requires_grad=True)
        # non-contiguous grad should be copied
        NonContGradFunc.apply(MyFunc.apply(a, b)).backward()
        self.assertFalse(a.grad.data_ptr() == MyFunc.static_grad_ptr)
        self.assertFalse(b.grad.data_ptr() == MyFunc.static_grad_ptr)
        # test case that should trigger no copy for one of a,b
        a.grad = b.grad = None
        MyFunc.apply(a, b)[1][0].backward()
        p_g = MyFunc.static_grad_ptr
        p_a = a.grad.data_ptr()
        p_b = b.grad.data_ptr()
        # check a,b uses different grad buffer
        self.assertFalse(p_a == p_b)
        # check one of them is using the computed buffer
        self.assertTrue(p_a == p_g or p_b == p_g)

    def test_no_grad_copy_sparse(self):
        # create autograd function that saves grad pointer as class static
        class MyFunc(Function):
            static_grad_ptr = None

            @staticmethod
            def forward(ctx, inp1, inp2):
                return inp1 + inp2

            @staticmethod
            def backward(ctx, grad):
                MyFunc.static_grad_ptr = grad._values().data_ptr()
                return grad, grad

        class NonContGradFunc(Function):
            static_grad_ptr = None

            @staticmethod
            def forward(ctx, inp1, inp2):
                return inp1 + inp2

            @staticmethod
            def backward(ctx, grad):
                # Create a sparse tensor with non-contigous indices and values
                # and return as grad.
                v = torch.rand(1, 3)
                i = torch.ones(1, 1, dtype=torch.long)
                nv = v.expand(8, 3)
                ni = i.expand(1, 8)
                ngrad = torch.sparse.FloatTensor(ni, nv, torch.Size([10, 3]))
                NonContGradFunc.static_grad_ptr = ngrad._values().data_ptr()
                return ngrad, ngrad

        a = torch.randn(10, 3, requires_grad=True)
        b = torch.randn(10, 3, requires_grad=True)
        input = torch.tensor([1, 2, 4, 5, 4, 3, 2, 9])
        offsets = torch.tensor([0, 4])
        import torch.nn.functional as F

        # test case that should trigger no copy for one of a,b
        emb_matrix = MyFunc.apply(a, b)
        loss = F.embedding_bag(emb_matrix, input, offsets, sparse=True).sum()
        loss.backward(retain_graph=True)
        p_g = MyFunc.static_grad_ptr
        p_a = a.grad._values().data_ptr()
        p_b = b.grad._values().data_ptr()
        # check a,b uses different grad buffer
        self.assertFalse(p_a == p_b)
        # check one of them is using the computed buffer
        self.assertTrue(p_a == p_g or p_b == p_g)

        # Run backwards multiple times to ensure accumulation works.
        for i in range(10):
            loss.backward(retain_graph=True)

        # non-contiguous indices and value, we should trigger a copy.
        a.grad = b.grad = None
        emb_matrix = NonContGradFunc.apply(a, b)
        loss = F.embedding_bag(emb_matrix, input, offsets, sparse=True).sum()
        loss.backward(retain_graph=True)
        p_g = NonContGradFunc.static_grad_ptr
        p_a = a.grad._values().data_ptr()
        p_b = b.grad._values().data_ptr()
        # check a,b uses different grad buffer
        self.assertFalse(p_a == p_b)
        # Verify we cloned both grads.
        self.assertFalse(p_a == p_g)
        self.assertFalse(p_b == p_g)

        # Run backwards multiple times to ensure accumulation works.
        for i in range(10):
            loss.backward(retain_graph=True)

    def test_gradcheck_single_input(self):
        def f(inp):
            return inp.mul(5)

        gradcheck(f, torch.rand(10, dtype=torch.float64, requires_grad=True))
        gradgradcheck(f, torch.rand(10, dtype=torch.float64, requires_grad=True))

    def test_gradcheck_sparse_input(self):
        def fn(sparse):
            return torch.sparse.sum(sparse)

        gradcheck(fn, torch.rand(10).to_sparse().requires_grad_(True), check_sparse_nnz=True)
        with self.assertRaisesRegex(RuntimeError, 'gradcheck expects all tensor inputs are dense'):
            gradcheck(fn, torch.rand(10).to_sparse().requires_grad_(True), check_sparse_nnz=False)

    def test_gradcheck_nondeterministic(self):
        class NonDetFunc(Function):
            @staticmethod
            def forward(ctx, x, jitter=0.0):
                ctx._jitter = jitter
                return x

            @staticmethod
            def backward(ctx, grad_out):
                return NonDetFunc.apply(grad_out, ctx._jitter) * (1 + torch.rand_like(grad_out) * ctx._jitter), None

        inp = torch.randn(5, 5, requires_grad=True)
        gradcheck(lambda x: NonDetFunc.apply(x, 0.0), inp)
        with self.assertRaisesRegex(RuntimeError, 'Backward is not reentrant'):
            gradcheck(lambda x: NonDetFunc.apply(x, 1e-6), inp)
        with self.assertRaisesRegex(RuntimeError, 'Backward is not reentrant'):
            gradgradcheck(lambda x: NonDetFunc.apply(x, 1e-12), inp)
        gradcheck(lambda x: NonDetFunc.apply(x, 0.0), inp, nondet_tol=1e-5)
        gradcheck(lambda x: NonDetFunc.apply(x, 1e-6), inp, nondet_tol=1e-5)
        gradgradcheck(lambda x: NonDetFunc.apply(x, 1e-12), inp, nondet_tol=1e-5)

    def test_version_counter(self):
        x = torch.randn(1, 2)

        # In-place op bumps version
        x_saved_version = x._version
        x.add_(1).add_(1)
        self.assertTrue(x._version > x_saved_version)

        # Differentiable view shares version counter
        xz = x[:]
        self.assertTrue(x._version == xz._version)
        xz.add_(1)
        self.assertTrue(x._version == xz._version)

        # `x.data = y` preserves version counter of `x`
        x_saved_version = x._version
        x.data = torch.randn(2, 3)
        self.assertTrue(x._version == x_saved_version)
        x.add_(1)
        self.assertTrue(x._version > x_saved_version)
        # Make sure `x` is still using the same version counter it shares with `xz`
        self.assertTrue(x._version == xz._version)

        # In-place op on `xz` also updates version of `x`,
        # because they share the version counter
        xz.add_(1)
        self.assertTrue(x._version == xz._version)

    def test_set_data_tensorimpl_type(self):
        # Dense tensor has impl of type `TensorImpl`, while sparse tensor has impl
        # of type `SparseTensorImpl`.
        x = torch.randn(1, 2)
        x_s = torch.sparse_coo_tensor(torch.zeros([1, 1]), torch.ones([1]))
        with self.assertRaisesRegex(RuntimeError, 'incompatible tensor type'):
            x.data = x_s

    def test_set_data_preserve_pyobj(self):
        a = torch.randn(1, 2)
        b = torch.randn(1, 2)
        b_id_saved = id(b)
        b.data = a
        self.assertTrue(b_id_saved == id(b))

    @unittest.skipIf(IS_WINDOWS, "Skipping because doesn't work for windows")
    def test_thread_shutdown(self):
        code = """import torch
from torch.autograd import Function
class MyFunction(Function):
    @staticmethod
    def forward(ctx, x):
        return x

    @staticmethod
    def backward(ctx, grad):
        return grad

for shape in [(1,), ()]:
    v = torch.ones(shape, requires_grad=True)
    MyFunction.apply(v).backward()
"""
        s = TestCase.runWithPytorchAPIUsageStderr(code)
        self.assertRegex(s, "PYTORCH_API_USAGE torch.autograd.thread_shutdown")

    @unittest.skipIf(IS_MACOS, "Fails with SIGBUS on macOS; https://github.com/pytorch/pytorch/issues/25941")
    def test_deep_reentrant(self):

        class DeepReentrant(Function):
            @staticmethod
            def forward(ctx, x):
                with torch.enable_grad():
                    ctx.x = Variable(x.detach(), requires_grad=True)
                    ctx.x = ctx.x - 1
                return ctx.x.detach()

            @staticmethod
            def backward(ctx, x):
                if ctx.x < 0:
                    return x
                with torch.enable_grad():
                    DeepReentrant.apply(ctx.x).sum().backward()
                return x

        # Test stack overflow escape mechanism
        v = torch.tensor(2000.0, requires_grad=True)
        # This will cause stack overflow if reentrant calls are handled
        # in the same thread recursively
        DeepReentrant.apply(v).sum().backward()

        # Test stack overflow escape mechanism multiple times
        # to ensure reusing workers in the pool works fine
        v2 = torch.tensor(200.0, requires_grad=True)
        DeepReentrant.apply(v2).sum().backward()

    def test_reentrant_priority(self):
        order = []

        class MyFunction(Function):
            @staticmethod
            def forward(ctx, x):
                return x

            @staticmethod
            def backward(ctx, x):
                order.append("MyFunction")
                return x

        class Reentrant(Function):
            @staticmethod
            def forward(ctx, x):
                with torch.enable_grad():
                    ctx.x = Variable(x.detach(), requires_grad=True)
                    ctx.x = ctx.x - 1
                return ctx.x.detach()

            @staticmethod
            def backward(ctx, x):
                order.append("Reentrant")
                if ctx.x < 0:
                    return x
                with torch.enable_grad():
                    Reentrant.apply(ctx.x).backward()
                return x

        a = MyFunction.apply(torch.tensor(6.0, requires_grad=True))
        b = Reentrant.apply(torch.tensor(9.0, requires_grad=True))
        v = a * b
        v.backward()
        # The tasks for the Reentrant and MyFunction backward() will be added
        # to the queue in the autograd engine at the same time. The backward
        # for Reentrant will be executed first, which will then add other
        # backward tasks to the queue. We want to ensure all the reentrant tasks
        # are prioritized over the MyFunction backward task regardless of their
        # sequence numbers
        self.assertEqual(len(order), 11)
        self.assertEqual(order.count("Reentrant"), 10)
        self.assertEqual(order[-1], "MyFunction")


    @slowTest
    def test_checkpointing(self):
        num_inp = 2000
        nz_inp = 10
        nz_out = 10
        nz_bottleneck = 1000

        # small proxy network for some complex reasoning we want to do per input
        module = nn.Sequential(
            nn.Linear(nz_inp, nz_bottleneck),
            nn.ReLU(),
            nn.Linear(nz_bottleneck, nz_inp)
        )

        feat_combined = []
        for r in range(num_inp):
            data_r = torch.Tensor(1, nz_inp)
            data_r.uniform_()
            data_r.requires_grad = True
            feat_r = checkpoint(module, data_r)
            feat_combined.append(feat_r)

        # compute mean as a proxy for some joint reasoning
        mean_combined = torch.stack(feat_combined).mean()
        mean_combined.backward()

    def _test_reentrant_with_callbacks(self, install_callbacks_in_depths):
        counter = {}
        counter["inner"] = 0
        counter["outer"] = 0

        def inc_inner_counter():
            counter["inner"] += 1

        def inc_outer_counter():
            counter["outer"] += 1

        class MyFunc(Function):
            @staticmethod
            def forward(ctx, input):
                return input

            @staticmethod
            @once_differentiable
            def backward(ctx, input):
                if 1 in install_callbacks_in_depths:
                    # Add a callback to execute.
                    Variable._execution_engine.queue_callback(inc_inner_counter)

                return input

        class MyReentrantFunc(Function):
            @staticmethod
            def forward(ctx, input):
                return input

            @staticmethod
            @once_differentiable
            def backward(ctx, input):
                if 0 in install_callbacks_in_depths:
                    # Add a callback to execute.
                    Variable._execution_engine.queue_callback(inc_outer_counter)
                # Reentrant backward call.
                tmp_inp = input.detach().requires_grad_()
                with torch.enable_grad():
                    tmp_out = (MyFunc.apply(tmp_inp)).sum()
                tmp_out.backward()
                return input

        t1 = torch.rand((3, 3), requires_grad=True)
        t2 = MyReentrantFunc.apply(t1)
        t3 = t2.sum()
        torch.autograd.backward([t3])

        return counter

    def test_reentrant_with_callbacks_depth_0(self):
        # Verify callback is called only once.
        ret = self._test_reentrant_with_callbacks([0])
        self.assertEqual(1, ret["outer"])
        self.assertEqual(0, ret["inner"])

    def test_reentrant_with_callbacks_depth_1(self):
        # Verify callback is called only once.
        ret = self._test_reentrant_with_callbacks([1])
        self.assertEqual(0, ret["outer"])
        self.assertEqual(1, ret["inner"])

    def test_reentrant_with_callbacks_both_depths(self):
        # Verify callback is called twice.
        ret = self._test_reentrant_with_callbacks([0, 1])
        self.assertEqual(1, ret["outer"])
        self.assertEqual(1, ret["inner"])

    def test_reentrant_with_leaf_variable_hook(self):
        handle = None
        param = torch.rand(10, requires_grad=True)

        def add_gradient_penalty_to_grad(grad):
            handle.remove()
            old_param_grad = grad
            param.grad = None
            # Add some sort of gradient penalty by directly updating the gradients
            with torch.enable_grad():
                g = grad.detach().requires_grad_()
                new_param = param.detach().requires_grad_()
                out = ((g * 2) + new_param).sum()
                out.backward()
            res = g.grad + grad
            param.grad = old_param_grad
            return res

        handle = param.register_hook(add_gradient_penalty_to_grad)
        # Forward pass
        tmp = (param * param)
        loss = tmp.sum()
        # Compute the gradients
        loss.backward()

    def test_reentrant_with_non_leaf_variable_hook(self):
        handle = None
        param = torch.rand(10, requires_grad=True)

        def manual_increase_gradient(grad):
            handle.remove()
            # Add some sort of gradient penalty by directly updating the gradients
            with torch.enable_grad():
                g = grad.detach().requires_grad_()
                out = ((g * 2) + 5).sum()
                out.backward()
            res = g.grad + grad
            return res

        # Forward pass
        tmp = (param * param)
        handle = tmp.register_hook(manual_increase_gradient)
        loss = tmp.sum()
        # Compute the gradients
        loss.backward()
        self.assertEqual(param.grad, 6 * param)

    def test_autograd_views_codegen(self):
        # This is not necessarily the absolute correct behavior, but this is the current
        # one. This test is here to make sure that any change to this behavior is detected
        # and not silent. The TODOs below mark the places with unexpected behavior.
        # Note that any change in these test will be BC-breaking and should be done carefully.

        # This test checks the behavior of two codegen functions (view_as and unbind)
        # with respect to view tracking and inplace operation on the output.

        def run_test(grad_mode, requires_grad, is_view, should_raise_tuple):
            def maybe_check_raise(fn, should_raise):
                self.assertTrue(should_raise is None or isinstance(should_raise, str))
                if should_raise is not None:
                    with self.assertRaisesRegex(RuntimeError, should_raise):
                        fn()
                else:
                    fn()

            inp = torch.rand(2, requires_grad=requires_grad).clone()
            with torch.set_grad_enabled(grad_mode):
                out = inp.view_as(inp)
            # Are they differentiable views?
            self.assertTrue(out._is_view() == is_view)
            # Are inplace allowed?
            maybe_check_raise(lambda: out.add_(1), should_raise_tuple[0])

            inp = torch.rand(2, requires_grad=requires_grad).clone()
            with torch.set_grad_enabled(grad_mode):
                out = inp.unbind()
            # Are they differentiable views?
            self.assertTrue(out[0]._is_view() == is_view)
            self.assertTrue(out[1]._is_view() == is_view)
            # Are inplace allowed?
            maybe_check_raise(lambda: out[0].add_(1), should_raise_tuple[1])
            maybe_check_raise(lambda: out[1].add_(1), should_raise_tuple[2])

        # should_raise contains None if it should not raise
        # should_raise contains a string of the error if it should raise
        # The 3 elements are for view_as, first output of unbind and second output of unbind
        run_test(grad_mode=True, requires_grad=False, is_view=True,
                 should_raise_tuple=(None, None, None))
        inp_change_err = "Output {} of UnbindBackward is a view and is being modified inplace."
        run_test(grad_mode=True, requires_grad=True, is_view=True,
                 should_raise_tuple=(None, inp_change_err.format("0"), inp_change_err.format("1")))
        leaf_grad_err = "A view was created in no_grad mode and is being modified inplace"
        run_test(grad_mode=False, requires_grad=True, is_view=True,
                 should_raise_tuple=(leaf_grad_err, leaf_grad_err, leaf_grad_err))
        run_test(grad_mode=False, requires_grad=False, is_view=True,
                 should_raise_tuple=(None, None, None))

    def test_inplace_not_requires_grad(self):
        class MyFn(torch.autograd.Function):
            @staticmethod
            def forward(ctx, inp):
                return inp.view_as(inp)

            @staticmethod
            def backward(ctx, grad):
                return grad

        # Original Tensor does not require grad
        a = torch.rand(1, 2)

        # Tensor being written does require grad
        b = torch.rand(1, requires_grad=True)

        # Take an invalid view on a that should raise an error
        view_a = MyFn.apply(a)

        with self.assertRaisesRegex(RuntimeError, 'Output 0 of a function created in no_grad mode is a view'):
            view_a += b

        # Extra test for copy_ that is a manual implementation and could be easily
        # forgotten when the codegen is updated
        a = torch.rand(1, 2)
        b = torch.rand(1, requires_grad=True)
        view_a = MyFn.apply(a)

        with self.assertRaisesRegex(RuntimeError, 'Output 0 of a function created in no_grad mode is a view'):
            view_a.copy_(b)

        # Functions that should throw must properly throw
        a = torch.rand(1, 2)
        b = torch.rand(1, requires_grad=True)
        view_a = a.unbind()[0]
        with self.assertRaisesRegex(RuntimeError, 'Output 0 of a function created in no_grad mode is a view'):
            view_a.copy_(b)

        # Sanity check that views that should work still work
        a = torch.rand(1, 2)
        b = torch.rand(1, requires_grad=True)
        a.select(1, 0).copy_(b)

    def _do_test_autograd_simple_views_python(self, dtype):
        # This is not necessarily the absolute correct behavior, but this is the current
        # one. This test is here to make sure that any change to this behavior is detected
        # and not silent. The TODOs below mark the places with unexpected behavior.
        # Note that any change in these test will be BC-breaking and should be done carefully.

        # This checks the autograd.Function behavior when we return one or multiple outputs
        # while one of these is an input, a view of an input or of a temporary tensor.

        # This indicator is used to track how many times the backward function was called
        bw_called = [0]
        # This indicator is used to check if the argument `ga` contains non-zero values
        ga_nz = [False]

        class IdOneOutput(Function):
            @staticmethod
            def forward(ctx, a, b, make_view):
                if make_view:
                    a = a.narrow(0, 0, 2)
                else:
                    a = a.clone()
                return a

            @staticmethod
            def backward(ctx, ga):
                bw_called[0] += 1
                return ga, None, None

        class IdTwoOutput(Function):
            @staticmethod
            def forward(ctx, a, b, make_view):
                if make_view:
                    a = a.narrow(0, 0, 2)
                else:
                    a = a.clone()
                return a, a + b

            @staticmethod
            def backward(ctx, ga, gab):
                bw_called[0] += 1
                if ga.eq(0).all():
                    ga_nz[0] = False
                else:
                    ga_nz[0] = True
                return ga + gab, gab, None

        err_msg_two_outputs = "Output 0 of IdTwoOutputBackward is a view and is being modified inplace."
        err_msg_two_outputs += " This view is the output of a function that returns multiple views."

        class ViewOfTemp(Function):
            @staticmethod
            def forward(ctx, a, make_view):
                ctx.save_for_backward(a)
                if make_view:
                    a = a.narrow(0, 0, 2)
                else:
                    a = a.clone()
                b = a.clone()
                return b.select(0, 0)

            @staticmethod
            def backward(ctx, grad):
                bw_called[0] += 1
                a, = ctx.saved_tensors
                res = torch.zeros_like(a)
                res.select(0, 0).copy_(grad)
                return res, None

        for fn_id in ["one_output", "two_output", "view_of_temp"]:
            for inplace in [True, False]:
                for make_view in [True, False]:
                    # Used for special casing the tests below
                    output_is_a_view = (make_view or fn_id == "view_of_temp")

                    def fn(a, b):
                        # never modify a, b inplace for gracheck
                        a = a.clone()
                        b = b.clone()
                        if fn_id == "two_output":
                            tmp1, tmp2 = IdTwoOutput.apply(a, b, make_view)
                            if inplace:
                                tmp1 += 3
                                tmp2 += 3
                            else:
                                tmp1 = tmp1 + 3
                                tmp2 = tmp2 + 3
                            tmp = tmp1 * tmp2
                        else:
                            if fn_id == "one_output":
                                tmp = IdOneOutput.apply(a, b, make_view)
                            else:
                                tmp = ViewOfTemp.apply(a + b, make_view)
                            if inplace:
                                tmp += 3
                            else:
                                tmp = tmp + 3

                        return tmp.sum()

                    a = torch.ones(2, dtype=dtype, requires_grad=True)
                    b = torch.ones(2, dtype=dtype, requires_grad=True)


                    if fn_id == "two_output" and inplace and output_is_a_view:
                        with self.assertRaisesRegex(RuntimeError, err_msg_two_outputs):
                            fn(a, b)
                    else:
                        # Are the computed gradients correct ?
                        if inplace and output_is_a_view:
                            with warnings.catch_warnings(record=True) as w:
                                if fn_id == "view_of_temp":
                                    # This will be fixed after the deprecation cycle and the warning becomes
                                    # an error.
                                    with self.assertRaisesRegex(RuntimeError,
                                                                "a view of a leaf Variable that requires grad "
                                                                "is being used in an in-place operation."):
                                        gradcheck(fn, (a, b))
                                else:
                                    # This works but the custom backward is not called (or called with partial)
                                    # gradients as tested below
                                    gradcheck(fn, (a, b))
                            self.assertTrue(len(w) > 0)
                        else:
                            gradcheck(fn, (a, b))

                        # Was the custom backward called properly
                        bw_called[0] = 0
                        ga_nz[0] = True  # For the case where the backward is called
                        with warnings.catch_warnings(record=True) as w:
                            if inplace and output_is_a_view and fn_id != "one_output":
                                with self.assertRaisesRegex(RuntimeError,
                                                            "a view of a leaf Variable that requires grad "
                                                            "is being used in an in-place operation."):
                                    fn(a, b).backward()
                            else:
                                fn(a, b).backward()

                        expected_called = 1
                        expected_ga_nz = True
                        expected_warning = False

                        if output_is_a_view and inplace:
                            expected_called = 0
                            expected_warning = True

                        self.assertTrue(bw_called[0] == expected_called)
                        self.assertTrue(ga_nz[0] == expected_ga_nz)
                        self.assertTrue((len(w) == 1) == expected_warning)

    def test_autograd_simple_views_python(self):
        self._do_test_autograd_simple_views_python(torch.double)
        self._do_test_autograd_simple_views_python(torch.cdouble)

    def test_autograd_complex_views_python(self):
        # This is not necessarily the absolute correct behavior, but this is the current
        # one. This test is here to make sure that any change to this behavior is detected
        # and not silent. The TODOs below mark the places with unexpected behavior.
        # Note that any change in these test will be BC-breaking and should be done carefully.

        # This checks that multiples views in the forward are properly traced and how they
        # behave with respect to inplace operations.

        # This indicator is used to track how many times the backward function was called
        bw_called = [0]

        class ComplexView(Function):
            @staticmethod
            def forward(ctx, a, idx):
                res = a.narrow(0, idx, 1)
                res = a.select(0, idx)
                ctx.save_for_backward(a)
                ctx.idx = idx
                return res

            @staticmethod
            def backward(ctx, grad):
                bw_called[0] += 1
                a, = ctx.saved_tensors
                res = torch.zeros_like(a)
                res.select(0, ctx.idx).copy_(grad)
                return res, None

        a = torch.ones(2, requires_grad=True)
        idx = 1

        bw_called[0] = 0
        out = ComplexView.apply(a.clone(), idx)
        out.sum().backward()
        self.assertTrue(bw_called[0] == 1)

        out = ComplexView.apply(a.clone(), idx)
        with warnings.catch_warnings(record=True) as w:
            out += 1
        self.assertEqual(len(w), 1)

    def test_autograd_inplace_views_python(self):
        # This is not necessarily the absolute correct behavior, but this is the current
        # one. This test is here to make sure that any change to this behavior is detected
        # and not silent. The TODOs below mark the places with unexpected behavior.
        # Note that any change in these test will be BC-breaking and should be done carefully.

        # This test checks custom autograd.Function that perform inplace operations

        bw_called = [0]

        # I) Single output
        class MyAdder(Function):
            @staticmethod
            def forward(ctx, a, b):
                a.add_(b)
                ctx.mark_dirty(a)
                return a

            @staticmethod
            def backward(ctx, grad):
                bw_called[0] += 1
                return grad, grad


        a = torch.ones(2, requires_grad=True)
        b = torch.ones(2, requires_grad=True)

        # No extra inplace
        c = MyAdder.apply(a.clone(), b)
        c.sum().backward()
        self.assertTrue(bw_called[0] == 1)

        # With extra inplace on the output
        bw_called[0] = 0
        c = MyAdder.apply(a.clone(), b)
        c += 2
        c.sum().backward()
        self.assertTrue(bw_called[0] == 1)

        # The input is a view
        bw_called[0] = 0
        c = MyAdder.apply(a.clone().view_as(a), b)
        c.sum().backward()
        self.assertTrue(bw_called[0] == 1)

        # Should not give non-inputs to mark_dirty
        class MyAdderBad(Function):
            @staticmethod
            def forward(ctx, a, b):
                c = 3 * a
                c.add_(b)
                ctx.mark_dirty(c)
                return c

            @staticmethod
            def backward(ctx, grad):
                bw_called[0] += 1
                grad = 3 * grad
                return grad, grad

        a = torch.ones(2, requires_grad=True)
        b = torch.ones(2, requires_grad=True)

        with warnings.catch_warnings(record=True) as w:
            MyAdderBad.apply(a.clone(), b)
        self.assertEqual(len(w), 1)

        # II) Multiple outputs
        class MyBadAdder(Function):
            @staticmethod
            def forward(ctx, a, b):
                a.add_(b)
                ctx.mark_dirty(a)
                return a, a + b

            @staticmethod
            def backward(ctx, ga, gab):
                bw_called[0] += 1
                return ga + gab, ga + gab

        # No extra inplace
        bw_called[0] = 0
        c, d = MyBadAdder.apply(a.clone(), b)
        (c * d).sum().backward()
        self.assertTrue(bw_called[0] == 1)

        # With extra inplace on the output
        bw_called[0] = 0
        c, d = MyBadAdder.apply(a.clone(), b)
        c += 2
        (c * d).sum().backward()
        self.assertTrue(bw_called[0] == 1)

        # The input is a view
        inplace_on_view_err = "your Function modifies inplace an input that is a view of another Tensor"
        with self.assertRaisesRegex(RuntimeError, inplace_on_view_err):
            c, d = MyBadAdder.apply(a.clone().view_as(a), b)

        # III) Inplace + other op
        class MyOutPlaceAdder(Function):
            @staticmethod
            def forward(ctx, a, b):
                a.add_(b)
                ctx.mark_dirty(a)
                return a.clone(), a + b

            @staticmethod
            def backward(ctx, ga, gab):
                bw_called[0] += 1
                return ga + gab, ga + 2 * gab

        # We don't reuse the input
        def fn(a, b):
            orig_a = a.clone().view_as(a)
            c, d = MyOutPlaceAdder.apply(orig_a, b)
            return (c * d).sum()

        bad_mark_dirty_err = "Some elements marked as dirty during the forward method were not returned as output."
        with self.assertRaisesRegex(RuntimeError, bad_mark_dirty_err):
            fn(a, b)

    def test_named_tensor_for_complex_views(self):
        names = ["batch", "height", "width", "complex"]
        z = torch.ones((5, 12, 14, 2), requires_grad=True)
        z_named = z.refine_names(*names)
        z_complex = torch.view_as_complex(z_named.rename(None)).refine_names(*names[:-1])
        z_complex.sum().backward()
        self.assertEqual(z.grad, torch.view_as_real(torch.ones_like(z_complex).rename(None)))

    def test_custom_function_return_view_in_nograd(self):
        class Alias(Function):
            @staticmethod
            def forward(ctx, x):
                return x[:]

            @staticmethod
            def backward(ctx, gx):
                return gx

        inp = torch.rand(2, requires_grad=True)

        with torch.no_grad():
            output = Alias.apply(inp)

        with torch.no_grad():
            expected_output = inp[:]

        # Calling the custom function should operate as if we called an equivalent op
        self.assertEqual(output.requires_grad, expected_output.requires_grad)

        # Check that in-place modification on view throws
        leaf_grad_err = "A view was created in no_grad mode and is being modified inplace"
        with self.assertRaisesRegex(RuntimeError, leaf_grad_err):
            output.zero_()

    def test_grad_mode_restored_reentrant(self):
        class MyFunction(Function):
            @staticmethod
            def forward(ctx, inp):
                return inp.clone()

            @staticmethod
            def backward(ctx, go):
                original = torch._C.is_grad_enabled()
                with torch.enable_grad():
                    self.assertTrue(torch._C.is_grad_enabled())
                    foo = torch.rand(go.size(), requires_grad=True)
                    grad, = torch.autograd.grad(
                        foo ** 3, foo, grad_outputs=go
                    )
                    self.assertTrue(torch._C.is_grad_enabled())
                self.assertTrue(torch._C.is_grad_enabled() == original)
                return grad

        inp = torch.rand(3, requires_grad=True)

        # Case where original==False
        MyFunction.apply(inp).sum().backward()
        # Case where original==True
        MyFunction.apply(inp).sum().backward(create_graph=True)

    def test_power_function(self):
        a = torch.tensor([0., 0., 0.])
        b = torch.tensor([-1., 0., 1.], requires_grad=True)
        c = torch.sum(a**b)
        c.backward()
        self.assertEqual(b.grad, torch.tensor([-inf, 0., 0.]))

        s = 0
        b = torch.tensor([-1., 0., 1.], requires_grad=True)
        c = torch.sum(s**b)
        c.backward()
        self.assertEqual(b.grad, torch.tensor([-inf, 0., 0.]))

    def test_nansum_with_nans(self):
        a = torch.randn(2, 2, 2, 2)
        with torch.no_grad():
            a[a < 0.2] = float('nan')
        a.requires_grad = True

        # No args
        gradcheck(lambda x: x.nansum(), a)
        gradgradcheck(lambda x: x.nansum(), a)

        # Single dim
        gradcheck(lambda x: x.nansum((0)), a)
        gradgradcheck(lambda x: x.nansum((0)), a)

        # Multi dim
        gradcheck(lambda x: x.nansum((0, 2)), a)
        gradgradcheck(lambda x: x.nansum((0, 2)), a)

        gradcheck(lambda x: x.nansum((0, -1)), a)
        gradgradcheck(lambda x: x.nansum((0, -1)), a)

        # With keep-dim
        gradcheck(lambda x: x.nansum((0, -1), True), a)
        gradgradcheck(lambda x: x.nansum((0, -1), True), a)

    def test_nansum_dtype(self):
        inp = torch.randn(2, 2, 2, 2)
        with torch.no_grad():
            inp[inp < 0.2] = float('nan')

        def test(inp, inp_dtype, out_dtype):
            with torch.no_grad():
                a = inp.to(inp_dtype)
            a.requires_grad = True
            b = torch.sum(a, dtype=out_dtype)
            b.backward()
            self.assertEqual(a.dtype, a.grad.dtype)

        test(inp, torch.float, torch.double)
        test(inp, torch.double, torch.float)

    def test_nan_to_num(self):
        a = torch.randn(3, 3, 3, 3)
        with torch.no_grad():
            a[torch.rand_like(a) < 0.2] = float('nan')
            a[torch.rand_like(a) < 0.2] = float('inf')
            a[torch.rand_like(a) < 0.2] = -float('inf')

        a.requires_grad = True

        gradcheck(lambda x: x.nan_to_num(), a)
        gradgradcheck(lambda x: x.nan_to_num(), a)

        gradcheck(lambda x: x.nan_to_num(nan=1.2), a)
        gradgradcheck(lambda x: x.nan_to_num(nan=1.2), a)

        gradcheck(lambda x: x.nan_to_num(nan=1.2, posinf=2.0), a)
        gradgradcheck(lambda x: x.nan_to_num(nan=1.2, posinf=2.0), a)

        gradcheck(lambda x: x.nan_to_num(nan=1.2, posinf=2.0, neginf=-2.0), a)
        gradgradcheck(lambda x: x.nan_to_num(nan=1.2, posinf=2.0, neginf=-2.0), a)

        gradcheck(lambda x: x.nan_to_num(posinf=2.0, neginf=-2.0), a)
        gradgradcheck(lambda x: x.nan_to_num(posinf=2.0, neginf=-2.0), a)

        gradcheck(lambda x: x.nan_to_num(neginf=-2.0), a)
        gradgradcheck(lambda x: x.nan_to_num(neginf=-2.0), a)

    def test_custom_function_error(self):
        class BadFw(Function):
            @staticmethod
            def backward(ctx, foo):
                return foo

        class BadBw(Function):
            @staticmethod
            def forward(ctx, foo):
                return foo.clone()

        inp = torch.rand(1, requires_grad=True)
        with self.assertRaisesRegex(NotImplementedError, "must implement the forward"):
            BadFw.apply(inp)

        with self.assertRaisesRegex(RuntimeError, "must implement the backward"):
            BadBw.apply(inp).sum().backward()

    def test_custom_function_local_inplace(self):
        class MyFn(torch.autograd.Function):
            @staticmethod
            def forward(ctx, inp, inplace):
                view = inp.clone()[:3]
                if inplace:
                    view += 2
                return view

            @staticmethod
            def backward(ctx, grad):
                return grad, None

        base = torch.rand(10, requires_grad=True)

        foo = MyFn.apply(base, False)
        self.assertEqual(foo.grad_fn.__class__.__name__, "MyFnBackward")

        foo = MyFn.apply(base, True)
        self.assertEqual(foo.grad_fn.__class__.__name__, "MyFnBackward")

    def test_integer_outputs(self):
        inp = torch.rand(4, requires_grad=True)

        out = inp.argmax()
        self.assertFalse(out.dtype.is_floating_point)
        self.assertFalse(out.requires_grad)

        out = inp.argmin()
        self.assertFalse(out.dtype.is_floating_point)
        self.assertFalse(out.requires_grad)

        out = inp.argsort()
        self.assertFalse(out.dtype.is_floating_point)
        self.assertFalse(out.requires_grad)

        val = torch.rand((), requires_grad=True)

        out = torch.searchsorted(inp, val)
        self.assertFalse(out.dtype.is_floating_point)
        self.assertFalse(out.requires_grad)

        bins = torch.linspace(0, 1.0, requires_grad=True)
        vals = torch.rand(5, 5, requires_grad=True)
        out = torch.bucketize(vals, bins)
        self.assertFalse(out.dtype.is_floating_point)
        self.assertFalse(out.requires_grad)

def index_variable(shape, max_indices):
    if not isinstance(shape, tuple):
        shape = (shape,)
    index = torch.rand(*shape).mul_(max_indices).floor_().long()
    return index


def index_perm_variable(shape, max_indices):
    if not isinstance(shape, tuple):
        shape = (shape,)

    index = torch.randperm(max_indices).narrow(0, 0, reduce(mul, shape)).view(shape)
    return index


def gather_variable(shape, index_dim, max_indices, duplicate=False):
    assert len(shape) == 2
    assert index_dim < 2
    batch_dim = 1 - index_dim
    index = torch.LongTensor(*shape)
    for i in range(shape[index_dim]):
        index.select(index_dim, i).copy_(
            torch.randperm(max_indices)[:shape[batch_dim]])
    if duplicate:
        index.select(batch_dim, 0).copy_(index.select(batch_dim, 1))
    return index


def bernoulli_scalar():
    return torch.tensor(0, dtype=torch.uint8).bernoulli_()


def gradgradcheck_method_precision_override(test_name):
    # these are just empirical observations, we should improve
    gradgradcheck_precision_override = {
        'test_norm': {'atol': 2e-2, 'rtol': 1e-2},
        'test_norm_1_5': {'atol': 1.5e-2, 'rtol': 1e-2},
        'test_norm_3': {'atol': 5e-2, 'rtol': 1e-2},
        'test_dist': {'atol': 5e-2, 'rtol': 1e-2},
        'test_dist_4': {'atol': 8e-2, 'rtol': 1e-2},
    }
    non_broadcasted_test_name = test_name.split("_broadcast")[0]
    override = gradgradcheck_precision_override.get(non_broadcasted_test_name)
    if override:
        if 'broadcast_lhs' in test_name or 'broadcast_rhs' in test_name:
            # errors accumulated across 1 dimension
            override = {'atol': override['atol'] * S, 'rtol': override['atol'] * S}
        elif 'broadcast_all' in test_name:
            # errors accumulated across multiple dimensions
            override = {'atol': override['atol'] * S * S, 'rtol': override['atol'] * S * S}
    return override

def run_grad_and_gradgrad_checks(test_case, name, test_name, apply_method, output_variable,
                                 input_variables, run_gradgradcheck=True):
    test_case.assertTrue(gradcheck(apply_method, input_variables, eps=1e-6, atol=PRECISION))
    if name in EXCLUDE_GRADGRADCHECK or test_name in EXCLUDE_GRADGRADCHECK_BY_TEST_NAME:
        return
    gradgradcheck_precision_override = gradgradcheck_method_precision_override(test_name)
    if gradgradcheck_precision_override is not None:
        atol = gradgradcheck_precision_override['atol']
        rtol = gradgradcheck_precision_override['rtol']
        test_case.assertTrue(gradgradcheck(apply_method, input_variables, None, atol=atol, rtol=rtol,
                                           gen_non_contig_grad_outputs=True))
    else:
        test_case.assertTrue(gradgradcheck(apply_method, input_variables, gen_non_contig_grad_outputs=True))


def run_functional_checks(test_case, test_name, name, apply_fn, run_grad_checks,
                          f_args_variable, f_args_tensor):
    output_variable = apply_fn(*f_args_variable)

    if run_grad_checks:
        run_grad_and_gradgrad_checks(test_case, name, test_name, apply_fn,
                                     output_variable, f_args_variable)

    self_variable = f_args_variable[0]
    if isinstance(output_variable, torch.Tensor) and output_variable.requires_grad and self_variable is not None:
        output_variable.backward(randn_like(output_variable))
        test_case.assertEqualTypeString(self_variable, self_variable.grad)
        test_case.assertEqual(self_variable.size(), self_variable.grad.size())

# this list corresponds to ops which have separate tests defined for complex dtypes in
# common_methods_invocations.py
# test for these ops with 'complex' in variant should only run for complex and
# the tests for these ops which do not have 'complex' in variant should not run for complex
# and only run for floating point

separate_complex_tests = ['view_as_real', 'real', 'imag', 'asin', 'acos', 'div', 'log',
                          'log10', 'log1p', 'log2', 'pow', 'tan', 'reciprocal', 'rsqrt',
                          '__rdiv__', 'add', 'sub']

# NOTE: Some non-holomorphic are separately tested in TestAutogradComplex until gradcheck works properly
# for non-holomorphic functions

# allow list for complex
complex_list = ['t', 'view', 'reshape', 'reshape_as', 'view_as', 'roll', 'clone',
                'repeat', 'expand', 'flip', 'fliplr', 'flipud', 'rot90', 'transpose',
                'permute', 'squeeze', 'unsqueeze', 'resize', 'resize_as', 'tril', 'triu',
                'chunk', 'split', 'split_with_sizes', 'repeat', 'expand', 'zero_',
                'eq_', 'ne_', 'add', '__radd__', 'sum', 'conj', 'sin', 'cos', 'mul', 'sinh',
                'cosh', '__rmul__', 'sgn', 'abs', 'dot', 'vdot', 'tensor_split', 'matmul',
                'bmm', 'mv', 'ger', 'diagonal', 'atan', 'angle', 'tanh', 'fill_', 'sub',
                'exp', 'mean'] + separate_complex_tests

# this list corresponds to cases that are not currently implemented
skip_cuda_list = ['bmm_complex', 'matmul_4d_4d_complex']

def add_test(
        name,
        self_size,
        args,
        variant_name='',
        check_ad=(),  # only used in test_jit
        dim_args_idx=(),
        skipTestIf=(),
        output_process_fn=lambda x: x,
        kwargs=None):
    kwargs = kwargs if kwargs else {}
    basic_test_name = 'test_' + name
    if variant_name != '':
        basic_test_name += '_' + variant_name

    if name in separate_complex_tests and 'complex' in variant_name:
        run_only_complex = True
    else:
        run_only_complex = False

    for dtype in [torch.double, torch.cdouble]:
        for dim_perm in product([-1, 1], repeat=len(dim_args_idx)):
            test_name = basic_test_name
            new_args = [arg * dim_perm[dim_args_idx.index(i)] if i in dim_args_idx else arg for i, arg in enumerate(args)]
            test_name = basic_test_name + ''.join('_neg' + str(i) for i, idx in enumerate(dim_perm) if idx < 0)

            if dtype.is_complex:
                # TODO: remove this. this is temporary while we ramp up the complex support.
                if name in complex_list:
                    if name in separate_complex_tests and 'complex' not in variant_name:
                        continue
                    if not run_only_complex:
                        test_name = test_name + '_complex'
                else:
                    continue
            elif run_only_complex:
                continue

            new_args = tuple(new_args)

            # for-loop bodies don't define scopes, so we have to save the variables
            # we want to close over in some way
            def do_test(self, device, dtype=dtype, name=name, self_size=self_size, args=new_args, test_name=test_name,
                        output_process_fn=output_process_fn):
                def check(name):
                    is_magic_method = name[:2] == '__' and name[-2:] == '__'
                    is_inplace = name[-1] == "_" and not is_magic_method
                    self_variable = create_input((self_size,), dtype=dtype, device=device)[0][0]
                    # FixMe: run grad checks on inplace self
                    if is_inplace:
                        self_variable.requires_grad = False
                    # need to record this because methods can change the size (e.g. unsqueeze)
                    args_variable, kwargs_variable = create_input(args, requires_grad=not is_inplace,
                                                                  call_kwargs=kwargs, dtype=dtype, device=device)
                    self_tensor = deepcopy(self_variable)
                    args_tensor = deepcopy(unpack_variables(args_variable))
                    if not exclude_tensor_method(name, test_name):
                        output_variable = getattr(self_variable, name)(*args_variable, **kwargs_variable)
                        output_tensor = getattr(self_tensor, name)(*args_tensor, **kwargs_variable)
                        if not isinstance(output_tensor, torch.Tensor) and not istuple(output_tensor):
                            if dtype.is_complex:
                                output_tensor = torch.tensor((output_tensor, ), dtype=torch.cfloat, device=device)
                            else:
                                output_tensor = torch.tensor((output_tensor, ), dtype=torch.float, device=device)
                        self.assertEqual(unpack_variables(output_variable), output_tensor)
                        # TODO: check that both have changed after adding all inplace ops

                        def fn(*inputs):
                            output = getattr(inputs[0], name)(*inputs[1:], **kwargs)
                            return output_process_fn(output)

                        if not is_inplace and name not in EXCLUDE_GRADCHECK:
                            run_grad_and_gradgrad_checks(self, name, test_name, fn,
                                                         output_variable, (self_variable,) + args_variable)

                    # functional interface tests
                    torch_fn = getattr_qualified(torch, name)
                    if torch_fn is not None and name not in EXCLUDE_FUNCTIONAL:
                        def fn(*inputs):
                            output = torch_fn(*inputs, **kwargs)
                            return output_process_fn(output)

                        f_args_variable = (self_variable,) + args_variable
                        f_args_tensor = (self_tensor,) + args_tensor
                        # could run the gradchecks again, but skip since we did it for the methods above.
                        run_gradcheck = exclude_tensor_method(name, test_name) and not is_inplace and name not in EXCLUDE_GRADCHECK
                        run_functional_checks(self, test_name, name, fn,
                                              run_gradcheck, f_args_variable, f_args_tensor)

                    # check for correct type of input and input.grad
                    if not is_inplace:
                        self_variable = create_input((self_size,), requires_grad=True, dtype=dtype)[0][0]
                        args_variable, kwargs_variable = create_input(args, requires_grad=False, call_kwargs=kwargs, dtype=dtype)
                        if hasattr(self_variable, name):
                            attribute_result = getattr(self_variable, name)
                            if callable(attribute_result):
                                output_variable = attribute_result(*args_variable, **kwargs_variable)
                            else:
                                self.assertTrue(len(args_variable) == 0)
                                self.assertTrue(len(kwargs_variable) == 0)
                                output_variable = attribute_result
                        else:
                            self_and_args_variable = (self_variable,) + args_variable
                            output_variable = torch_fn(*self_and_args_variable, **kwargs_variable)
                        if isinstance(output_variable, torch.autograd.Variable):
                            if output_variable.is_sparse:
                                rand = randn_like(output_variable.to_dense()).to_sparse()
                            else:
                                rand = randn_like(output_variable)
                            output_variable.backward(rand)
                            self.assertTrue(type(self_variable) == type(self_variable.grad))
                            self.assertTrue(self_variable.size() == self_variable.grad.size())

                        # compare grads to inplace grads
                        inplace_name = name + '_'
                        # can't broadcast inplace to left hand side
                        skip_inplace = ('broadcast_lhs' in test_name or
                                        'broadcast_all' in test_name or
                                        'atanh' in test_name or
                                        'acosh' in test_name or
                                        'asinh' in test_name or
                                        'abs_complex' in test_name or
                                        'abs_scalar_complex' in test_name)
                        if hasattr(torch.ones(1), inplace_name) and not skip_inplace:
                            output_variable = getattr(self_variable, name)(*args_variable, **kwargs_variable)
                            if not isinstance(output_variable, tuple):
                                output_variable = (output_variable,)
                            inplace_self_variable = deepcopy(self_variable)
                            inplace_self_variable_copy = tuple(i.clone() if isinstance(i, torch.Tensor) else i
                                                               for i in (inplace_self_variable,))
                            inplace_args_variable = deepcopy(args_variable)
                            inplace_args_variable_copy = tuple(i.clone() if isinstance(i, torch.Tensor) else i
                                                               for i in inplace_args_variable)

                            inplace_output_variable = (
                                getattr(inplace_self_variable_copy[0], inplace_name)(*inplace_args_variable_copy,
                                                                                     **kwargs_variable))
                            if not isinstance(inplace_output_variable, tuple):
                                inplace_output_variable = (inplace_output_variable,)
                            self.assertEqual(inplace_output_variable, output_variable)
                            # Check that gradient is the same
                            for inp_i, i in zip((inplace_self_variable,) + inplace_args_variable,
                                                (self_variable,) + args_variable):
                                if not isinstance(inp_i, torch.Tensor):
                                    assert not isinstance(i, torch.Tensor)
                                    continue
                                if inp_i.grad is not None:
                                    with torch.no_grad():
                                        inp_i.grad.zero_()
                                if i.grad is not None:
                                    with torch.no_grad():
                                        i.grad.zero_()
                            for i_o, o in zip(inplace_output_variable, output_variable):
                                if dtype.is_complex:
                                    grad = randn_like(i_o).to(torch.cdouble)
                                else:
                                    grad = randn_like(i_o).double()
                                i_o.backward(grad)
                                o.backward(grad)
                            for inp_i, i in zip((inplace_self_variable,) + inplace_args_variable,
                                                (self_variable,) + args_variable):
                                if not isinstance(inp_i, torch.Tensor):
                                    continue
                                self.assertEqual(inp_i.grad, i.grad)

                check(name)
                inplace_name = name + '_'
                # can't broadcast inplace to left hand side
                broadcast_skip_inplace = 'broadcast_lhs' in test_name or 'broadcast_all' in test_name
                # skip C -> R inplace tests
                skip_c_to_r_inplace = 'abs_complex' in test_name or 'abs_scalar_complex' in test_name
                skip_inplace = broadcast_skip_inplace or skip_c_to_r_inplace
                if hasattr(torch.ones(1), inplace_name) and not skip_inplace:
                    check(inplace_name)

            assert not hasattr(TestAutograd, test_name), 'Two tests have the same name: ' + test_name

            for skip in skipTestIf:
                do_test = skip(do_test)

            # TODO: remove this once tests from skip_cuda_list work
            do_test = skipCUDAIf(
                any(skip_test in test_name for skip_test in skip_cuda_list),
                "not implemented for CUDA yet")(do_test)

            setattr(TestAutogradDeviceType, test_name, do_test)

class TestAutogradComplex(TestCase):
    def test_view_func_for_complex_views(self):
        # case 1: both parent and child have view_func
        x = torch.randn(2, 2, 2, dtype=torch.double, requires_grad=True)
        y = x.detach().requires_grad_(True)

        x0 = x.clone()
        x1 = torch.view_as_complex(x0)
        x2 = torch.view_as_real(x1)
        x2.mul_(2)
        x2.sum().backward()

        y0 = y.clone()
        y0.mul_(2)
        y0.sum().backward()

        self.assertEqual(x.grad, y.grad)

        # case 2: parent has view_func but child does not
        x = torch.randn(2, 2, 2, dtype=torch.double, requires_grad=True)
        y = x.detach().requires_grad_(True)

        def fn(a):
            b = a.clone()
            b1 = torch.view_as_complex(b)
            b2 = b1.reshape(b1.numel())
            return b2

        x0 = fn(x)
        x0.mul_(2)
        x0.sum().backward()

        y0 = fn(y)
        y1 = y0.mul(2)
        y1.sum().backward()

        self.assertEqual(x.grad, y.grad)

        # case 3: parent does not have a view_func but child does
        x = torch.randn(10, dtype=torch.cdouble, requires_grad=True)
        y = x.detach().requires_grad_(True)

        def fn(a, dim0_size=5):
            b = a.clone()
            b1 = b.reshape(dim0_size, 2)
            b2 = torch.view_as_real(b1)
            return b2

        x0 = fn(x)
        x0.mul_(2)
        x0.sum().backward()

        y0 = fn(y)
        y1 = y0.mul(2)
        y1.sum().backward()

        self.assertEqual(x.grad, y.grad)

    def as_identity(self):
        # view_as_real and view_as_complex behavior should be like an identity
        def func(z):
            z_ = torch.view_as_complex(z)
            z_select = torch.select(z_, z_.dim() - 1, 0)
            z_select_real = torch.view_as_real(z_select)
            return z_select_real.sum()

        z = torch.randn(10, 2, 2, dtype=torch.double, requires_grad=True)
        gradcheck(func, [z])
        func(z).backward()

        z1 = z.clone().detach().requires_grad_(True)
        torch.select(z1, z1.dim() - 2, 0).sum().backward()

        self.assertEqual(z.grad, z1.grad)

class TestAutogradFunctional(TestCase):
    def _assert_same_struct(self, res, base):
        # base and res should be Tensors or tuple of Tensors with the same size
        if isinstance(base, torch.Tensor):
            self.assertTrue(isinstance(res, torch.Tensor))
            self.assertEqual(base.size(), res.size())
        elif isinstance(base, tuple):
            self.assertTrue(isinstance(res, tuple))
            self.assertEqual(len(base), len(res))
            for el_base, el_res in zip(base, res):
                self.assertTrue(isinstance(el_base, torch.Tensor))
                self.assertTrue(isinstance(el_res, torch.Tensor))
                self.assertEqual(el_base.size(), el_res.size())
        else:
            # Wrong base
            raise RuntimeError("The base given to `_assert_same_struct` doesn't have"
                               " the right structure.")

    def _assert_interleaved_struct(self, res, base1, base2):
        # base1 and base2 can be Tensors or tuples of Tensors.
        # If they are tuples, res should be a tuple as well.
        # The indexing works as follows for base1, base2 being
        # - tuple, tuple: res[i][j][k][l] = (base1[i][k], base2[j][l])
        # - tuple, Tensor: res[i][k][l] = (base1[i][k], base2[l])
        # - Tensor, tuple: res[i][j][l] = (base1[i], base2[j][l])
        # - Tensor, Tensor: res[k][l] = (base1[k], base2[l])
        if isinstance(base1, torch.Tensor) and isinstance(base2, torch.Tensor):
            self.assertTrue(isinstance(res, torch.Tensor))
            self.assertEqual(res.size(), base1.size() + base2.size())
        elif isinstance(base1, tuple) and isinstance(base2, torch.Tensor):
            self.assertTrue(isinstance(res, tuple))
            self.assertEqual(len(res), len(base1))
            for el_res, el_base1 in zip(res, base1):
                self.assertTrue(isinstance(el_res, torch.Tensor))
                self.assertTrue(isinstance(el_base1, torch.Tensor))
                self.assertEqual(el_res.size(), el_base1.size() + base2.size())
        elif isinstance(base1, torch.Tensor) and isinstance(base2, tuple):
            self.assertTrue(isinstance(res, tuple))
            self.assertEqual(len(res), len(base2))
            for el_res, el_base2 in zip(res, base2):
                self.assertTrue(isinstance(el_res, torch.Tensor))
                self.assertTrue(isinstance(el_base2, torch.Tensor))
                self.assertEqual(el_res.size(), base1.size() + el_base2.size())
        elif isinstance(base1, tuple) and isinstance(base2, tuple):
            self.assertTrue(isinstance(res, tuple))
            self.assertEqual(len(res), len(base1))
            for el_res, el_base1 in zip(res, base1):
                self.assertTrue(isinstance(el_res, tuple))
                self.assertEqual(len(res), len(base2))
                for el_el_res, el_base2 in zip(el_res, base2):
                    self.assertTrue(isinstance(el_el_res, torch.Tensor))
                    self.assertTrue(isinstance(el_base2, torch.Tensor))
                    self.assertEqual(el_el_res.size(), el_base1.size() + el_base2.size())
        else:
            # Wrong bases
            raise RuntimeError("The bases given to `_assert_interleaved_struct` don't have"
                               " the right structure.")

    def test_vjp_err_check(self):
        def foo(a):
            return 3 * a.narrow(0, 0, 3)

        def bar(a):
            return 3 * a.narrow(0, 0, 3), "bar"

        inp = torch.rand(4)
        v = torch.ones(3)
        with self.assertRaisesRegex(TypeError, "The inputs given to vjp must be either a Tensor"):
            res = autogradF.vjp(foo, (inp, 2), v)

        with self.assertRaisesRegex(TypeError, "The outputs of the user-provided function given to vjp must"):
            res = autogradF.vjp(bar, inp, v)

        with self.assertRaisesRegex(RuntimeError, "The vector v can only be None if the user-provided function returns"):
            res = autogradF.vjp(foo, inp)

        with self.assertRaisesRegex(RuntimeError, "The given v should contain a single Tensor."):
            res = autogradF.vjp(foo, inp, (torch.ones_like(inp), torch.ones_like(inp)))

        with self.assertRaisesRegex(RuntimeError, "v has invalid size: should be torch.Size"):
            res = autogradF.vjp(foo, inp, v[:2])

        res = autogradF.vjp(foo, inp, v)[1]
        self._assert_same_struct(res, inp)

    def test_vjp_err_check_strict(self):
        def foo(a):
            return a.detach()

        def bar(a):
            # Make a non-leaf Tensor that requires_grad but that is not connected to the input
            return a.long().float().requires_grad_().clone()

        inp = torch.rand(4)
        v = torch.rand(4)
        with self.assertRaisesRegex(RuntimeError, "Output 0 of the user-provided function does not require gradients."):
            res = autogradF.vjp(foo, inp, v, strict=True)
        res = autogradF.vjp(foo, inp, v, strict=False)
        self._assert_same_struct(res[1], inp)
        self.assertEqual(res[1].abs().sum(), 0.)

        with self.assertRaisesRegex(RuntimeError, "The output of the user-provided function is independent of input 0"):
            res = autogradF.vjp(bar, inp, v, strict=True)
        res = autogradF.vjp(bar, inp, v, strict=False)
        self._assert_same_struct(res[1], inp)
        self.assertEqual(res[1].abs().sum(), 0.)

        # The Jacobian does not depend on the input
        def foo(a):
            return a.clone()

        inp.requires_grad_()
        with self.assertRaisesRegex(RuntimeError, "jacobian of the user-provided function is independent of input 0."):
            res = autogradF.vjp(foo, inp, v, create_graph=True, strict=True)
        res = autogradF.vjp(foo, inp, v, create_graph=True, strict=False)
        self._assert_same_struct(res[1], inp)
        self.assertEqual(res[1], v)

    def test_vjp_output(self):
        def reducer(x):
            return x.sum(dim=1)
        inputs = torch.rand(4, 4)
        v = torch.ones(4)
        res = autogradF.vjp(reducer, inputs, v)
        self._assert_same_struct(res[1], inputs)
        self.assertIsNone(res[0].grad_fn)
        self.assertIsNone(res[1].grad_fn)

        def adder(x, y):
            return 2 * x + 3 * y

        inputs = (torch.rand(2), torch.rand(2))
        v = torch.ones(2)
        out, vjp_val = autogradF.vjp(adder, inputs, v)
        self._assert_same_struct(vjp_val, inputs)
        self.assertIsNone(out.grad_fn)
        self.assertIsNone(vjp_val[0].grad_fn)
        self.assertIsNone(vjp_val[1].grad_fn)

        def adder(x, y):
            return 2 * x + 3 * y, x + y

        inputs = (torch.rand(2), torch.rand(2))
        v = (torch.tensor([1., 0.]), torch.tensor([1., 0.]))
        out, vjp_val = autogradF.vjp(adder, inputs, v)
        self._assert_same_struct(vjp_val, inputs)
        self.assertIsNone(out[0].grad_fn)
        self.assertIsNone(out[1].grad_fn)
        self.assertIsNone(vjp_val[0].grad_fn)
        self.assertIsNone(vjp_val[1].grad_fn)

    def test_vjp_scalar(self):
        def reducer(x):
            return x.sum()
        inputs = torch.rand(4, 4)
        v = torch.ones([])
        res = autogradF.vjp(reducer, inputs, v)
        self._assert_same_struct(res[0], v)
        self._assert_same_struct(res[1], inputs)

        res = autogradF.vjp(reducer, inputs)
        self._assert_same_struct(res[0], v)
        self._assert_same_struct(res[1], inputs)

        def expander(x):
            return x.unsqueeze(0).repeat(4)
        inputs = torch.rand([])
        v = torch.ones(4)
        res = autogradF.vjp(expander, inputs, v)
        self._assert_same_struct(res[0], v)
        self._assert_same_struct(res[1], inputs)

    def test_vjp_create_graph(self):
        def reducer(x):
            return x.sum(dim=1)
        inputs = torch.rand(2, 2)
        v = torch.ones(2)

        inputs.requires_grad_()
        v.requires_grad_()
        res = autogradF.vjp(reducer, inputs, v, create_graph=True)
        self._assert_same_struct(res[1], inputs)
        self.assertIsNotNone(res[0].grad_fn)
        self.assertIsNotNone(res[1].grad_fn)

        gradcheck(lambda inp, v: autogradF.vjp(reducer, inputs, v, create_graph=True), (inputs, v))
        gradgradcheck(lambda inp, v: autogradF.vjp(reducer, inputs, v, create_graph=True), (inputs, v))

        def adder(x, y):
            return 2 * x + 3 * y, x * y

        inputs = (torch.rand(2, requires_grad=True), torch.rand(2, requires_grad=True))
        v = (torch.tensor([1., 0.], requires_grad=True), torch.tensor([1., 0.], requires_grad=True))

        gradcheck(lambda *args: autogradF.vjp(adder, args[:2], args[2:], create_graph=True)[1], inputs + v)
        gradgradcheck(lambda *args: autogradF.vjp(adder, args[:2], args[2:], create_graph=True)[1], inputs + v)

        def foo(*args):
            x, y = args[:2]
            v = args[2:]

            x = x.cos()
            val, grad = autogradF.vjp(adder, (x, y), v, create_graph=True)

            return val[0].exp() + val[1].exp() + grad[0].exp() + grad[1].exp() + x.exp() + y.exp()

        gradcheck(foo, inputs + v)
        gradgradcheck(foo, inputs + v)

    def test_jvp_err_check(self):
        def foo(a):
            return 3 * a.narrow(0, 0, 3)

        def bar(a):
            return 3 * a.narrow(0, 0, 3), "bar"

        inp = torch.rand(4)
        v = torch.rand(4)
        with self.assertRaisesRegex(TypeError, "The inputs given to jvp must be either a Tensor"):
            res = autogradF.jvp(foo, (inp, 2), v)

        with self.assertRaisesRegex(TypeError, "The outputs of the user-provided function given to jvp must"):
            res = autogradF.jvp(bar, inp, v)

        with self.assertRaisesRegex(RuntimeError, "The vector v can only be None if the input to the user-provided function"):
            res = autogradF.jvp(foo, inp)

        with self.assertRaisesRegex(RuntimeError, "The given v should contain a single Tensor."):
            res = autogradF.jvp(foo, inp, (v, v))

        with self.assertRaisesRegex(RuntimeError, "v has invalid size: should be torch.Size"):
            res = autogradF.jvp(foo, inp, v[:2])

        res = autogradF.jvp(foo, inp, v)[1]
        self._assert_same_struct(res, foo(inp))

    def test_jvp_err_check_strict(self):
        def foo(a):
            return a.detach()

        def bar(a):
            # Make a non-leaf Tensor that requires_grad but that is not connected to the input
            return a.long().float().requires_grad_().clone()

        inp = torch.rand(4)
        v = torch.rand(4)
        with self.assertRaisesRegex(RuntimeError, "Output 0 of the user-provided function does not require gradients."):
            res = autogradF.jvp(foo, inp, v, strict=True)
        res = autogradF.jvp(foo, inp, v, strict=False)
        self._assert_same_struct(res[1], res[0])
        self.assertEqual(res[1].abs().sum(), 0.)

        with self.assertRaisesRegex(RuntimeError, "The output of the user-provided function is independent of input 0"):
            res = autogradF.jvp(bar, inp, v, strict=True)
        res = autogradF.jvp(bar, inp, v, strict=False)
        self._assert_same_struct(res[1], res[0])
        self.assertEqual(res[1].abs().sum(), 0.)

        # The Jacobian does not depend on the input
        def foo(a):
            return a.clone()

        inp.requires_grad_()
        with self.assertRaisesRegex(RuntimeError, "jacobian of the user-provided function is independent of input 0."):
            res = autogradF.jvp(foo, inp, v, create_graph=True, strict=True)
        res = autogradF.jvp(foo, inp, v, create_graph=True, strict=False)
        self._assert_same_struct(res[1], inp)
        self.assertEqual(res[1], v)

    def test_jvp_output(self):
        def reducer(x):
            return x.sum(dim=1)
        inputs = torch.rand(4, 4)
        v = torch.ones(4, 4)
        res = autogradF.jvp(reducer, inputs, v)
        self._assert_same_struct(res[1], res[0])
        self.assertIsNone(res[0].grad_fn)
        self.assertIsNone(res[1].grad_fn)

        def adder(x, y):
            return 2 * x + 3 * y

        inputs = (torch.rand(2), torch.rand(2))
        v = (torch.ones(2), torch.ones(2))
        out, jvp_val = autogradF.jvp(adder, inputs, v)
        self._assert_same_struct(jvp_val, out)
        self.assertIsNone(out.grad_fn)
        self.assertIsNone(jvp_val[0].grad_fn)
        self.assertIsNone(jvp_val[1].grad_fn)

        def adder(x, y):
            return 2 * x + 3 * y, x + y

        inputs = (torch.rand(2), torch.rand(2))
        v = (torch.tensor([1., 0.]), torch.tensor([1., 0.]))
        out, jvp_val = autogradF.jvp(adder, inputs, v)
        self._assert_same_struct(jvp_val, out)
        self.assertIsNone(out[0].grad_fn)
        self.assertIsNone(out[1].grad_fn)
        self.assertIsNone(jvp_val[0].grad_fn)
        self.assertIsNone(jvp_val[1].grad_fn)

    def test_jvp_scalar(self):
        def reducer(x):
            return x.sum()
        inputs = torch.rand(4, 4)
        v = torch.ones(4, 4)
        res = autogradF.jvp(reducer, inputs, v)
        self._assert_same_struct(res[0], torch.zeros([]))
        self._assert_same_struct(res[1], res[0])

        def expander(x):
            return x.unsqueeze(0).repeat(4)
        inputs = torch.rand([])
        v = torch.ones([])
        res = autogradF.jvp(expander, inputs, v)
        self._assert_same_struct(res[0], torch.zeros(4))
        self._assert_same_struct(res[1], res[0])

        res = autogradF.jvp(expander, inputs)
        self._assert_same_struct(res[0], torch.zeros(4))
        self._assert_same_struct(res[1], res[0])

    def test_jvp_create_graph(self):
        def reducer(x):
            return x.sum(dim=1)
        inputs = torch.rand(2, 2)
        v = torch.ones(2, 2)

        inputs.requires_grad_()
        v.requires_grad_()
        res = autogradF.jvp(reducer, inputs, v, create_graph=True)
        self._assert_same_struct(res[1], res[0])
        self.assertIsNotNone(res[0].grad_fn)
        self.assertIsNotNone(res[1].grad_fn)

        gradcheck(lambda inp, v: autogradF.jvp(reducer, inp, v, create_graph=True), (inputs, v))
        gradgradcheck(lambda inp, v: autogradF.jvp(reducer, inp, v, create_graph=True), (inputs, v))

        def adder(x, y):
            return 2 * x + 3 * y, x * y

        inputs = (torch.rand(2, requires_grad=True), torch.rand(2, requires_grad=True))
        v = (torch.tensor([1., 0.], requires_grad=True), torch.tensor([1., 0.], requires_grad=True))

        gradcheck(lambda *args: autogradF.jvp(adder, args[:2], args[2:], create_graph=True)[1], inputs + v)
        gradgradcheck(lambda *args: autogradF.jvp(adder, args[:2], args[2:], create_graph=True)[1], inputs + v)

        def foo(*args):
            x, y = args[:2]
            v = args[2:]

            x = x.cos()
            val, grad = autogradF.jvp(adder, (x, y), v, create_graph=True)

            return val[0].exp() + val[1].exp() + grad[0].exp() + grad[1].exp() + x.exp() + y.exp()

        gradcheck(foo, inputs + v)
        gradgradcheck(foo, inputs + v)

    def test_jacobian_err_check(self):
        def foo(a):
            return 3 * a.narrow(0, 0, 3)

        def bar(a):
            return 3 * a.narrow(0, 0, 3), "bar"

        inp = torch.rand(4)
        with self.assertRaisesRegex(TypeError, "The inputs given to jacobian must be either a Tensor"):
            res = autogradF.jacobian(foo, (inp, 2))

        with self.assertRaisesRegex(TypeError, "The outputs of the user-provided function given to jacobian must"):
            res = autogradF.jacobian(bar, inp)

        res = autogradF.jacobian(foo, inp)
        self._assert_interleaved_struct(res, foo(inp), inp)

        def foo(a, b):
            return b, 3 * a.narrow(0, 0, 3)

        inp = (torch.rand(4), torch.rand(5))

        res = autogradF.jacobian(foo, inp)
        self._assert_interleaved_struct(res, foo(*inp), inp)

    def test_jacobian_err_check_strict(self):
        def foo(a):
            return a.detach()

        def bar(a):
            # Make a non-leaf Tensor that requires_grad but that is not connected to the input
            return a.long().float().requires_grad_().clone()

        inp = torch.rand(4)
        with self.assertRaisesRegex(RuntimeError, "Output 0 of the user-provided function does not require gradients."):
            res = autogradF.jacobian(foo, inp, strict=True)
        res = autogradF.jacobian(foo, inp, strict=False)
        self._assert_interleaved_struct(res, foo(inp), inp)
        self.assertEqual(res.abs().sum(), 0.)

        with self.assertRaisesRegex(RuntimeError, "Output 0 of the user-provided function is independent of input 0."):
            res = autogradF.jacobian(bar, inp, strict=True)
        res = autogradF.jacobian(bar, inp, strict=False)
        self._assert_interleaved_struct(res, foo(inp), inp)
        self.assertEqual(res.abs().sum(), 0.)

        # The Jacobian does not depend on the input
        def foo(a):
            return a.clone()

        inp.requires_grad_()
        with self.assertRaisesRegex(RuntimeError, "jacobian of the user-provided function is independent of input 0."):
            res = autogradF.jacobian(foo, inp, create_graph=True, strict=True)
        res = autogradF.jacobian(foo, inp, create_graph=True, strict=False)
        self._assert_interleaved_struct(res, inp, inp)
        self.assertEqual(res, torch.eye(4))

    def test_jacobian_output(self):
        def exp_reducer(x):
            return x.exp().sum(dim=1)

        inputs = torch.rand(4, 4)
        res = autogradF.jacobian(exp_reducer, inputs)
        self._assert_interleaved_struct(res, exp_reducer(inputs), inputs)
        self.assertIsNone(res.grad_fn)

        def identity(x):
            return x.clone()

        inputs = torch.rand(4)
        res = autogradF.jacobian(identity, inputs)
        self._assert_interleaved_struct(res, identity(inputs), inputs)
        self.assertIsNone(res.grad_fn)
        self.assertEqual(res, torch.eye(4))

        def add_exp_reducer(x, y):
            return (x + y.exp()).sum(dim=1)

        inputs = (torch.rand(4, 4), torch.rand(4, 4))
        res = autogradF.jacobian(add_exp_reducer, inputs)
        self._assert_interleaved_struct(res, add_exp_reducer(*inputs), inputs)
        self.assertIsNone(res[0].grad_fn)
        self.assertIsNone(res[1].grad_fn)

    def test_jacobian_scalar(self):
        def reducer(x):
            return x.sum()
        inputs = torch.rand(4, 4)
        res = autogradF.jacobian(reducer, inputs)
        self._assert_same_struct(res, inputs)

        def expander(x):
            return x.unsqueeze(0).repeat(4)
        inputs = torch.rand([])
        res = autogradF.jacobian(expander, inputs)
        self._assert_same_struct(res, torch.zeros(4))

    def test_jacobian_create_graph(self):
        def exp_reducer(x):
            return x.exp().sum(dim=1)

        inputs = torch.rand(4, 4, requires_grad=True)
        res = autogradF.jacobian(exp_reducer, inputs, create_graph=True)
        self._assert_interleaved_struct(res, exp_reducer(inputs), inputs)
        self.assertIsNotNone(res.grad_fn)

        gradcheck(lambda inp: autogradF.jacobian(exp_reducer, inp, create_graph=True), inputs)
        gradgradcheck(lambda inp: autogradF.jacobian(exp_reducer, inp, create_graph=True), inputs)

        def add_exp_reducer(x, y):
            return (x + y).exp().sum(dim=1)

        inputs = (torch.rand(4, 4, requires_grad=True), torch.rand(4, 4, requires_grad=True))
        res = autogradF.jacobian(add_exp_reducer, inputs, create_graph=True)
        self._assert_interleaved_struct(res, add_exp_reducer(*inputs), inputs)
        self.assertIsNotNone(res[0].grad_fn)
        self.assertIsNotNone(res[1].grad_fn)

        gradcheck(lambda *inp: autogradF.jacobian(add_exp_reducer, inp, create_graph=True), inputs)
        gradgradcheck(lambda *inp: autogradF.jacobian(add_exp_reducer, inp, create_graph=True), inputs)

        def foo(x, y):
            x = x.cos()
            val, jac = autogradF.jacobian(add_exp_reducer, (x, y), create_graph=True)

            res = val[0].exp().sum() + val[1].exp().sum() + jac[0].exp().sum()
            res = res + jac[1].exp().sum() + x.exp().sum() + y.exp().sum()
            return res

        gradcheck(foo, inputs)
        gradgradcheck(foo, inputs)

    def test_hessian_err_check(self):
        def foo(a):
            return 3 * a.narrow(0, 0, 3).exp().sum()

        def bar(a):
            return 3 * a.narrow(0, 0, 3), "bar"

        def bar2(a):
            return 3 * a.narrow(0, 0, 3)

        def bar3(a):
            return 3 * a.narrow(0, 0, 3), 3 * a.narrow(0, 0, 3)

        inp = torch.rand(4)
        with self.assertRaisesRegex(TypeError, "The inputs given to hessian must be either a Tensor"):
            res = autogradF.hessian(foo, (inp, 2))

        with self.assertRaisesRegex(TypeError, "The outputs of the user-provided function given to hessian must"):
            res = autogradF.hessian(bar, inp)

        err_msg_out = "The Tensor returned by the function given to hessian should contain a single element"
        with self.assertRaisesRegex(RuntimeError, err_msg_out):
            res = autogradF.hessian(bar2, inp)

        with self.assertRaisesRegex(RuntimeError, "The function given to hessian should return a single Tensor"):
            res = autogradF.hessian(bar3, inp)

        res = autogradF.hessian(foo, inp)
        self._assert_interleaved_struct(res, inp, inp)

        def foo(a, b):
            return (3 * b.narrow(0, 0, 3) * a.narrow(0, 0, 3)).sum()

        inp = (torch.rand(4), torch.rand(5))

        res = autogradF.hessian(foo, inp)
        self._assert_interleaved_struct(res, inp, inp)

    def test_hessian_err_check_strict(self):
        def foo(a):
            return a.detach().sum()

        def bar(a):
            # Make a non-leaf Tensor that requires_grad but that is not connected to the input
            return a.long().float().requires_grad_().clone().sum()

        def bar2(a):
            # A Linear function for which the jacobian is independent of the input
            return (3 * a).sum()

        inp = torch.rand(4)
        with self.assertRaisesRegex(RuntimeError, "Output 0 of the user-provided function does not require gradients."):
            res = autogradF.hessian(foo, inp, strict=True)
        res = autogradF.hessian(foo, inp, strict=False)
        self._assert_interleaved_struct(res, inp, inp)
        self.assertEqual(res.abs().sum(), 0.)

        with self.assertRaisesRegex(RuntimeError, "jacobian of the user-provided function with respect to input 0"):
            res = autogradF.hessian(bar, inp, strict=True)
        res = autogradF.hessian(bar, inp, strict=False)
        self._assert_interleaved_struct(res, inp, inp)
        self.assertEqual(res.abs().sum(), 0.)

        with self.assertRaisesRegex(RuntimeError, "jacobian of the user-provided function with respect to input 0 is"):
            res = autogradF.hessian(bar2, inp, strict=True)
        res = autogradF.hessian(bar2, inp, strict=False)
        self._assert_interleaved_struct(res, inp, inp)
        self.assertEqual(res.abs().sum(), 0.)

    def test_hessian_output(self):
        def pow_reducer(x):
            return x.pow(3).sum()

        inputs = torch.rand(2, 2)
        res = autogradF.hessian(pow_reducer, inputs)
        self._assert_interleaved_struct(res, inputs, inputs)
        self.assertIsNone(res.grad_fn)

        def add_pow_reducer(x, y):
            return (x + y).pow(3).sum()

        inputs = (torch.rand(2, 2), torch.rand(2, 2))
        res = autogradF.hessian(add_pow_reducer, inputs)
        self._assert_interleaved_struct(res, inputs, inputs)
        self.assertIsNone(res[0][0].grad_fn)
        self.assertIsNone(res[0][1].grad_fn)
        self.assertIsNone(res[1][0].grad_fn)
        self.assertIsNone(res[1][1].grad_fn)

    def test_hessian_scalar(self):
        def reducer(x):
            return x.sum()
        inputs = torch.rand(4, 4)
        res = autogradF.hessian(reducer, inputs)
        self._assert_interleaved_struct(res, inputs, inputs)

        inputs = torch.rand([])
        res = autogradF.hessian(reducer, inputs)
        self._assert_same_struct(res, inputs)

        def bad_reducer(x):
            return x.sum().view(1, 1, 1)
        inputs = torch.rand(4, 4)
        res = autogradF.hessian(bad_reducer, inputs)
        self._assert_interleaved_struct(res, inputs, inputs)

    def test_hessian_create_graph(self):
        def pow_reducer(x):
            return x.pow(3).sum()

        inputs = torch.rand(2, 2, requires_grad=True)
        res = autogradF.hessian(pow_reducer, inputs, create_graph=True)
        self._assert_interleaved_struct(res, inputs, inputs)
        self.assertIsNotNone(res.grad_fn)

        gradcheck(lambda inp: autogradF.hessian(pow_reducer, inp, create_graph=True), inputs)
        gradgradcheck(lambda inp: autogradF.hessian(pow_reducer, inp, create_graph=True), inputs)

        def add_pow_reducer(x, y):
            return (x + y).pow(3).sum()

        inputs = (torch.rand(2, 2, requires_grad=True), torch.rand(2, 2, requires_grad=True))
        res = autogradF.hessian(add_pow_reducer, inputs, create_graph=True)
        self._assert_interleaved_struct(res, inputs, inputs)
        self.assertIsNotNone(res[0][0].grad_fn)
        self.assertIsNotNone(res[0][1].grad_fn)
        self.assertIsNotNone(res[1][0].grad_fn)
        self.assertIsNotNone(res[1][1].grad_fn)

        def flatten(inp):
            return tuple(el_lvl2 for el_lvl1 in inp for el_lvl2 in el_lvl1)

        gradcheck(lambda *inp: flatten(autogradF.hessian(add_pow_reducer, inp, create_graph=True)), inputs)
        gradgradcheck(lambda *inp: flatten(autogradF.hessian(add_pow_reducer, inp, create_graph=True)), inputs)

        def foo(x, y):
            x = x.cos()
            val, hess = autogradF.hessian(add_pow_reducer, (x, y), create_graph=True)

            res = val[0].cos().sum() + val[1].cos().sum() + hess[0].cos().sum()
            res = res + hess[1].cos().sum() + x.cos().sum() + y.cos().sum()
            return res

        gradcheck(foo, inputs)
        gradgradcheck(foo, inputs)

    def test_vhp_err_check(self):
        def foo(a):
            return 3 * a.narrow(0, 0, 3).exp().sum()

        def bar(a):
            return 3 * a.narrow(0, 0, 3), "bar"

        def bar2(a):
            return 3 * a.narrow(0, 0, 3)

        inp = torch.rand(4)
        v = torch.rand(4)
        with self.assertRaisesRegex(TypeError, "The inputs given to vhp must be either a Tensor"):
            res = autogradF.vhp(foo, (inp, 2), v)

        with self.assertRaisesRegex(TypeError, "The outputs of the user-provided function given to vhp must"):
            res = autogradF.vhp(bar, inp, v)

        err_msg_out = "The Tensor returned by the function given to vhp should contain a single element"
        with self.assertRaisesRegex(RuntimeError, err_msg_out):
            res = autogradF.vhp(bar2, inp, v)

        with self.assertRaisesRegex(RuntimeError, "v has invalid size:"):
            res = autogradF.vhp(foo, inp, torch.rand(5))

        with self.assertRaisesRegex(TypeError, "The v given to vhp must be either a Tensor or a tuple of Tensors"):
            res = autogradF.vhp(foo, inp, (v, 2))

        res = autogradF.vhp(foo, inp, v)
        self._assert_same_struct(res[1], inp)

        def foo(a, b):
            return (3 * b.narrow(0, 0, 3) * a.narrow(0, 0, 3)).sum()

        inp = (torch.rand(4), torch.rand(5))
        v = (torch.rand(4), torch.rand(5))

        res = autogradF.vhp(foo, inp, v)
        self._assert_same_struct(res[1], inp)

    def test_vhp_err_check_strict(self):
        def foo(a):
            return a.detach().sum()

        def bar(a):
            # Make a non-leaf Tensor that requires_grad but that is not connected to the input
            return a.long().float().requires_grad_().clone().sum()

        def bar2(a):
            # A Linear function for which the jacobian is independent of the input
            return (3 * a).sum()

        inp = torch.rand(4)
        v = torch.rand(4)
        with self.assertRaisesRegex(RuntimeError, "Output 0 of the user-provided function does not require gradients."):
            res = autogradF.vhp(foo, inp, v, strict=True)
        res = autogradF.vhp(foo, inp, v, strict=False)
        self._assert_same_struct(res[1], inp)
        self.assertEqual(res[1].abs().sum(), 0.)

        with self.assertRaisesRegex(RuntimeError, "The output of the user-provided function is independent of input 0"):
            res = autogradF.vhp(bar, inp, v, strict=True)
        res = autogradF.vhp(bar, inp, v, strict=False)
        self._assert_same_struct(res[1], inp)
        self.assertEqual(res[1].abs().sum(), 0.)

        with self.assertRaisesRegex(RuntimeError, "jacobian of the user-provided function with respect to input 0 is"):
            res = autogradF.vhp(bar2, inp, v, strict=True)
        res = autogradF.vhp(bar2, inp, v, strict=False)
        self._assert_same_struct(res[1], inp)
        self.assertEqual(res[1].abs().sum(), 0.)

    def test_vhp_output(self):
        def foo(a):
            return 3 * a.narrow(0, 0, 3).exp().sum()

        inputs = torch.rand(4, 4)
        v = torch.ones(4, 4)
        res = autogradF.vhp(foo, inputs, v)
        self._assert_same_struct(res[1], inputs)
        self.assertIsNone(res[0].grad_fn)
        self.assertIsNone(res[1].grad_fn)

        def bar(a, b):
            return (a + 3 * b.narrow(0, 0, 3)).exp().sum()

        inputs = (torch.rand(3), torch.rand(4))
        v = (torch.ones(3), torch.ones(4))
        out, vhp_val = autogradF.vhp(bar, inputs, v)
        self._assert_same_struct(vhp_val, inputs)
        self.assertIsNone(out.grad_fn)
        self.assertIsNone(vhp_val[0].grad_fn)
        self.assertIsNone(vhp_val[1].grad_fn)

    def test_vhp_scalar(self):
        def reducer(x):
            return x.sum()
        inputs = torch.rand(4, 4)
        v = torch.ones(4, 4)
        res = autogradF.vhp(reducer, inputs, v)
        self._assert_same_struct(res[1], inputs)

        inputs = torch.rand([])
        v = torch.rand([])
        res = autogradF.vhp(reducer, inputs, v)
        self._assert_same_struct(res[1], inputs)

        res = autogradF.vhp(reducer, inputs)
        self._assert_same_struct(res[1], inputs)

        def bad_reducer(x):
            return x.sum().view(1, 1, 1)
        inputs = torch.rand(4, 4)
        v = torch.rand(4, 4)
        res = autogradF.vhp(bad_reducer, inputs, v)
        self._assert_same_struct(res[1], inputs)

    def test_vhp_create_graph(self):
        def foo(a):
            return 3 * a.narrow(0, 0, 3).exp().sum()

        inputs = torch.rand(4, 4, requires_grad=True)
        v = torch.ones(4, 4, requires_grad=True)
        res = autogradF.vhp(foo, inputs, v, create_graph=True)
        self._assert_same_struct(res[1], inputs)
        self.assertIsNotNone(res[0].grad_fn)
        self.assertIsNotNone(res[1].grad_fn)

        gradcheck(lambda inp, v: autogradF.vhp(foo, inp, v, create_graph=True), (inputs, v))
        gradgradcheck(lambda inp, v: autogradF.vhp(foo, inp, v, create_graph=True), (inputs, v))

        def bar(a, b):
            return (a + 3 * b.narrow(0, 0, 3)).exp().sum()

        inputs = (torch.rand(3, requires_grad=True), torch.rand(4, requires_grad=True))
        v = (torch.ones(3, requires_grad=True), torch.ones(4, requires_grad=True))
        out, vhp_val = autogradF.vhp(bar, inputs, v, create_graph=True)
        self._assert_same_struct(vhp_val, inputs)
        self.assertIsNotNone(out.grad_fn)
        self.assertIsNotNone(vhp_val[0].grad_fn)
        self.assertIsNotNone(vhp_val[1].grad_fn)

        gradcheck(lambda *args: autogradF.vhp(bar, args[:2], args[2:], create_graph=True)[1], inputs + v)
        gradgradcheck(lambda *args: autogradF.vhp(bar, args[:2], args[2:], create_graph=True)[1], inputs + v)

        def foo(*args):
            x, y = args[:2]
            v = args[2:]

            x = x.cos()
            val, grad = autogradF.vhp(bar, (x, y), v, create_graph=True)

            return val.cos() + grad[0].cos().sum() + grad[1].cos() + x.cos().sum() + y.cos()

        gradcheck(foo, inputs + v)
        gradgradcheck(foo, inputs + v)

    def test_hvp_err_check(self):
        def foo(a):
            return 3 * a.narrow(0, 0, 3).exp().sum()

        def bar(a):
            return 3 * a.narrow(0, 0, 3), "bar"

        def bar2(a):
            return 3 * a.narrow(0, 0, 3)

        inp = torch.rand(4)
        v = torch.rand(4)
        res = autogradF.hvp(foo, inp, v)
        with self.assertRaisesRegex(TypeError, "The inputs given to hvp must be either a Tensor"):
            res = autogradF.hvp(foo, (inp, 2), v)

        with self.assertRaisesRegex(TypeError, "The outputs of the user-provided function given to hvp must"):
            res = autogradF.hvp(bar, inp, v)

        err_msg_out = "The Tensor returned by the function given to hvp should contain a single element"
        with self.assertRaisesRegex(RuntimeError, err_msg_out):
            res = autogradF.hvp(bar2, inp, v)

        with self.assertRaisesRegex(RuntimeError, "v has invalid size:"):
            res = autogradF.hvp(foo, inp, torch.rand(5))

        with self.assertRaisesRegex(TypeError, "The v given to hvp must be either a Tensor or a tuple of Tensors"):
            res = autogradF.hvp(foo, inp, (v, 2))

        res = autogradF.hvp(foo, inp, v)
        self._assert_same_struct(res[1], inp)

        def foo(a, b):
            return (3 * b.narrow(0, 0, 3) * a.narrow(0, 0, 3)).sum()

        inp = (torch.rand(4), torch.rand(5))
        v = (torch.rand(4), torch.rand(5))

        res = autogradF.hvp(foo, inp, v)
        self._assert_same_struct(res[1], inp)

    def test_hvp_err_check_strict(self):
        def foo(a):
            return a.detach().sum()

        def bar(a):
            # Make a non-leaf Tensor that requires_grad but that is not connected to the input
            return a.long().float().requires_grad_().clone().sum()

        def bar2(a):
            # A Linear function for which the jacobian is independent of the input
            return (3 * a).sum()

        inp = torch.rand(4)
        v = torch.rand(4)
        with self.assertRaisesRegex(RuntimeError, "Output 0 of the user-provided function does not require gradients."):
            res = autogradF.hvp(foo, inp, v, strict=True)
        res = autogradF.hvp(foo, inp, v, strict=False)
        self._assert_same_struct(res[1], inp)
        self.assertEqual(res[1].abs().sum(), 0.)

        with self.assertRaisesRegex(RuntimeError, "The output of the user-provided function is independent of input 0"):
            res = autogradF.hvp(bar, inp, v, strict=True)
        res = autogradF.hvp(bar, inp, v, strict=False)
        self._assert_same_struct(res[1], inp)
        self.assertEqual(res[1].abs().sum(), 0.)

        with self.assertRaisesRegex(RuntimeError, "jacobian of the user-provided function with respect to input 0 is"):
            res = autogradF.hvp(bar2, inp, v, strict=True)
        res = autogradF.hvp(bar2, inp, v, strict=False)
        self._assert_same_struct(res[1], inp)
        self.assertEqual(res[1].abs().sum(), 0.)

    def test_hvp_output(self):
        def foo(a):
            return 3 * a.narrow(0, 0, 3).exp().sum()

        inputs = torch.rand(4, 4)
        v = torch.ones(4, 4)
        res = autogradF.hvp(foo, inputs, v)
        self._assert_same_struct(res[1], inputs)
        self.assertIsNone(res[0].grad_fn)
        self.assertIsNone(res[1].grad_fn)

        def bar(a, b):
            return (a + 3 * b.narrow(0, 0, 3)).exp().sum()

        inputs = (torch.rand(3), torch.rand(4))
        v = (torch.ones(3), torch.ones(4))
        out, hvp_val = autogradF.hvp(bar, inputs, v)
        self._assert_same_struct(hvp_val, inputs)
        self.assertIsNone(out.grad_fn)
        self.assertIsNone(hvp_val[0].grad_fn)
        self.assertIsNone(hvp_val[1].grad_fn)

    def test_hvp_scalar(self):
        def reducer(x):
            return x.exp().sum()
        inputs = torch.rand(4, 4)
        v = torch.ones(4, 4)
        res = autogradF.hvp(reducer, inputs, v)
        self._assert_same_struct(res[1], inputs)

        inputs = torch.rand([])
        v = torch.rand([])
        res = autogradF.hvp(reducer, inputs, v)
        self._assert_same_struct(res[1], inputs)

        res = autogradF.hvp(reducer, inputs)
        self._assert_same_struct(res[1], inputs)

        def bad_reducer(x):
            return x.exp().sum().view(1, 1, 1)
        inputs = torch.rand(4, 4)
        v = torch.rand(4, 4)
        res = autogradF.hvp(bad_reducer, inputs, v)
        self._assert_same_struct(res[1], inputs)

    def test_hvp_create_graph(self):
        def foo(a):
            return 3 * a.narrow(0, 0, 3).exp().sum()

        inputs = torch.rand(4, 4, requires_grad=True)
        v = torch.ones(4, 4, requires_grad=True)
        res = autogradF.hvp(foo, inputs, v, create_graph=True)
        self._assert_same_struct(res[1], inputs)
        self.assertIsNotNone(res[0].grad_fn)
        self.assertIsNotNone(res[1].grad_fn)

        gradcheck(lambda inp, v: autogradF.hvp(foo, inp, v, create_graph=True), (inputs, v))
        gradgradcheck(lambda inp, v: autogradF.hvp(foo, inp, v, create_graph=True), (inputs, v))

        def bar(a, b):
            return (a + 3 * b.narrow(0, 0, 3)).exp().sum()

        inputs = (torch.rand(3, requires_grad=True), torch.rand(4, requires_grad=True))
        v = (torch.ones(3, requires_grad=True), torch.ones(4, requires_grad=True))
        out, hvp_val = autogradF.hvp(bar, inputs, v, create_graph=True)
        self._assert_same_struct(hvp_val, inputs)
        self.assertIsNotNone(out.grad_fn)
        self.assertIsNotNone(hvp_val[0].grad_fn)
        self.assertIsNotNone(hvp_val[1].grad_fn)

        gradcheck(lambda *args: autogradF.hvp(bar, args[:2], args[2:], create_graph=True)[1], inputs + v)
        gradgradcheck(lambda *args: autogradF.hvp(bar, args[:2], args[2:], create_graph=True)[1], inputs + v)

        def foo(*args):
            x, y = args[:2]
            v = args[2:]

            x = x.cos()
            val, grad = autogradF.hvp(bar, (x, y), v, create_graph=True)

            return val.cos() + grad[0].cos().sum() + grad[1].cos() + x.cos().sum() + y.cos()

        gradcheck(foo, inputs + v)
        gradgradcheck(foo, inputs + v)

    def test_jacobian_match_vjp_jvp(self):
        def foo(x):
            return x ** 3 + x.sum()

        inputs = torch.rand(4)
        v = torch.rand(4)

        jac = autogradF.jacobian(foo, inputs)
        jvp = autogradF.jvp(foo, inputs, v)[1]
        vjp = autogradF.vjp(foo, inputs, v)[1]

        self.assertEqual(jvp, torch.mm(jac, v.unsqueeze(1)).squeeze(1))
        self.assertEqual(vjp, torch.mm(v.unsqueeze(0), jac).squeeze(0))

    def test_hessian_match_vhp_hvp(self):
        def foo(a):
            return 3 * a.narrow(0, 0, 3).exp().sum()

        inputs = torch.rand(4)
        v = torch.rand(4)

        hes = autogradF.hessian(foo, inputs)
        hvp = autogradF.hvp(foo, inputs, v)[1]
        vhp = autogradF.vhp(foo, inputs, v)[1]

        self.assertEqual(hvp, torch.mm(hes, v.unsqueeze(1)).squeeze(1))
        self.assertEqual(vhp, torch.mm(v.unsqueeze(0), hes).squeeze(0))


# Generic device type autograd tests.
class TestAutogradDeviceType(TestCase):

    def test_min_max_median_backprops_to_all_values(self, device):
        for f in [torch.min, torch.max, torch.median, torch.nanmedian]:
            x1 = torch.tensor([1., 0., 1., 0., 1., 0.], device=device, requires_grad=True)
            x2 = torch.tensor([float('nan'), float('nan'), float('nan')], requires_grad=True)
            for x in [x1, x2]:
                y = f(x)
                y.backward()
                self.assertEqual(x.grad.sum(), 1.)
                self.assertEqual((x.grad == 1 / 3).sum(), 3)

    # skip this test if running on rocm, because in cdist
    # we use __shfl_down_sync on CUDA for fast reduction
    # and it gives incorrect results on rocm platform
    @skipCUDAIfRocm
    def test_cdist(self, device):
        def _test_cdist_for_size(sizex, sizey=None):
            if sizey is None:
                sizey = sizex
            for p in [0, 1, 2, 3, 1.5, 2.5, float('inf')]:
                x = torch.randn(sizex, device=device, dtype=torch.double)
                y = torch.randn(sizey, device=device, dtype=torch.double)
                eps = 1e-6
                # to avoid extremum
                x = x - (((x - y) < eps).double() * 2 * eps)
                x.requires_grad = True
                y.requires_grad = True
                f_args_variable = (x, y)

                def f(a, b):
                    return torch.cdist(a, b, p)
                f_args_tensor = deepcopy(unpack_variables(f_args_variable))
                run_functional_checks(self, "test_cdist", "cdist", f,
                                      True, f_args_variable, f_args_tensor)

        def _test_euclidean_large_cdist(sizex, sizey=None):
            if sizey is None:
                sizey = sizex
            x = torch.randn(sizex, device=device, dtype=torch.float)
            y = torch.randn(sizey, device=device, dtype=torch.float)
            eps = 1e-6
            # to avoid extremum
            x = x - (((x - y) < eps).float() * 2 * eps)
            x.requires_grad = True
            y.requires_grad = True
            dist = torch.cdist(x, y, p=2)
            # Do a backward pass to check that it is valid for large
            # matrices
            loss = dist.sum()
            loss.backward()

        _test_cdist_for_size((S, S))
        _test_cdist_for_size((S, S, S))
        _test_cdist_for_size((3, 5))
        _test_cdist_for_size((2, 3, 5))
        _test_cdist_for_size((1, 2, 3))
        _test_cdist_for_size((1, 1), (S, 1))
        _test_euclidean_large_cdist((2000, 5))

    # Ensure that cdist backward with p<1 does not produce NaNs
    def test_cdist_grad_p_lt_1_no_nan(self, device):
        for p in [0.99, 0.7, 0.5, 0.1, 0.01]:
            x = torch.randn(1, 2, device=device)
            y = x.clone().detach() + torch.tensor([[1., 0.]], device=device)
            x.requires_grad = True
            y.requires_grad = True
            result = torch.cdist(x, y, p=p)
            result.backward(torch.ones_like(result))
            self.assertFalse(torch.isnan(x.grad).any())
            self.assertFalse(torch.isnan(y.grad).any())

    def test_cdist_same_inputs(self, device):
        # Test to detect issues in cdist gradient calculation
        # When the distances are 0
        sizex = (1, 27, 32)
        for p in [0, 1, 2, 3, 1.5, 2.5, float('inf')]:
            x = torch.randn(sizex, device=device, dtype=torch.float)
            dist_grad = torch.randn((1, 27, 27), device=device, dtype=torch.float)
            y = x.clone()
            eps = 1e-6
            x.requires_grad = True
            d = torch.cdist(x, y)
            d.backward(dist_grad)
            # Check that the backward passs does not contain invalid
            # values such as nan or inf
            assert torch.isfinite(x.grad).all()

    def test_parameter_resize(self, device):
        asd = torch.nn.Parameter(torch.ones(16, device=device))

        for i in range(2):
            with torch.no_grad():
                asd.set_(asd[1:])
                asd.grad = None

            m = torch.cat((asd, asd))
            m.sum().backward()

    # NOTE: flaky on ROCm CI
    @skipCUDAIfRocm
    def test_sparse_ctor_getter_backward(self, device):
        # See NOTE [ Sparse: autograd and API ] on the expected behavior of this test
        def _test(size, sparse_dim, nnz, device):
            v_size = [nnz] + list(size[sparse_dim:])
            i = torch.rand(sparse_dim, nnz)
            i.mul_(torch.tensor(size[:sparse_dim]).unsqueeze(1).to(i))
            i = i.to(torch.long)

            inp = torch.randn(v_size, requires_grad=True)
            other = self.genSparseTensor(size, sparse_dim, nnz, is_uncoalesced=True)[0]
            other = other.to(device)

            def fn(v):
                x = torch.sparse_coo_tensor(i, v, size, device=device)
                y = (x + other).coalesce()
                yv = y.values()
                new_v = yv.tanh()
                z = torch.sparse_coo_tensor(y.indices(), new_v, y.size())
                return z.coalesce().values()

            gradcheck(fn, (inp,))
            # FIXME: make gradgradcheck work.
            # gradgradcheck(fn, (inp,))

            # assert that _values is non-differentiable
            with self.assertRaisesRegex(RuntimeError, "does not have a grad_fn"):
                other.detach().requires_grad_()._values().backward(torch.ones_like(other._values()))

        for empty_i, empty_v, empty_nnz in product([True, False], repeat=3):
            sparse_size = [] if empty_i else [2, 1]
            dense_size = [1, 0, 2] if empty_v else [1, 2]
            nnz = 0 if empty_nnz else 5
            _test(sparse_size + dense_size, len(sparse_size), nnz, device)

    # autograd tests via common_method_invocations don't allow input tensors to
    # be sparse (RuntimeError: gradcheck expects all tensor inputs are dense when
    # check_sparse_nnz is set to False.)
    def test_sparse_mask_autograd(self, device):
        tensor = torch.randn(3, requires_grad=True, device=device)
        mask = torch.ones(3, device=device)
        mask[1] = 0
        mask = mask.to_sparse()
        converted = tensor.sparse_mask(mask).to_dense()
        converted.sum().backward()
        self.assertEqual(tensor.grad, mask.to_dense())

    def test_pyscalar_conversions(self, device):
        def _test_pyscalar_conversions(t, integral_conv):
            # integral -> integral
            l = t(torch.zeros(1, 1, 1, dtype=torch.long))
            pyscalar = -12345
            l[0] = pyscalar
            self.assertEqual(integral_conv(l), pyscalar)

            # floating point -> floating point
            f = Variable(t(torch.randn(1, 1)))
            pyscalar = -12345.1
            f[0] = pyscalar
            self.assertEqual(float(f), pyscalar)
            f[0] = nan
            self.assertTrue(math.isnan(float(f)))
            f[0] = inf
            self.assertEqual(float(f), inf)
            f[0] = -inf
            self.assertEqual(float(f), -inf)

            # integral -> floating point
            # check we can convert something that loses precision
            pyscalar = 1234567890123456789
            self.assertNotEqual(pyscalar, integral_conv(float(pyscalar)))
            l[0] = pyscalar
            self.assertEqual(float(l), float(pyscalar))

            # floating point -> integral
            f[0] = nan
            self.assertRaises(ValueError, lambda: integral_conv(f[0]))
            f[0] = inf
            self.assertRaises(OverflowError, lambda: integral_conv(f[0]))
            f[0] = -inf
            self.assertRaises(OverflowError, lambda: integral_conv(f[0]))
            f[0] = sys.float_info.max
            self.assertEqual(integral_conv(f), sys.float_info.max)

            # bool, nonzero
            def test_nonzero(tensor, value, expected):
                tensor[0] = value
                self.assertEqual(expected, bool(tensor))
                self.assertEqual(expected, True if tensor else False)

            test_nonzero(l, 0, False)
            test_nonzero(l, -2, True)
            test_nonzero(f, 0.0, False)
            test_nonzero(f, sys.float_info.min, True)
            test_nonzero(f, nan, bool(nan))
            test_nonzero(f, inf, bool(inf))
            test_nonzero(f, -inf, bool(-inf))


        _test_pyscalar_conversions(lambda x: x.to(device), lambda x: int(x))

    @dtypesIfCUDA(torch.half, torch.float, torch.double, torch.int8, torch.int16, torch.int32, torch.int64)
    @dtypes(torch.float, torch.double, torch.int8, torch.int16, torch.int32, torch.int64)
    def test_set_requires_grad_only_for_floats(self, device, dtype):
        def f1():
            a = torch.ones(1, dtype=dtype, device=device)
            a.requires_grad_()

        def f2():
            a = torch.ones(1, dtype=dtype, device=device)
            a.requires_grad = True

        def f3():
            torch.ones(1, dtype=dtype, device=device, requires_grad=True)

        a = torch.ones(1, dtype=dtype, device=device)
        a.requires_grad = False  # should always work
        a.requires_grad_(False)

        for f in [f1, f2, f3]:
            if dtype.is_floating_point:
                f()
            else:
                with self.assertRaisesRegex(RuntimeError, 'floating point', msg="dt: {} device: {}".format(a.dtype, a.device)):
                    f()

    @onlyCUDA
    def test_advanced_indexing_backwards_large(self, device):
        # See https://github.com/pytorch/pytorch/issues/22843
        n = (1 << 16)
        x = torch.rand(n, 1, device=device, requires_grad=True)
        a = x[:, [0]]
        a.sum().backward()
        self.assertEqual(x.grad, torch.ones(n, 1, device=device))

    def test_advanced_indexing_backwards_memory_format(self, device):
        # See https://github.com/pytorch/pytorch/issues/36956
        shape = (2, 8, 1, 2)
        i = torch.randint(1, shape, device=device).contiguous(memory_format=torch.channels_last)
        x = torch.randn(shape, requires_grad=True, device=device)
        x[i].sum().backward()

    def _test_reentrant_parent_error_on_cpu(self, device):
        t1 = torch.rand([3, 3], requires_grad=True)
        t2 = torch.rand([3, 3], device=device, requires_grad=True)
        t3 = torch.rand([3, 3], device=device, requires_grad=True)

        # Parent graph cpu graph.
        t4 = t1 * t1
        t5 = TestAutograd.SimulateBackwardError.apply(t4)

        # Child gpu graph (much longer than parent graph).
        prev = t2 * t2
        for i in range(10):
            prev = prev * t2
        reentrant_root = prev

        class ReentrantFunc(Function):
            @staticmethod
            def forward(ctx, inp):
                return inp.clone()

            @staticmethod
            def backward(ctx, grad):
                # Reentrant backward in child will take much longer.
                reentrant_root.backward()
                return grad

        # Parent gpu graph.
        t6 = ReentrantFunc.apply(t3)
        t7 = t6 * t6

        # Parent graph will error out first, while child graph will continue executing.
        with self.assertRaisesRegex(Exception, "Simulate error"):
            torch.autograd.backward([t5.sum(), t7.sum()])

        # No grads should be accumulated since child graph will stop execution
        # after parent receives error.
        self.assertIsNone(t2.grad)
        self.assertIsNone(t1.grad)
        self.assertIsNone(t3.grad)

    @onlyCUDA
    def test_reentrant_parent_error_on_cpu(self, device):
        before = CudaMemoryLeakCheck.get_cuda_memory_usage()

        # Run as separate function so that gc can clean up everything when we
        # check for memory usage.
        self._test_reentrant_parent_error_on_cpu(device)

        # Wait for autograd thread to cleanup failed tasks.
        after = CudaMemoryLeakCheck.get_cuda_memory_usage()
        start = time.time()
        while before != after and time.time() - start < 30:
            time.sleep(0.1)
            after = CudaMemoryLeakCheck.get_cuda_memory_usage()

        self.assertEqual(before, after)

    # test for backward in https://github.com/pytorch/pytorch/issues/15511
    def test_pdist_large(self, device):
        def func(x):
            return torch.pdist(x, p=2)

        # shape[0] should be able to be (roughly) arbitrarily large, but the kernel
        # is currently limited to smaller sizes (see issue above); this is just testing
        # a floor.
        shape = (1000, 1)
        x = torch.randn(shape, device=device).requires_grad_()
        output = torch.pdist(x, p=2)
        # just run a single backward, as gradcheck/gradgradcheck is expensive here
        output.sum().backward()

    def test_where_functional(self, device):
        x = torch.randn(5, 5, device=device, requires_grad=True)
        y = torch.randn(5, 5, device=device, requires_grad=True)
        cond = mask_not_all_zeros((5, 5)).to(device=device)

        def where(cond, x, y):
            return torch.where(cond, x, y)

        gradcheck(where, [cond, x, y], raise_exception=True)
        gradgradcheck(where, [cond, x, y], [torch.randn(5, 5, device=device)])

        x = torch.randn(5, 1, 5, device=device, requires_grad=True)
        y = torch.randn(5, 5, 1, device=device, requires_grad=True)
        gradcheck(where, [cond, x, y], raise_exception=True)
        gradgradcheck(where, [cond, x, y], [torch.randn(5, 5, 5, device=device)])

    def test_where_scalar(self, device):
        x = torch.randn(5, 5, device=device, requires_grad=True)
        scalar = 4.
        cond = mask_not_all_zeros((5, 5)).to(device=device)

        def where_scalar_first(cond, x):
            return torch.where(cond, scalar, x)

        def where_scalar_second(cond, x):
            return torch.where(cond, x, scalar)

        gradcheck(where_scalar_first, (cond, x))
        gradgradcheck(where_scalar_first, (cond, x))

        gradcheck(where_scalar_second, (cond, x))
        gradgradcheck(where_scalar_second, (cond, x))

    @skipCUDAIf(True, """Test is flaky on Linux and Windows, typical error message:
            https://github.com/pytorch/pytorch/issues/34870""")
    def test_ctc_loss(self, device):
        batch_size = 64
        num_labels = 101
        target_length = 15
        gradcheck_input_size = 10

        ZERO_NONE = 0
        ZERO_SOME = 1
        ZERO_ALL = 2

        # input_length, vary_lengths, zero_lengths
        tests = [(150, False, ZERO_NONE),
                 (150, True, ZERO_NONE),
                 (50, True, ZERO_SOME),
                 (50, True, ZERO_ALL)]

        if 'cuda' in device:
            tests += [(50, False, ZERO_NONE),
                      (50, True, ZERO_NONE),
                      (150, True, ZERO_SOME),
                      (150, True, ZERO_ALL)]

        for input_length, vary_lengths, zero_mode in tests:
            targets = torch.randint(1, num_labels, (batch_size, target_length),
                                    device=device, dtype=torch.long)
            x = torch.randn(gradcheck_input_size, device=device, requires_grad=True)
            tile_factors = torch.randn(input_length * batch_size * num_labels // gradcheck_input_size + 1,
                                       device=device)
            input_lengths = [(torch.randint(input_length // 2, input_length + 1, ()).item()
                              if vary_lengths or i == 0 else input_length) for i in range(batch_size)]
            if zero_mode == ZERO_ALL:
                target_lengths = [0 for _ in range(batch_size)]
            else:
                target_lengths = [(torch.randint(target_length // 2, target_length + 1, ()).item()
                                   if vary_lengths else target_length) for _ in range(batch_size)]
                if zero_mode == ZERO_SOME:
                    idxes = torch.randint(0, batch_size, (10,))
                    for i in idxes:
                        target_lengths[i] = 0

            def ctc_after_softmax(x):
                x_full = ((x[:, None] * tile_factors[None, :]).view(-1)[:input_length * batch_size * num_labels]
                          .view(input_length, batch_size, num_labels))
                log_probs = torch.log_softmax(x_full, 2)
                return torch.nn.functional.ctc_loss(log_probs, targets, input_lengths, target_lengths)

            gradcheck(ctc_after_softmax, [x])

    @onlyCUDA
    @skipCUDAIfRocm
    @skipCUDAIfCudnnVersionLessThan(7600)
    def test_ctc_loss_cudnn(self, device):
        batch_size = 16
        input_length = 30
        num_labels = 101
        target_length = 15
        targets = torch.randint(1, num_labels, (batch_size * target_length,),
                                device='cuda', dtype=torch.long)
        log_probs = torch.log_softmax(torch.randn(input_length, batch_size, num_labels, device='cuda', dtype=torch.float), 2)
        log_probs.requires_grad_()

        input_lengths = batch_size * [input_length]
        target_lengths = batch_size * [target_length]
        grad_out = torch.randn(batch_size, device='cuda', dtype=torch.float)
        with torch.backends.cudnn.flags(enabled=False):
            loss_native = torch.nn.functional.ctc_loss(log_probs, targets, input_lengths, target_lengths, reduction='none')
            grad_native, = torch.autograd.grad(loss_native, log_probs, grad_out)
        loss_cudnn = torch.nn.functional.ctc_loss(log_probs, targets.to('cpu', torch.int32),
                                                  input_lengths, target_lengths, reduction='none')
        self.assertTrue("Cudnn" in str(loss_cudnn.grad_fn))
        grad_cudnn, = torch.autograd.grad(loss_cudnn, log_probs, grad_out)
        self.assertEqual(grad_cudnn, grad_native, atol=1e-4, rtol=0)

    @skipCUDAIfRocm
    def test_leaky_relu_inplace_with_neg_slope(self, device):
        a = torch.tensor([-1., 1.], device=device, requires_grad=True)
        b = torch.nn.functional.leaky_relu_(a.clone(), -2)
        with self.assertRaisesRegex(RuntimeError, "call out-of-place version"):
            b.backward(torch.ones(2, device=device))

        a = torch.tensor([-1., 1.], device=device, requires_grad=True)
        b = torch.nn.functional.rrelu_(a.clone(), -5.0, 1.0)
        with self.assertRaisesRegex(RuntimeError, "call out-of-place version"):
            b.backward(torch.ones(2, device=device))

    @skipCUDAIfRocm
    def test_leaky_relu_inplace_with_zero_slope(self, device):
        a = torch.tensor([-2., 0., 2.], device=device, requires_grad=True)
        b = torch.nn.functional.leaky_relu_(a.clone(), 0.0)
        b.backward(torch.ones(3, device=device))
        expected = torch.tensor([0., 0., 1.], device=device)
        self.assertEqual(a.grad, expected)

    @onlyCUDA
    def test_free_unneeded_tensor(self, device):
        x = torch.randn(2, 3, 10, 10, device=device, requires_grad=True)
        m = torch.randn(1, 3, 1, 1, device=device)

        z = x.sum()
        base_mem = torch.cuda.memory_allocated()
        z = ((x + 2) * m).sum()
        end_mem = torch.cuda.memory_allocated()

        # In the end the memory usage should remain equal, because neither of
        # (x + 2) and ((x + 2) * m) should be kept alive for backward, while the
        # previous allocation of z had the same size as the current one.
        self.assertEqual(base_mem, end_mem)

    @onlyCUDA
    def test_pin_memory(self, device):
        x = torch.randn(2, 2, requires_grad=True)
        self.assertEqual(x, x.pin_memory())
        self.assertIsNot(x, x.pin_memory())
        self.assertTrue(x.pin_memory().requires_grad)
        gradcheck(lambda x: x.pin_memory(), [x])
        gradgradcheck(lambda x: x.pin_memory(), [x])

    @skipCUDAIfRocm
    @onlyCUDA
    def test_profiler_emit_nvtx(self, device):
        # This test is not intended to ensure correctness of nvtx ranges.
        # That would require something a great deal more complex (you'd have to create a
        # profile in a subprocess, open it, and parse the sql somehow).
        # This test is merely intended to catch if emit_nvtx breaks on construction.
        a = torch.tensor([1, 2, 3], dtype=torch.float32, device=device)
        with torch.cuda.profiler.profile():
            with emit_nvtx():
                a.add(1.0)

    @onlyCUDA
    def test_rnn_backward_to_input_but_not_parameters(self, device):
        # this checks whether it is possible to not require
        # weight parameters, but require inputs, see #7722
        l = torch.nn.LSTM(2, 3).to(device)
        for p in l.parameters():
            p.requires_grad = False
        s = torch.randn(1, 1, 2, requires_grad=True, device=device)
        out, _ = l(s)
        out.sum().backward()
        self.assertFalse(s.grad is None or s.grad.abs().sum().item() == 0)

    @onlyCUDA
    def test_lstmcell_backward_only_one_output_grad(self, device):
        # checks that undefined gradients doen't hamper the backward
        # see #11872
        l = torch.nn.LSTMCell(2, 3).to(device).double()
        s = torch.randn(1, 2, device=device, dtype=torch.double, requires_grad=True)
        for i in range(2):
            out = l(s)[i]
            out.sum().backward()
            self.assertFalse(s.grad is None or s.grad.abs().sum().item() == 0)

    def _test_rnn_mod(self, mod, inp):
        from functools import partial

        def flatten_out(mod, inp):
            out = mod(inp)
            return tuple([t if isinstance(t, torch.Tensor) else tt for t in out for tt in t])
        gradcheckfunc = partial(flatten_out, mod)
        with torch.backends.cudnn.flags(enabled=False):
            torch.autograd.gradcheck(gradcheckfunc, inp)
            torch.autograd.gradgradcheck(gradcheckfunc, inp)

    def test_LSTM_grad_and_gradgrad(self, device):
        hsize = 4
        inp = torch.rand(1, 3, hsize, device=device, dtype=torch.float64, requires_grad=True)
        for bias in [True, False]:
            mod = torch.nn.LSTM(hsize, hsize, bias=bias).to(device).to(torch.float64)
            self._test_rnn_mod(mod, inp)

    def test_GRU_grad_and_gradgrad(self, device):
        hsize = 4
        inp = torch.rand(1, 3, hsize, device=device, dtype=torch.float64, requires_grad=True)
        for bias in [True, False]:
            mod = torch.nn.GRU(hsize, hsize, bias=bias).to(device).to(torch.float64)
            self._test_rnn_mod(mod, inp)

    def test_copysign_subgradient(self, device):
        # Input is 0.0
        x = torch.tensor([0.0, 0.0, 0.0], dtype=torch.float, device=device, requires_grad=True)
        y = torch.tensor([-1.0, 0.0, 1.0], dtype=torch.float, device=device, requires_grad=True)
        out = torch.copysign(x, y)
        out.sum().backward()
        self.assertEqual(x.grad.tolist(), [0.0, 0.0, 0.0])
        self.assertEqual(y.grad.tolist(), [0.0] * 3)

        # Input is -0.0
        x = torch.tensor([-0.0, -0.0, -0.0], dtype=torch.float, device=device, requires_grad=True)
        y = torch.tensor([-1.0, 0.0, 1.0], dtype=torch.float, device=device, requires_grad=True)
        out = torch.copysign(x, y)
        out.sum().backward()
        self.assertEqual(x.grad.tolist(), [0.0, 0.0, 0.0])
        self.assertEqual(y.grad.tolist(), [0.0] * 3)

        # Other is 0.0
        x = torch.tensor([-1.0, 0.0, 1.0], dtype=torch.float, device=device, requires_grad=True)
        y = torch.tensor([0.0, 0.0, 0.0], dtype=torch.float, device=device, requires_grad=True)
        out = torch.copysign(x, y)
        out.sum().backward()
        self.assertEqual(x.grad.tolist(), [-1.0, 0.0, 1.0])
        self.assertEqual(y.grad.tolist(), [0.0] * 3)

        # Other is -0.0
        x = torch.tensor([-1.0, 0.0, 1.0], dtype=torch.float, device=device, requires_grad=True)
        y = torch.tensor([-0.0, -0.0, -0.0], dtype=torch.float, device=device, requires_grad=True)
        out = torch.copysign(x, y)
        out.sum().backward()
        self.assertEqual(x.grad.tolist(), [1.0, 0.0, -1.0])
        self.assertEqual(y.grad.tolist(), [0.0] * 3)

    @deviceCountAtLeast(1)
    def test_grad_assignment(self, devices):
        x = torch.randn(5, 5, device=devices[0])

        # Tests that the wrong shape raises
        with self.assertRaises(RuntimeError):
            x.grad = torch.randn(2, 2, device=devices[0])

        # Tests that the wrong dtype raises
        with self.assertRaises(RuntimeError):
            x.grad = torch.randn(5, 5, dtype=torch.long, device=devices[0])

        # Tests that self-assignment raises
        with self.assertRaises(RuntimeError):
            x.grad = x

        # Tests device -> cpu grad assignment raises
        if self.device_type != 'cpu':
            with self.assertRaises(RuntimeError):
                t_cpu = torch.rand(5, 5)
                t_cpu.grad = torch.randn(5, 5, device=devices[0])

        # Tests half type on CUDA
        if self.device_type == 'cuda':
            x = x.to(dtype=torch.half, device=devices[0])
            x.grad = torch.zeros_like(x)

        # Tests cross-device assignment raises
        if len(devices) > 1:
            x = torch.randn(5, 5, device=devices[0])
            with self.assertRaises(RuntimeError):
                x.grad = torch.randn(5, 5, device=devices[1])

    @deviceCountAtLeast(1)
    @dtypes(torch.float, torch.double)
    def test_requires_grad_factory(self, devices, dtype):
        fns = [torch.ones_like, torch.testing.randn_like]
        x = torch.randn(2, 3, dtype=dtype, device=devices[0])

        for fn in fns:
            for requires_grad in [True, False]:
                output = fn(x, dtype=dtype, device=devices[0], requires_grad=requires_grad)
                self.assertEqual(requires_grad, output.requires_grad)
                self.assertIs(dtype, output.dtype)
                self.assertEqual(devices[0], str(x.device))

    @deviceCountAtLeast(2)
    def test_unused_output_device(self, devices):
        from torch.nn.parallel._functions import Broadcast
        x = torch.randn(5, 5, dtype=torch.float, device=devices[0], requires_grad=True)
        outputs = Broadcast.apply(list(range(len(devices))), x)
        y = outputs[-1] * 2
        y.sum().backward()
        # TODO(#38095): Replace assertEqualIgnoreType. See issue #38095
        self.assertEqualIgnoreType(x.grad, torch.ones(5, 5) * 2)

    @deviceCountAtLeast(2)
    def test_backward_device(self, devices):
        # check that current device matches the variable's device
        device = [None]

        class Identity(torch.autograd.Function):
            @staticmethod
            def forward(ctx, x):
                return x.clone()

            @staticmethod
            def backward(ctx, grad_output):
                device[0] = grad_output.device
                return grad_output.clone()

        v = torch.randn(1, device=devices[1], requires_grad=True)
        Identity.apply(v).backward()
        self.assertEqual(str(device[0]), devices[1])

    @deviceCountAtLeast(2)
    def test_inputbuffer_add_multidevice(self, devices):
        input = torch.randn(1, device=devices[0], requires_grad=True)
        output = input.to(device=devices[1]) + input.to(device=devices[1])
        output.backward()

    @onlyCPU
    def test_copy_(self, device):
        # At the time of writing this test, copy_ is not generated from native_functions.yaml
        # there was a bug that bfloat16 was not recognized as floating.
        x = torch.randn(10, device=device, requires_grad=True)
        floating_dt = [dt for dt in torch.testing.get_all_dtypes() if dt.is_floating_point]
        for dt in floating_dt:
            y = torch.empty(10, device=device, dtype=dt)
            y.copy_(x)
            self.assertTrue(y.requires_grad)
            z = x.to(torch.bfloat16)
            self.assertTrue(z.requires_grad)

    @onlyCUDA
    def test_simple_reentrant_cross_device(self, device):
        class ReentrantFunc(Function):
            _cpu_mode = True

            @staticmethod
            def forward(ctx, x):
                return x * (x + 2)

            @staticmethod
            def backward(ctx, grad_output):
                with torch.enable_grad():
                    if ReentrantFunc._cpu_mode:
                        new_param = torch.randn(2, 2, requires_grad=True)
                        (new_param ** 2).sum().backward()
                    else:
                        new_param = torch.randn(2, 2, device=device, requires_grad=True)
                        (new_param ** 2).sum().backward()
                return grad_output

        # Reentrant starts on GPU thread, finishs on GPU thread
        x = torch.randn(2, 2, device=device, requires_grad=True)
        out = ReentrantFunc.apply(x)
        out.sum().backward()

        # Reentrant starts on CPU thread, finishs on GPU thread
        x = torch.randn(2, 2, requires_grad=True)
        # set ReentrantFunc node to GPU to emit tasks to GPU queue
        ReentrantFunc._cpu_mode = False
        out = ReentrantFunc.apply(x)
        out.sum().backward()

        # Reentrant starts on GPU thread, finishs on CPU thread
        x = torch.randn(2, 2, device=device, requires_grad=True)
        # set ReentrantFunc node to CPU to emit tasks to CPU queue
        ReentrantFunc._cpu_mode = True
        out = ReentrantFunc.apply(x)
        out.sum().backward()

    @onlyCUDA
    def test_cross_device_reentrant_autograd(self, device):
        # Output on gpu so that this task will be associated with the gpu thread
        def fn_on_gpu(inp):
            # Artificially increase the priority of the next op to make sure it runs
            # as soon as we reach it before the ops of branch1.
            dummy = inp * 2 * 2 * 2 * 2
            return inp.to(device=device)

        def parent_on_cpu(inp):
            # Slow branch of ops on gpu so that the work queue for the gpu thread
            # won't empty too quickly. They also have smaller priorities than the
            # ones created by fn_on_gpu
            branch1 = inp.to(device=device)
            branch1 = branch1 / branch1
            branch1 = branch1 / branch1
            branch1 = branch1 / branch1
            # Perform checkpoint on cpu tensors. So the last op performed in the reentrant
            # autograd is an AccumulateGrad that runs on the cpu thread for the gpu thread.
            # So the cpu thread will notify the gpu thread with an empty NodeTask.
            branch2 = checkpoint(fn_on_gpu, inp)
            out = branch2 + branch1
            return out

        inp = torch.rand(2, requires_grad=True)
        out = parent_on_cpu(inp)
        # This will segfault if the empty NodeTask is not handled properly in the
        # gpu thread ReadyQueue
        out.sum().backward()

    def test_inplace_view_backprop_base(self, device):
        # modify view and back-prop through base
        root = torch.randn(2, 2, device=device, requires_grad=True)
        x = root.clone()
        v1 = x.narrow(0, 0, 1)
        v1.mul_(2)
        x.sum().backward()
        self.assertEqual(root.grad.tolist(), [[2, 2], [1, 1]])

    def test_inplace_view_backprop_view_of_view(self, device):
        # modify view and backprop through view-of-view
        root = torch.randn(2, 2, device=device, requires_grad=True)
        x = root.clone()
        v1 = x.narrow(0, 0, 1)
        v2 = x.narrow(0, 0, 1)
        v1.mul_(2)
        v2.sum().backward()
        self.assertEqual(root.grad.tolist(), [[2, 2], [0, 0]])

    def test_inplace_view_of_view(self, device):
        # modify view-of-view and backprop through base
        root = torch.randn(2, 2, device=device, requires_grad=True)
        x = root.clone()
        v1 = x.narrow(0, 0, 1)
        v2 = v1.narrow(1, 1, 1)
        v2.mul_(2)
        x.sum().backward()
        self.assertEqual(root.grad.tolist(), [[1, 2], [1, 1]])

    def test_inplace_view_gradcheck(self, device):
        # gradcheck modifications to views
        a = torch.randn(4, 4, device=device, requires_grad=True)
        b = torch.randn(2, 2, device=device, requires_grad=True)

        def func(root, b):
            x = root.clone()
            x.narrow(1, 2, 2).narrow(0, 1, 2).mul_(b)
            x.narrow(1, 0, 2).narrow(0, 1, 2).mul_(b)
            return x

        gradcheck(func, [a, b], raise_exception=True)
        go = torch.randn(a.size(), device=device, requires_grad=True)
        gradgradcheck(func, (a, b), (go,))

    def test_inplace_view_multiple_outputs(self, device):
        root = torch.arange(9.).reshape(3, 3).requires_grad_()
        x = root.clone()
        v1 = x.unbind()
        with self.assertRaises(RuntimeError):
            v1[0].mul_(2)

    def test_inplace_view_makes_base_require_grad(self, device):
        # in-place modification to view makes base require grad
        a = torch.randn(4, 4, device=device, requires_grad=False)
        b = torch.randn(4, 2, device=device, requires_grad=True)

        def func(root, b):
            x = root.clone()
            self.assertFalse(x.requires_grad)
            x.narrow(1, 2, 2).mul_(b)
            self.assertTrue(x.requires_grad)
            return x

        gradcheck(func, [a, b], raise_exception=True)
        go = torch.randn(a.size(), device=device, requires_grad=True)
        gradgradcheck(func, (a, b), (go,))

    def test_inplace_view_backprop_view(self, device):
        # modify view and backprop through view
        a = torch.tensor([2., 5.], device=device, requires_grad=False)
        b = torch.tensor([3.], device=device, requires_grad=True)
        res = a.narrow(0, 1, 1).mul_(b)
        res.sum().backward()
        self.assertEqual(b.grad.tolist(), [5])
        self.assertIsNone(a.grad)

    def test_inplace_view_modify_base(self, device):
        # Test that an in-place operation on a base that forced it to require
        # grad also forces any previous views to require grad and backprop
        # correctly
        r = torch.ones(1, device=device, requires_grad=True)

        def fn(r):
            x = torch.ones(5, device=device)
            v = x.select(0, 1)
            self.assertFalse(v.requires_grad)
            self.assertIsNone(v.grad_fn)
            x.add_(r)  # v is now dependent on r due to the in-place op on x
            self.assertTrue(v.requires_grad)
            return v

        gradcheck(fn, [r])
        gradgradcheck(fn, [r])

    def test_inplace_view_python(self, device):
        # in-place modifications of Python-autograd created view
        a = torch.randn(4, 4, device=device, requires_grad=True)
        b = torch.randn(2, 2, device=device, requires_grad=True)

        class PyAdd(torch.autograd.Function):
            @staticmethod
            def forward(ctx, x, y):
                ctx.mark_dirty(x)
                x.add_(y)
                return x

            @staticmethod
            def backward(ctx, grad):
                return grad, grad

        def func(root, b):
            x = root.clone()
            PyAdd.apply(x.narrow(1, 2, 2).narrow(0, 1, 2), b)
            PyAdd.apply(x.narrow(1, 0, 2).narrow(0, 1, 2), b)
            return x

        gradcheck(func, [a, b], raise_exception=True)
        go = torch.randn(a.size(), device=device, requires_grad=True)
        gradgradcheck(func, (a, b), (go,))

    def test_inplace_view_non_contig(self, device):
        root = torch.ones(2, 3, 2, device=device).select(2, 1).t().requires_grad_(True)
        x = root.clone()
        v1 = x.narrow(0, 0, 1)
        v2 = v1.narrow(1, 1, 1)
        v2.mul_(2)
        x.sum().backward()
        self.assertEqual(root.grad.tolist(), [[1, 2], [1, 1], [1, 1]])

    def test_inplace_view_multi_output_unsafe(self, device):
        for f in [lambda t: t.unsafe_split(1),
                  lambda t: t.unsafe_split_with_sizes((1, 1, 1)),
                  lambda t: t.unsafe_chunk(3)]:
            a = torch.randn(3, 3, device=device, requires_grad=True)
            b = a + a
            s1, s2, s3 = f(b)
            s1.mul_(s2)
            s1.sum().backward()

    def test_inplace_view_multi_output_safe(self, device):
        for f in [lambda t: t.split(1),
                  lambda t: t.split_with_sizes((1, 1, 1)),
                  lambda t: t.chunk(3)]:
            a = torch.randn(3, 3, device=device, requires_grad=True)
            b = a + a
            s1, s2, s3 = f(b)
            with warnings.catch_warnings(record=True) as w:
                s1.mul_(s2)
            self.assertIn('Consider using `unsafe_` version', str(w[0].message))

    def test_mv_grad_stride_0(self, device):
        # Reference: https://github.com/pytorch/pytorch/issues/38315
        mat = torch.randn(2, 2, device=device)
        vec = torch.randn(1, device=device).requires_grad_(True)

        def fn(vec):
            # Expand inside the function to make sure the input to
            # gradcheck does not have overlapping memory
            vec = vec.expand(2)
            return (mat @ vec).sum()

        gradcheck(fn, (vec))
        gradgradcheck(fn, (vec))

    def test_logcumsumexp_large_value(self, device):
        a = torch.rand(4, 4, 4, dtype=torch.double, requires_grad=True)
        with torch.no_grad():
            # Large Number
            a[0] = 10000

        gradcheck(lambda x: x.logcumsumexp(0), a)
        gradgradcheck(lambda x: x.logcumsumexp(0), a)

        gradcheck(lambda x: x.logcumsumexp(1), a)
        gradgradcheck(lambda x: x.logcumsumexp(1), a)

        gradcheck(lambda x: x.logcumsumexp(2), a)
        gradgradcheck(lambda x: x.logcumsumexp(2), a)

    @slowTest
    def test_lu_backward(self, device):
        def run_test(*sizes):
            x = torch.rand(*sizes, device=device, dtype=torch.double).requires_grad_(True)

            gradcheck(lambda x: x.lu(get_infos=True), x)
            gradgradcheck(lambda x: x.lu(get_infos=True), x)

            gradcheck(lambda x: x.lu(get_infos=False), x)
            gradgradcheck(lambda x: x.lu(get_infos=False), x)

            # there is no pivot-less LU factorization on CPU
            if x.device.type == 'cuda':
                gradcheck(lambda x: x.lu(pivot=False, get_infos=True), x)
                gradgradcheck(lambda x: x.lu(pivot=False, get_infos=True), x)

                gradcheck(lambda x: x.lu(pivot=False, get_infos=False), x)
                gradgradcheck(lambda x: x.lu(pivot=False, get_infos=False), x)

        run_test(3, 3)
        run_test(3, 3, 3)
        run_test(3, 3, 3, 3)
        run_test(5, 5)
        run_test(3, 5, 5)
        run_test(3, 3, 5, 5)

    def test_strided_leaf_grad_layout(self, device):
        # (1) If leaf is non-overlapping and dense, grad's layout should match its leaf.
        for fmt_a in (torch.contiguous_format, torch.channels_last):
            for fmt_b in (torch.contiguous_format, torch.channels_last):
                a = torch.rand((2, 3, 4, 5), device=device).to(memory_format=fmt_a)
                b = torch.rand((2, 3, 4, 5), device=device).to(memory_format=fmt_b)
                a.requires_grad_()
                b.requires_grad_()
                # checks (1) for broadcasted gradients
                a.sum().backward()
                self.assertEqual(a.grad.stride(), a.stride())
                b.sum().backward()
                self.assertEqual(b.grad.stride(), b.stride())
                # checks (1) for non-broadcasted gradients
                a.grad = None
                b.grad = None
                (a * b).sum().backward()
                self.assertEqual(a.grad.stride(), a.stride())
                self.assertEqual(b.grad.stride(), b.stride())

        # (2) If leaf isn't dense, checks that grads are rowmajor contiguous.
        c = torch.empty_strided((2, 2), (4, 2), device=device).copy_(torch.rand((2, 2), device=device))
        c.requires_grad_()
        d = torch.rand((2, 2), device=device)
        # checks (2) for broadcasted gradients
        c.sum().backward()
        self.assertEqual(c.grad.stride(), (2, 1))
        # checks (2) for non-broadcasted gradients
        c.grad = None
        (c * d).sum().backward()
        self.assertEqual(c.grad.stride(), (2, 1))

    def test_movedim(self, device):
        x = torch.randn(4, 3, 2, 1, dtype=torch.double, device=device, requires_grad=True)

        # Positive axis
        gradcheck(lambda x: torch.movedim(x, (0, 1, 2, 3), (3, 2, 1, 0)), x)
        gradgradcheck(lambda x: torch.movedim(x, (0, 1, 2, 3), (3, 2, 1, 0)), x)

        # Negative axis
        gradcheck(lambda x: torch.movedim(x, (0, -1, -2, -3), (-3, -2, -1, -0)), x)
        gradgradcheck(lambda x: torch.movedim(x, (0, -1, -2, -3), (-3, -2, -1, -0)), x)

    def _test_atleast(self, device, torch_fn):
        # 0-dim
        s = torch.tensor(0.5, dtype=torch.double, requires_grad=True)

        gradcheck(lambda x: torch_fn(x), s)
        gradgradcheck(lambda x: torch_fn(x), s)

        # 1-dim
        a = torch.rand(4, dtype=torch.double, requires_grad=True)

        gradcheck(lambda x: torch_fn(x), a)
        gradgradcheck(lambda x: torch_fn(x), a)

        # 2,3,4-dim
        b = torch.rand(4, 3, dtype=torch.double, requires_grad=True)
        c = torch.rand(4, 3, 2, dtype=torch.double, requires_grad=True)
        d = torch.rand(4, 3, 2, 1, dtype=torch.double, requires_grad=True)

        input_tuple = (s, a, b, c, d)
        gradcheck(lambda s, w, x, y, z: torch_fn(s, w, x, y, z), input_tuple)
        gradgradcheck(lambda s, w, x, y, z: torch_fn(s, w, x, y, z), input_tuple)

    def test_atleast(self, device):
        self._test_atleast(device, torch.atleast_1d)
        self._test_atleast(device, torch.atleast_2d)
        self._test_atleast(device, torch.atleast_3d)

class TestMultithreadAutograd(TestCase):
    def _run_py_multithread_fn(self, fn, args=(), num_threads=10, kwargs=None):
        threads = []
        for _ in range(num_threads):
            p = threading.Thread(target=fn, args=(args))
            p.start()
            threads.append(p)

        for p in threads:
            p.join()

    def test_simple_backward(self):
        # simple multithreaded backward that create threads in the beginning of training
        # and everything else is training separately, i.e. inputs, operations, etc.
        def train_fn():
            x = torch.ones(5, 5, requires_grad=True)
            y = (x + 3) * (x + 4) * 0.5
            y.sum().backward()
            self.assertEqual(x.grad, x + 3.5)

        self._run_py_multithread_fn(train_fn)

    def test_simple_backward_same_input(self):
        # simple multithreaded backward with only shared inputs (i.e. This is common
        # for things like Hogwild multithreaded training with multiple CPU threads)
        def train_fn_backward(x):
            y = (x + 3) * (x + 4) * 0.5
            y.sum().backward()

        x = torch.ones(5, 5, requires_grad=True)
        self._run_py_multithread_fn(train_fn_backward, (x,))
        # Since we are calling backward from multiple threads
        # and all threads share the same input, when we do backward
        # concurrently, different backwards will all accumulate to
        # the same .grad for each input, and the gradients should
        # be equal to num_threads * gradient
        self.assertEqual(x.grad, 10 * (x + 3.5))

        def train_fn_grad(x):
            y = (x + 3) * (x + 4) * 0.5
            grads = torch.autograd.grad(y.sum(), x)
            self.assertEqual(len(grads), 1)
            self.assertEqual(grads[0], x + 3.5)

        # since we use functional grad() api, gradients will not
        # be accumulate to the same place and should be the same
        self._run_py_multithread_fn(train_fn_grad, (x,))

    def test_python_thread_in_middle(self):
        # User might write a network that starts on one CPU thread, then runs its second half
        # concurrently with other threads (either via python threading or fork/join calls),
        # then calls backward()/grad() on BOTH threads, like a Y pattern from input at the
        # bottom to output at the top. This way part of the GraphTask is being shared across
        # different threads and we need to ensure user specify retain_graph=True, otherwise
        # error out with the correct error message

        # Case 1: multiple backward with python threads, retain_graph=False
        # should throw error in some threads with no retain_graph.
        success_vs_raises = [0, 0]

        def train_fn_no_retain_graph(x):
            y = x + x ** 2
            try:
                y.sum().backward()
                success_vs_raises[0] += 1
            except RuntimeError as error:
                success_vs_raises[1] += 1
                self.assertRegex(str(error), "Specify retain_graph=True")

        x_no_retain = torch.ones(5, 5, requires_grad=True)
        y_no_retain = x_no_retain + x_no_retain ** 2
        self._run_py_multithread_fn(train_fn_no_retain_graph, (y_no_retain,), num_threads=5)
        # at least one thread will be success in this case, all other threads should raise
        # with the error that throw to user to recommend them specify retain_graph=True
        self.assertTrue(success_vs_raises[0] >= 1)

        # multiple backward with python threads, no error with retain_graph=True
        def train_fn_retain_graph(x):
            y = x + x ** 2
            y.sum().backward(retain_graph=True)

        x_retain = torch.ones(5, 5, requires_grad=True)
        y_retain = x_retain + x_retain ** 2
        self._run_py_multithread_fn(train_fn_retain_graph, (y_retain,), num_threads=5)
        # result should equal to num_thread * gradients
        self.assertEqual(x_retain.grad, 5 * (4 * x_retain ** 3 + 6 * (x_retain ** 2) + 4 * x_retain + 1))

    def test_fork_join_in_middle(self):
        # multiple backward with jit threads (fork/join primitive)
        # similar to test_python_thread_in_middle, we test with retain_graph=False/True

        # Case 1: multiple grad() calls with jit threads, retain_graph=False
        # should throw error in some threads with no retain_graph.
        @torch.jit.script
        def train_fn_jit_no_retain(middle, orig_x):
            y = middle + middle ** 2
            return torch.autograd.grad([y.sum()], [orig_x])

        @torch.jit.script
        def train_fn_fork_join_calls_no_retain(x):
            y_no_retain = (x + 3) * (x + 4) * 0.5

            fut = torch.jit._fork(train_fn_jit_no_retain, y_no_retain, x)
            grad_hat = train_fn_jit_no_retain(y_no_retain, x)
            grad = torch.jit._wait(fut)
            return grad, grad_hat

        try:
            train_fn_fork_join_calls_no_retain(torch.randn(5, 5, requires_grad=True))
        except RuntimeError as error:
            self.assertRegex(str(error), "Specify retain_graph=True")

        # Case 2: no error with retain_graph=True
        @torch.jit.script
        def train_fn_jit_retain(middle, orig_x):
            y = middle + middle ** 2
            return torch.autograd.grad([y.sum()], [orig_x], retain_graph=True)

        @torch.jit.script
        def train_fn_fork_join_calls_retain(x):
            y_retain = (x + 3) * (x + 4) * 0.5
            fut1 = torch.jit._fork(train_fn_jit_retain, y_retain, x)
            fut2 = torch.jit._fork(train_fn_jit_retain, y_retain, x)
            grad = train_fn_jit_retain(y_retain, x)
            grad1 = torch.jit._wait(fut1)
            grad2 = torch.jit._wait(fut2)
            return grad, grad1, grad2

        grad, grad1, grad2 = train_fn_fork_join_calls_retain(torch.randn(5, 5, requires_grad=True))
        self.assertEqual(grad, grad1)
        self.assertEqual(grad, grad2)

    def test_preserve_backtrace(self):
        class Foo(torch.autograd.Function):
            @staticmethod
            def forward(ctx, input):
                return input

            @staticmethod
            def backward(ctx, *grad):
                raise ValueError("something")

        t = torch.rand(10, requires_grad=True)
        try:
            Foo.apply(t).sum().backward()
        except Exception:
            import traceback
            tb = sys.exc_info()[2]
            tb_str = "\n".join(traceback.format_tb(tb))
            self.assertTrue('raise ValueError("something")' in tb_str)

for test in method_tests():
    add_test(*test)


# e.g., TestAutogradDeviceTypeCPU and TestAutogradDeviceTypeCUDA
instantiate_device_type_tests(
    TestAutogradDeviceType,
    globals(),
    # Exclude ROCM for now, there are a lot of failures.  See
    # https://github.com/pytorch/pytorch/issues/30845
    except_for='cuda' if TEST_WITH_ROCM else None
)

if __name__ == '__main__':
    run_tests()