backprop.py

# Copyright 2017 The TensorFlow Authors. All Rights Reserved.
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
#     http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
# ==============================================================================
"""Code for backpropagation using the tape utilities."""

# TODO(b/159343581): Properly support CompositeTensor in all functions in this
# file.

import functools
import operator

from tensorflow.python import pywrap_tfe
from tensorflow.python.eager import backprop_util
from tensorflow.python.eager import context
from tensorflow.python.eager import execute
from tensorflow.python.eager import imperative_grad
from tensorflow.python.eager import tape
from tensorflow.python.framework import composite_tensor
from tensorflow.python.framework import composite_tensor_gradient
from tensorflow.python.framework import constant_op
from tensorflow.python.framework import dtypes
from tensorflow.python.framework import indexed_slices
from tensorflow.python.framework import ops
from tensorflow.python.framework import tensor_shape
from tensorflow.python.framework import tensor_util
from tensorflow.python.framework import type_spec
from tensorflow.python.ops import array_ops
from tensorflow.python.ops import check_ops
from tensorflow.python.ops import control_flow_util
from tensorflow.python.ops import default_gradient
from tensorflow.python.ops import gen_array_ops
from tensorflow.python.ops import gen_math_ops
from tensorflow.python.ops import resource_variable_ops
from tensorflow.python.ops.parallel_for import control_flow_ops as pfor_ops
from tensorflow.python.ops.unconnected_gradients import UnconnectedGradients
from tensorflow.python.platform import tf_logging as logging
from tensorflow.python.util import _pywrap_utils
from tensorflow.python.util import nest
from tensorflow.python.util import tf_contextlib
from tensorflow.python.util import tf_inspect
from tensorflow.python.util import variable_utils
from tensorflow.python.util.tf_export import tf_export


_op_attr_type_cache = {}


def op_attr_type(op_type, attr_name):
  try:
    return _op_attr_type_cache[(op_type, attr_name)]
  except KeyError:
    context.ensure_initialized()
    h = context.context()._handle  # pylint: disable=protected-access
    attr_type = pywrap_tfe.TFE_OpNameGetAttrType(h, op_type, attr_name)
  _op_attr_type_cache[(op_type, attr_name)] = attr_type
  return attr_type


def make_attr(attr_type, value):
  # pybind11 enums do not return the raw value like SWIG enums do. They are
  # useful when comparing amongst each other but not direct integers as we are
  # doing in most tests.
  # https://pybind11.readthedocs.io/en/stable/classes.html#enumerations-and-internal-types
  # TODO(amitpatankar): After all SWIG transitions, convert the enum comparisons
  # from integer value to class.
  if attr_type == int(pywrap_tfe.TF_ATTR_TYPE):
    return dtypes.as_dtype(value)
  if attr_type == [int(pywrap_tfe.TF_ATTR_TYPE)]:
    return [dtypes.as_dtype(v) for v in value]
  if attr_type == int(pywrap_tfe.TF_ATTR_SHAPE):
    return tensor_shape.as_shape(value).as_proto()
  if attr_type == [int(pywrap_tfe.TF_ATTR_SHAPE)]:
    return [tensor_shape.as_shape(v).as_proto() for v in value]
  return nest.map_structure(
      lambda v: v.encode() if isinstance(v, str) else v,
      value)


class _MockOp(object):
  """Pretends to be a tf.Operation for the gradient functions."""

  def __init__(self, attrs, inputs, outputs, typ, skip_input_indices):
    self.attrs = attrs
    self.inputs = inputs
    self.outputs = outputs
    self.type = typ
    self.skip_input_indices = skip_input_indices

  def get_attr(self, attr):
    typ = op_attr_type(self.type, attr)
    for i in range(0, len(self.attrs), 2):
      if self.attrs[i] == attr:
        return make_attr(typ, self.attrs[i + 1])
    raise KeyError(attr)

  def _get_control_flow_context(self):
    raise NotImplementedError(
        "tf.GradientTape.gradients() does not support graph control flow "
        "operations like tf.cond or tf.while at this time. Use tf.gradients() "
        "instead. If you need this feature, please file a feature request at "
        "https://github.com/tensorflow/tensorflow/issues/new"
    )


def _gradient_function(op_name, attr_tuple, num_inputs, inputs, outputs,
                       out_grads, skip_input_indices, forward_pass_name_scope):
  """Calls the gradient function of the op.

  Args:
    op_name: the name of the op to be differentiated.
    attr_tuple: the attrs, as a tuple.
    num_inputs: the number of inputs to the op.
    inputs: inputs to the original operation.
    outputs: outputs to the original operation.
    out_grads: gradients of the operation wrt its outputs.
    skip_input_indices: a tuple that is passed to the gradient function,
      indicating which inputs to skip calculating the gradient for
    forward_pass_name_scope: the namescope of the op in the forward pass.

  Returns:
    The gradients with respect to the inputs of the function, as a list.
  """
  mock_op = _MockOp(attr_tuple, inputs, outputs, op_name, skip_input_indices)
  grad_fn = ops._gradient_registry.lookup(op_name)  # pylint: disable=protected-access
  if grad_fn is None:
    return [None] * num_inputs

  # This does not work with v1 TensorArrays.
  if ops.executing_eagerly_outside_functions(
  ) or control_flow_util.EnableControlFlowV2(ops.get_default_graph()):
    gradient_name_scope = "gradient_tape/"
    if forward_pass_name_scope:
      gradient_name_scope += forward_pass_name_scope + "/"
    with ops.name_scope(gradient_name_scope):
      return grad_fn(mock_op, *out_grads)
  else:
    return grad_fn(mock_op, *out_grads)


pywrap_tfe.TFE_Py_RegisterGradientFunction(_gradient_function)


def _must_record_gradient():
  return not pywrap_tfe.TFE_Py_TapeSetIsEmpty()


@tf_export("__internal__.record_gradient", v1=[])
def record_gradient(op_name, inputs, attrs, outputs):
  """Explicitly record the gradient for a given op.

  Args:
    op_name: The op name as listed in the `OpDef` for the op.
    inputs: A list of tensor inputs to the op.
    attrs: The op attributes as a flattened list of alternating attribute names
      and attribute values.
    outputs: A list of tensor outputs from the op.
  """
  pywrap_tfe.TFE_Py_RecordGradient(op_name, inputs, attrs, outputs,
                                   ops.get_name_scope())


execute.must_record_gradient = _must_record_gradient
execute.record_gradient = record_gradient


def implicit_val_and_grad(f):
  """Returns a function which differentiates f with respect to variables.

  The wrapped function returns the value and the gradient of f when called with
  the same arguments. The gradient is with respect to all trainable TFE
  variables accessed by `f`.

  This function is useful when the exact set of variables to differentiate with
  is not known ahead of time.

  Example:

  ```python
  dense_layer = tf.compat.v1.layers.Dense(1)
  def loss(x, y):
    return tf.reduce_sum(tf.square(dense_layer(x) - y))

  # Obtain the gradient function.
  val_grad_fn = tfe.implicit_value_and_gradients(loss)

  # Invoke the gradient function with concrete values of x and y.
  x = tf.constant([[1.0, 2.0, 3.0], [4.0, 5.0, 6.0]])
  y = tf.constant([[10.0], [20.0]])
  value, grads_and_vars = val_grad_fn(x, y)
  print('Value of loss: %s' % value)

  # Apply the gradients to Variables.
  optimizer = tf.compat.v1.train.GradientDescentOptimizer(0.1)
  optimizer.apply_gradients(grads_and_vars)
  ```

  Args:
    f: function to be differentiated. If `f` returns a scalar, this scalar will
      be differentiated. If `f` returns a tensor or list of tensors, by default
      a scalar will be computed by adding all their values to produce a single
      scalar.

  Returns:
    A function which, when called, returns a tuple pair.
    Its first element is the value to which the function evaluates.
    Its second element is list of (gradient, variable) pairs.

  Raises:
    ValueError: if `f` returns None.
  """
  # TODO(cais): Remove calls to tf.constant() once the gradients functions
  # accept lists and np.ndarrays.

  def grad_fn(*args, **kwds):
    """Computes the gradient of the wrapped function."""
    this_tape = tape.push_new_tape()
    try:
      end_node = f(*args, **kwds)
      if end_node is None:
        raise ValueError("Cannot differentiate a function that returns None; "
                         "did you forget to return a value from {}?".format(
                             f.__name__))
    finally:
      tape.pop_tape(this_tape)
    # Note: variables are returned in construction order. This ensures unique
    # order across executions.
    variables = this_tape.watched_variables()
    if not variables:
      raise ValueError("No trainable variables were accessed while the "
                       "function was being computed.")

    sources = [v.handle for v in variables]
    for s in sources:
      if getattr(s, "is_packed", False):
        raise ValueError(
            "GradientTape.gradient is not supported on packed EagerTensors yet."
        )
    grad = imperative_grad.imperative_grad(this_tape, nest.flatten(end_node),
                                           sources)
    return end_node, list(zip(grad, variables))

  return grad_fn


def implicit_grad(f):
  """Returns a function which differentiates f with respect to variables.

  The wrapped function returns the gradient of f when called with the same
  arguments. The gradient is with respect to all trainable TFE variables
  accessed by `f`.

  This function is useful when the exact set of variables to differentiate with
  is not known ahead of time.

  Example:

  ```python
  dense_layer = tf.compat.v1.layers.Dense(1)
  def loss(x, y):
    return tf.reduce_sum(tf.square(dense_layer(x) - y))

  # Obtain the gradient function.
  grad_fn = tfe.implicit_gradients(loss)

  # Invoke the gradient function with concrete values of x and y.
  x = tf.constant([[1.0, 2.0, 3.0], [4.0, 5.0, 6.0]])
  y = tf.constant([[10.0], [20.0]])
  grads_and_vars = grad_fn(x, y)

  # Apply the gradients to Variables.
  optimizer = tf.compat.v1.train.GradientDescentOptimizer(0.1)
  optimizer.apply_gradients(grads_and_vars)
  ```

  Args:
    f: function to be differentiated. If `f` returns a scalar, this scalar will
      be differentiated. If `f` returns a tensor or list of tensors, by default
      a scalar will be computed by adding all their values to produce a single
      scalar.

  Returns:
    A function which, when called, returns a list of (gradient, variable) pairs.
  """
  # TODO(cais): Remove calls to tf.constant() once the gradients functions
  # accept lists and np.ndarrays.

  def grad_fn(*args, **kwds):
    """Computes the gradient of the wrapped function."""
    return implicit_val_and_grad(f)(*args, **kwds)[1]

  return grad_fn


def _get_arg_spec(f, params, param_args):
  """The positions of the parameters of f to be differentiated in param_args."""
  try:
    args = tf_inspect.getfullargspec(f).args
  except TypeError as e:
    # TypeError can happen when f is a callable object.
    if params is None:
      return range(len(param_args))
    elif all(isinstance(x, int) for x in params):
      return params
    raise ValueError("Either callable provided is not a function or could not "
                     "inspect its arguments by name: %s. Original error: %s"
                     % (f, e))
  if params is None:
    if not args:
      return range(len(param_args))
    if args[0] == "self":
      return range(len(args) - 1)
    else:
      return range(len(args))
  elif all(isinstance(x, str) for x in params):
    return [args.index(n) for n in params]
  elif all(isinstance(x, int) for x in params):
    return params
  else:
    raise ValueError(
        "params must be all strings or all integers; got %s." % params)


def gradients_function(f, params=None):
  """Returns a function which differentiates f with respect to params.

  Example:
  ```python
  # f(x, y) = (x ^ 3) * y - x * (y ^ 2)
  # Therefore, the 1st order derivatives are:
  #   df / dx = 3 * (x ^ 2) * y - y ^ 2
  #   df / dy = x ^ 3 - 2 * x * y
  # The 2nd order derivatives with respect to x is:
  #   d^2 f / (dx)^2 = 6 * x * y
  def f(x, y):
    return x * x * x * y - x * y * y

  # Obtain a function that returns 1st order gradients.
  grad_fn = tfe.gradients_function(f)

  x = 2.0
  y = 3.0

  # Invoke the 1st order gradient function.
  x_grad, y_grad = grad_fn(x, y)
  assert x_grad.numpy() == 3 * (2 ** 2) * 3 - 3 ** 2
  assert y_grad.numpy() == (2 ** 3) - 2 * 2 * 3

  # Obtain a function that returns the 2nd order gradient with respect to x.
  gradgrad_fn = tfe.gradients_function(lambda x, y: grad_fn(x, y)[0])

  # Invoke the 2nd order gradient function.
  x_gradgrad = gradgrad_fn(x, y)[0]
  assert x_gradgrad.numpy() == 6 * 2 * 3

  # To obtain a callable that returns the gradient(s) of `f` with respect to a
  # subset of its inputs, use the `params` keyword argument with
  # `gradients_function()`.
  ygrad_fn = tfe.gradients_function(f, params=[1])

  (y_grad,) = ygrad_fn(x, y)
  assert y_grad.numpy() == (2 ** 3) - 2 * 2 * 3
  ```

  Note that only tensors with real or complex dtypes are differentiable.

  Args:
    f: function to be differentiated. If `f` returns a scalar, this scalar will
      be differentiated. If `f` returns a tensor or list of tensors, by default
      a scalar will be computed by adding all their values to produce a single
      scalar. If desired, the tensors can be elementwise multiplied by the
      tensors passed as the `dy` keyword argument to the returned gradient
      function.
    params: list of parameter names of f or list of integers indexing the
      parameters with respect to which we'll differentiate. Passing None
      differentiates with respect to all parameters.

  Returns:
    function which, when called, returns the value of f and the gradient
    of `f` with respect to all of `params`. The function takes an extra optional
    keyword argument `dy`. Setting it allows computation of vector jacobian
    products for vectors other than the vector of ones.

  Raises:
    ValueError: if the params are not all strings or all integers.
  """

  def decorated(*args, **kwds):
    """Computes the gradient of the decorated function."""

    _, grad = val_and_grad_function(f, params=params)(*args, **kwds)
    return grad

  return decorated


def _ensure_unique_tensor_objects(parameter_positions, args):
  """Make each of the parameter_positions in args a unique ops.Tensor object.

  Ensure that each parameter is treated independently.
  For example:

  def f(x, y): return x * y
  g = gradients_function(f)
  one = tf.constant(1.)

  g(one, one) should return [1., 1.]
  (even though the two arguments are the same Tensor object).

  Args:
    parameter_positions: List of indices into args defining the arguments to
      differentiate against.
    args: A list of arguments to the function to be differentiated.

  Returns:
    args, possibly edited in-place.
  """
  s = set()
  for (i, t) in enumerate(args):
    if i in parameter_positions:
      tid = ops.tensor_id(t)
      if tid in s:
        args[i] = gen_array_ops.identity(args[i])
      else:
        s.add(tid)
  return args


def val_and_grad_function(f, params=None):
  """Returns a function that computes f and its derivative w.r.t. params.

  Example:
  ```python
  # f(x, y) = (x ^ 3) * y - x * (y ^ 2)
  # Therefore, the 1st order derivatives are:
  #   df / dx = 3 * (x ^ 2) * y - y ^ 2
  #   df / dy = x ^ 3 - 2 * x * y
  def f(x, y):
    return x * x * x * y - x * y * y

  # Obtain a function that returns the function value and the 1st order
  # gradients.
  val_grads_fn = tfe.value_and_gradients_function(f)

  x = 2.0
  y = 3.0

  # Invoke the value-and-gradients function.
  f_val, (x_grad, y_grad) = val_grads_fn(x, y)
  assert f_val.numpy() == (2 ** 3) * 3 - 2 * (3 ** 2)
  assert x_grad.numpy() == 3 * (2 ** 2) * 3 - 3 ** 2
  assert y_grad.numpy() == (2 ** 3) - 2 * 2 * 3

  # To obtain a callable that returns the value of `f` and the gradient(s) of
  # `f` with respect to a subset of its inputs, use the `params` keyword
  # argument with `value_and_gradients_function()`.
  val_ygrad_fn = tfe.value_and_gradients_function(f, params=[1])

  f_val, (y_grad,) = val_ygrad_fn(x, y)
  assert f_val.numpy() == (2 ** 3) * 3 - 2 * (3 ** 2)
  assert y_grad.numpy() == (2 ** 3) - 2 * 2 * 3
  ```

  Args:
    f: function to be differentiated. If `f` returns a scalar, this scalar will
      be differentiated. If `f` returns a tensor or list of tensors, by default
      a scalar will be computed by adding all their values to produce a single
      scalar. If desired, the tensors can be elementwise multiplied by the
      tensors passed as the `dy` keyword argument to the returned gradient
      function.
    params: list of parameter names of f or list of integers indexing the
      parameters with respect to which we'll differentiate. Passing `None`
      differentiates with respect to all parameters.

  Returns:
    function which, when called, returns the value of f and the gradient
    of f with respect to all of `params`. The function takes an extra optional
    keyword argument "dy". Setting it allows computation of vector jacobian
    products for vectors other than the vector of ones.

  Raises:
    ValueError: if the params are not all strings or all integers.
  """

  def decorated(*args, **kwds):
    """Computes the value and gradient of the decorated function."""
    dy = kwds.pop("dy", None)
    if kwds:
      raise ValueError("Functions to be differentiated cannot "
                       "receive keyword arguments.")
    val, vjp = make_vjp(f, params)(*args, **kwds)
    return val, vjp(dy=dy)

  return decorated


def make_vjp(f, params=None, persistent=True):
  """Returns a function that computes f and its vjp w.r.t.

  params.

  The term "vjp" here is an abbreviation for vector-jacobian product.

  Args:
    f: the function to be differentiated.
    params: the parameters (numbers or names) to differentiate with respect to.
      A value of None will differentiate with respect to all parameters.
    persistent: Boolean controlling whether the VJP function can be re-used.
      Must be True or False.

  Returns:
    A function, which when called, returns a tuple (value, vjp), where:
    - value is the result of calling f.
    - vjp is a function, which takes a vector as an argument and
      returns the product of that vector with the Jacobian of f.
      Providing no argument to vjp is equivalent to providing a
      vector of ones.

    For example,
    ```python
    def f(x):
      return x * x

    wrapped_fn = tfe.make_vjp(f)
    result, vjp = wrapped_fn(tf.constant(3.0))
    # result is 9.0
    vjp()  # the vjp function returns 6.0

  Raises:
    ValueError: if `f` returns None.
  """

  def decorated(*args, **kwds):
    """Computes the value and gradient of the decorated function."""
    parameter_positions = _get_arg_spec(f, params, args)
    assert not kwds, "The gradient function can't take keyword arguments."
    this_tape = tape.push_new_tape(persistent=persistent)
    try:
      sources = []
      args = [
          ops.convert_to_tensor(arg) if i in parameter_positions else arg
          for i, arg in enumerate(args)
      ]
      args = _ensure_unique_tensor_objects(parameter_positions, args)
      for i in parameter_positions:
        if getattr(args[i], "is_packed", False):
          raise ValueError(
              "GradientTape.gradient is not supported on packed EagerTensors"
              "yet.")
        sources.append(args[i])
        tape.watch(this_tape, args[i])
      result = f(*args)
      if result is None:
        raise ValueError("Cannot differentiate a function that returns None; "
                         "did you forget to return a value from {}?".format(
                             f.__name__))
      flat_result = nest.flatten(result)
      flat_result = [gen_array_ops.identity(x) for x in flat_result]
      result = nest.pack_sequence_as(result, flat_result)
    finally:
      tape.pop_tape(this_tape)
    def vjp(dy=None):
      if dy is not None:
        dy = [ops.convert_to_tensor(x) for x in nest.flatten(dy)]
      return imperative_grad.imperative_grad(
          this_tape, nest.flatten(result), sources, output_gradients=dy)

    return result, vjp

  return decorated


def _aggregate_grads(gradients):
  """Aggregate gradients from multiple sources.

  Args:
    gradients: A list of 'Tensor' or 'IndexedSlices' gradients.

  Returns:
    If 'gradients' only has 'Tensor', returns an aggregated 'Tensor'.
    Otherwise returns an aggregated 'IndexedSlices'.
  """
  assert gradients, "No gradients to aggregate"

  if len(gradients) == 1:
    return gradients[0]
  if all(isinstance(g, ops.Tensor) for g in gradients):
    return gen_math_ops.add_n(gradients)
  else:
    assert all(
        isinstance(g, (ops.Tensor, indexed_slices.IndexedSlices))
        for g in gradients)
    return backprop_util.AggregateIndexedSlicesGradients(gradients)


def _num_elements(grad):
  """The number of elements in the `grad` tensor."""
  if isinstance(grad, ops.Tensor):
    shape_tuple = grad._shape_tuple()  # pylint: disable=protected-access
  elif isinstance(grad, indexed_slices.IndexedSlices):
    shape_tuple = grad.values._shape_tuple()  # pylint: disable=protected-access
  else:
    raise ValueError("`grad` not a Tensor or IndexedSlices.")
  if shape_tuple is None or None in shape_tuple:
    return 0
  return functools.reduce(operator.mul, shape_tuple, 1)


def _fast_fill(value, shape, dtype):
  return array_ops.fill(
      constant_op.constant(shape, dtype=dtypes.int32),
      constant_op.constant(value, dtype=dtype))


def _zeros(shape, dtype):
  """Helper to return (possibly cached) zero tensors in eager mode."""
  # Note: variants will use _zeros_like
  if dtype == dtypes.string or dtype == dtypes.resource:
    return None

  ctx = context.context()
  if not ctx.executing_eagerly():
    return array_ops.zeros(shape, dtype)

  device = ctx.device_name

  if tensor_util.is_tf_type(shape):
    shape_key = shape.ref()
  else:
    shape_key = shape
  cache_key = shape_key, dtype, device
  cached = ctx.zeros_cache().get(cache_key)
  if cached is None:
    if dtypes.as_dtype(dtype).is_bool:
      value = False
    else:
      value = 0
    cached = _fast_fill(value, shape, dtype)
    ctx.zeros_cache().put(cache_key, cached)
  return cached


def _ones(shape, dtype):
  as_dtype = dtypes.as_dtype(dtype)
  if as_dtype == dtypes.string:
    return None

  if not context.executing_eagerly():
    return array_ops.ones(shape, dtype)

  if as_dtype.is_bool:
    value = True
  else:
    value = 1

  if shape == ():  # pylint: disable=g-explicit-bool-comparison
    return constant_op.constant(value, dtype=dtype)
  return _fast_fill(value, shape, dtype)


_default_vspace = imperative_grad.VSpace(
    num_elements_fn=_num_elements,
    aggregate_fn=_aggregate_grads,
    zeros_fn=_zeros,
    ones_fn=_ones,
    zeros_like_fn=default_gradient.zeros_like,
    ones_like_fn=default_gradient.ones_like,
    graph_shape_fn=gen_array_ops.shape)
pywrap_tfe.TFE_Py_RegisterVSpace(_default_vspace)


def _handle_or_self(x):
  """Unwrap resource variable/ndarray to return tensors."""
  if resource_variable_ops.is_resource_variable(x):
    return x.handle
  return x


def _extract_tensors_and_variables(tensor):
  """Extracts tensors and variables from the input object."""
  for obj in nest.flatten(tensor):
    if _pywrap_utils.IsTensor(obj) or _pywrap_utils.IsVariable(obj):
      yield obj
    elif isinstance(obj, composite_tensor.CompositeTensor):
      components = type_spec.type_spec_from_value(obj)._to_components(obj)  # pylint: disable=protected-access
      yield from _extract_tensors_and_variables(components)
    else:
      raise ValueError(f"Passed in object {obj} of type {type(obj).__name__!r}"
                       f", not tf.Tensor or tf.Variable or ExtensionType.")


@tf_export("GradientTape", "autodiff.GradientTape", v1=["GradientTape"])
class GradientTape:
  """Record operations for automatic differentiation.

  Operations are recorded if they are executed within this context manager and
  at least one of their inputs is being "watched".

  Trainable variables (created by `tf.Variable` or `tf.compat.v1.get_variable`,
  where `trainable=True` is default in both cases) are automatically watched.
  Tensors can be manually watched by invoking the `watch` method on this context
  manager.

  For example, consider the function `y = x * x`. The gradient at `x = 3.0` can
  be computed as:

  >>> x = tf.constant(3.0)
  >>> with tf.GradientTape() as g:
  ...   g.watch(x)
  ...   y = x * x
  >>> dy_dx = g.gradient(y, x)
  >>> print(dy_dx)
  tf.Tensor(6.0, shape=(), dtype=float32)

  GradientTapes can be nested to compute higher-order derivatives. For example,

  >>> x = tf.constant(5.0)
  >>> with tf.GradientTape() as g:
  ...   g.watch(x)
  ...   with tf.GradientTape() as gg:
  ...     gg.watch(x)
  ...     y = x * x
  ...   dy_dx = gg.gradient(y, x)  # dy_dx = 2 * x
  >>> d2y_dx2 = g.gradient(dy_dx, x)  # d2y_dx2 = 2
  >>> print(dy_dx)
  tf.Tensor(10.0, shape=(), dtype=float32)
  >>> print(d2y_dx2)
  tf.Tensor(2.0, shape=(), dtype=float32)

  By default, the resources held by a GradientTape are released as soon as
  GradientTape.gradient() method is called. To compute multiple gradients over
  the same computation, create a persistent gradient tape. This allows multiple
  calls to the gradient() method as resources are released when the tape object
  is garbage collected. For example:

  >>> x = tf.constant(3.0)
  >>> with tf.GradientTape(persistent=True) as g:
  ...   g.watch(x)
  ...   y = x * x
  ...   z = y * y
  >>> dz_dx = g.gradient(z, x)  # (4*x^3 at x = 3)
  >>> print(dz_dx)
  tf.Tensor(108.0, shape=(), dtype=float32)
  >>> dy_dx = g.gradient(y, x)
  >>> print(dy_dx)
  tf.Tensor(6.0, shape=(), dtype=float32)

  By default GradientTape will automatically watch any trainable variables that
  are accessed inside the context. If you want fine grained control over which
  variables are watched you can disable automatic tracking by passing
  `watch_accessed_variables=False` to the tape constructor:

  >>> x = tf.Variable(2.0)
  >>> w = tf.Variable(5.0)
  >>> with tf.GradientTape(
  ...     watch_accessed_variables=False, persistent=True) as tape:
  ...   tape.watch(x)
  ...   y = x ** 2  # Gradients will be available for `x`.
  ...   z = w ** 3  # No gradients will be available as `w` isn't being watched.
  >>> dy_dx = tape.gradient(y, x)
  >>> print(dy_dx)
  tf.Tensor(4.0, shape=(), dtype=float32)
  >>> # No gradients will be available as `w` isn't being watched.
  >>> dz_dw = tape.gradient(z, w)
  >>> print(dz_dw)
  None

  Note that when using models you should ensure that your variables exist when
  using `watch_accessed_variables=False`. Otherwise it's quite easy to make your
  first iteration not have any gradients:

  ```python
  a = tf.keras.layers.Dense(32)
  b = tf.keras.layers.Dense(32)

  with tf.GradientTape(watch_accessed_variables=False) as tape:
    tape.watch(a.variables)  # Since `a.build` has not been called at this point
                             # `a.variables` will return an empty list and the
                             # tape will not be watching anything.
    result = b(a(inputs))
    tape.gradient(result, a.variables)  # The result of this computation will be
                                        # a list of `None`s since a's variables
                                        # are not being watched.
  ```

  Note that only tensors with real or complex dtypes are differentiable.
  """

  def __init__(self, persistent=False, watch_accessed_variables=True):
    """Creates a new GradientTape.

    Args:
      persistent: Boolean controlling whether a persistent gradient tape
        is created. False by default, which means at most one call can
        be made to the gradient() method on this object.
      watch_accessed_variables: Boolean controlling whether the tape will
        automatically `watch` any (trainable) variables accessed while the tape
        is active. Defaults to True meaning gradients can be requested from any
        result computed in the tape derived from reading a trainable `Variable`.
        If False users must explicitly `watch` any `Variable`s they want to
        request gradients from.
    """
    self._tape = None
    self._persistent = persistent
    self._watch_accessed_variables = watch_accessed_variables
    self._watched_variables = ()
    self._recording = False

  def __enter__(self):
    """Enters a context inside which operations are recorded on this tape."""
    self._push_tape()
    return self

  def __exit__(self, typ, value, traceback):
    """Exits the recording context, no further operations are traced."""
    if self._recording:
      self._pop_tape()

  def _push_tape(self):
    """Pushes a new tape onto the tape stack."""
    if self._recording:
      raise ValueError("Tape is still recording, This can happen if you try to "
                       "re-enter an already-active tape.")
    if self._tape is None:
      self._tape = tape.push_new_tape(
          persistent=self._persistent,
          watch_accessed_variables=self._watch_accessed_variables)
    else:
      tape.push_tape(self._tape)
    self._recording = True

  def _pop_tape(self):
    if not self._recording:
      raise ValueError("Tape is not recording.")
    tape.pop_tape(self._tape)
    self._recording = False

  @tf_contextlib.contextmanager
  def _ensure_recording(self):
    """Ensures that this tape is recording."""
    if not self._recording:
      try:
        self._push_tape()
        yield
      finally:
        self._pop_tape()
    else:
      yield

  # TODO(b/209081027): Add a variable in composite tensor test case after
  # variables become composite tensors.
  def watch(self, tensor):
    """Ensures that `tensor` is being traced by this tape.

    Args:
      tensor: a Tensor/Variable or list of Tensors/Variables.

    Raises:
      ValueError: if it encounters something that is not a tensor.
    """
    for t in _extract_tensors_and_variables(tensor):
      if not backprop_util.IsTrainable(t):
        logging.log_first_n(
            logging.WARN, "The dtype of the watched tensor must be "
            "floating (e.g. tf.float32), got %r", 5, t.dtype)
      if hasattr(t, "handle"):
        # There are many variable-like objects, all of them currently have
        # `handle` attribute that points to a tensor. If this changes,
        # internals of watch_variable need to change as well.
        tape.watch_variable(self._tape, t)
      else:
        tape.watch(self._tape, t)

  @tf_contextlib.contextmanager
  def stop_recording(self):
    """Temporarily stops recording operations on this tape.

    Operations executed while this context manager is active will not be
    recorded on the tape. This is useful for reducing the memory used by tracing
    all computations.

    For example:

    >>> x = tf.constant(4.0)
    >>> with tf.GradientTape() as tape:
    ...   with tape.stop_recording():
    ...     y = x ** 2
    >>> dy_dx = tape.gradient(y, x)
    >>> print(dy_dx)
    None

    Yields:
      None
    Raises:
      RuntimeError: if the tape is not currently recording.
    """
    if self._tape is None:
      raise RuntimeError(
          "Trying to stop recording a tape which is not recording.")
    self._pop_tape()
    try:
      yield
    finally:
      self._push_tape()

  def reset(self):
    """Clears all information stored in this tape.

    Equivalent to exiting and reentering the tape context manager with a new
    tape. For example, the two following code blocks are equivalent:

    ```
    with tf.GradientTape() as t:
      loss = loss_fn()
    with tf.GradientTape() as t:
      loss += other_loss_fn()
    t.gradient(loss, ...)  # Only differentiates other_loss_fn, not loss_fn


    # The following is equivalent to the above
    with tf.GradientTape() as t:
      loss = loss_fn()
      t.reset()
      loss += other_loss_fn()
    t.gradient(loss, ...)  # Only differentiates other_loss_fn, not loss_fn
    ```

    This is useful if you don't want to exit the context manager for the tape,
    or can't because the desired reset point is inside a control flow construct:

    ```
    with tf.GradientTape() as t:
      loss = ...
      if loss > k:
        t.reset()
    ```
    """
    self._pop_tape()
    self._tape = None
    self._push_tape()

  def watched_variables(self):
    """Returns variables watched by this tape in order of construction."""
    if self._tape is not None:
      self._watched_variables = self._tape.watched_variables()
    return self._watched_variables

  def gradient(self,
               target,
               sources,
               output_gradients=None,
               unconnected_gradients=UnconnectedGradients.NONE):
    """Computes the gradient using operations recorded in context of this tape.

    Note: Unless you set `persistent=True` a GradientTape can only be used to
    compute one set of gradients (or jacobians).

    In addition to Tensors, gradient also supports RaggedTensors. For example,

    >>> x = tf.ragged.constant([[1.0, 2.0], [3.0]])
    >>> with tf.GradientTape() as g:
    ...   g.watch(x)
    ...   y = x * x
    >>> g.gradient(y, x)
    <tf.RaggedTensor [[2.0, 4.0], [6.0]]>

    Args:
      target: a list or nested structure of Tensors or Variables or
        CompositeTensors to be differentiated.
      sources: a list or nested structure of Tensors or Variables or
        CompositeTensors. `target` will be differentiated against elements in
        `sources`.
      output_gradients: a list of gradients, one for each differentiable
        element of target. Defaults to None.
      unconnected_gradients: a value which can either hold 'none' or 'zero' and
        alters the value which will be returned if the target and sources are
        unconnected. The possible values and effects are detailed in
        'UnconnectedGradients' and it defaults to 'none'.

    Returns:
      a list or nested structure of Tensors (or IndexedSlices, or None, or
      CompositeTensor), one for each element in `sources`. Returned structure
      is the same as the structure of `sources`.

    Raises:
      RuntimeError: If called on a used, non-persistent tape.
      RuntimeError: If called inside the context of the tape.
      TypeError: If the target is a None object.
      ValueError: If the target is a variable or if unconnected gradients is
       called with an unknown value.
    """
    if self._tape is None:
      raise RuntimeError("A non-persistent GradientTape can only be used to "
                         "compute one set of gradients (or jacobians)")
    if self._recording:
      if not self._persistent:
        self._pop_tape()
      else:
        logging.log_first_n(
            logging.WARN, "Calling GradientTape.gradient on a persistent "
            "tape inside its context is significantly less "
            "efficient than calling it outside the context (it "
            "causes the gradient ops to be recorded on the "
            "tape, leading to increased CPU and memory usage). "
            "Only call GradientTape.gradient inside the "
            "context if you actually want to trace the "
            "gradient in order to compute higher order "
            "derivatives.", 1)

    if target is None:
      raise TypeError("Argument `target` should be a list or nested structure"
                      " of Tensors, Variables or CompositeTensors to be "
                      "differentiated, but received None.")

    flat_targets = []
    for t in nest.flatten(target):
      flat_targets.append(_handle_or_self(t))
    flat_targets = composite_tensor_gradient.get_flat_tensors_for_gradients(
        flat_targets)
    for t in flat_targets:
      if not backprop_util.IsTrainable(t):
        logging.vlog(
            1, "The dtype of the target tensor must be "
            "floating (e.g. tf.float32) when calling GradientTape.gradient, "
            "got %r", t.dtype)

    flat_sources_raw = nest.flatten(sources)
    flat_sources = []
    for t in flat_sources_raw:
      flat_sources.append(_handle_or_self(t))
    flat_sources = composite_tensor_gradient.get_flat_tensors_for_gradients(
        flat_sources)
    for t in flat_sources:
      if not backprop_util.IsTrainable(t):
        logging.vlog(
            1, "The dtype of the source tensor must be "
            "floating (e.g. tf.float32) when calling GradientTape.gradient, "
            "got %r", t.dtype)
      if getattr(t, "is_packed", False):
        raise ValueError(
            "GradientTape.gradient is not supported on packed EagerTensors yet."
        )

    if output_gradients is not None:
      output_gradients = nest.flatten(
          variable_utils.convert_variables_to_tensors(output_gradients))
      output_gradients = (
          composite_tensor_gradient.get_flat_tensors_for_gradients(
              output_gradients))
      output_gradients = [None if x is None else ops.convert_to_tensor(x)
                          for x in output_gradients]

    flat_grad = imperative_grad.imperative_grad(
        self._tape,
        flat_targets,
        flat_sources,
        output_gradients=output_gradients,
        sources_raw=flat_sources_raw,
        unconnected_gradients=unconnected_gradients)

    if not self._persistent:
      # Keep track of watched variables before setting tape to None
      self._watched_variables = self._tape.watched_variables()
      self._tape = None

    flat_sources_raw = nest.map_structure(_handle_or_self, flat_sources_raw)
    flat_grad = composite_tensor_gradient.replace_flat_tensors_for_gradients(
        flat_sources_raw, flat_grad)
    grad = nest.pack_sequence_as(sources, flat_grad)
    return grad

  def jacobian(self,
               target,
               sources,
               unconnected_gradients=UnconnectedGradients.NONE,
               parallel_iterations=None,
               experimental_use_pfor=True):
    """Computes the jacobian using operations recorded in context of this tape.

    Note: Unless you set `persistent=True` a GradientTape can only be used to
    compute one set of gradients (or jacobians).

    Note: By default the jacobian implementation uses parallel for (pfor), which
    creates a tf.function under the hood for each jacobian call. For better
    performance, and to avoid recompilation and vectorization rewrites on each
    call, enclose GradientTape code in @tf.function.

    See[wikipedia
    article](http://en.wikipedia.org/wiki/jacobian_matrix_and_determinant)
    for the definition of a Jacobian.

    Example usage:

    ```python
    with tf.GradientTape() as g:
      x  = tf.constant([1.0, 2.0])
      g.watch(x)
      y = x * x
    jacobian = g.jacobian(y, x)
    # jacobian value is [[2., 0.], [0., 4.]]
    ```

    Args:
      target: Tensor to be differentiated.
      sources: a list or nested structure of Tensors or Variables. `target`
        will be differentiated against elements in `sources`.
      unconnected_gradients: a value which can either hold 'none' or 'zero' and
        alters the value which will be returned if the target and sources are
        unconnected. The possible values and effects are detailed in
        'UnconnectedGradients' and it defaults to 'none'.
      parallel_iterations: A knob to control how many iterations are dispatched
        in parallel. This knob can be used to control the total memory usage.
      experimental_use_pfor: If true, vectorizes the jacobian computation. Else
        falls back to a sequential while_loop. Vectorization can sometimes fail
        or lead to excessive memory usage. This option can be used to disable
        vectorization in such cases.

    Returns:
      A list or nested structure of Tensors (or None), one for each element in
      `sources`. Returned structure is the same as the structure of `sources`.
      Note if any gradient is sparse (IndexedSlices), jacobian function
      currently makes it dense and returns a Tensor instead. This may change in
      the future.


    Raises:
      RuntimeError: If called on a used, non-persistent tape.
      RuntimeError: If called on a non-persistent tape with eager execution
        enabled and without enabling experimental_use_pfor.
      ValueError: If vectorization of jacobian computation fails.
    """
    if self._tape is None:
      raise RuntimeError("A non-persistent GradientTape can only be used to "
                         "compute one set of gradients (or jacobians)")

    flat_sources = nest.flatten(sources)
    target_static_shape = target.shape
    target_shape = array_ops.shape(target)
    # Note that we push and pop the tape here and below. This is needed since we
    # need gradients through the enclosed operations.
    with self._ensure_recording():
      target = array_ops.reshape(target, [-1])

    def loop_fn(i):
      with self._ensure_recording():
        y = array_ops.gather(target, i)
      return self.gradient(y, flat_sources,
                           unconnected_gradients=unconnected_gradients)

    try:
      target_size = int(target.shape[0])
    except TypeError:
      target_size = array_ops.shape(target)[0]

    if experimental_use_pfor:
      try:
        output = pfor_ops.pfor(loop_fn, target_size,
                               parallel_iterations=parallel_iterations)
      except ValueError as err:
        raise ValueError(
            "Encountered an exception while vectorizing the "
            "jacobian computation. Vectorization can be disabled by setting"
            " experimental_use_pfor to False.") from err
    else:
      if context.executing_eagerly() and not self._persistent:
        raise RuntimeError(
            "GradientTape must be created with persistent=True"
            " to compute the jacobian with eager execution enabled and with "
            " experimental_use_pfor set to False.")
      output = pfor_ops.for_loop(
          loop_fn, [target.dtype] * len(flat_sources), target_size,
          parallel_iterations=parallel_iterations)

    for i, out in enumerate(output):
      if out is not None:
        new_shape = array_ops.concat(
            [target_shape, array_ops.shape(out)[1:]], axis=0)
        out = array_ops.reshape(out, new_shape)
        if context.executing_eagerly():
          out.set_shape(target_static_shape.concatenate(flat_sources[i].shape))
      output[i] = out

    return nest.pack_sequence_as(sources, output)

  def batch_jacobian(self,
                     target,
                     source,
                     unconnected_gradients=UnconnectedGradients.NONE,
                     parallel_iterations=None,
                     experimental_use_pfor=True):
    """Computes and stacks per-example jacobians.

    See [wikipedia article](http://en.wikipedia.org/wiki/jacobian_matrix_and_determinant)
    for the definition of a Jacobian. This function is essentially an efficient
    implementation of the following:

    `tf.stack([self.jacobian(y[i], x[i]) for i in range(x.shape[0])])`.

    Note that compared to `GradientTape.jacobian` which computes gradient of
    each output value w.r.t each input value, this function is useful when
    `target[i,...]` is independent of `source[j,...]` for `j != i`. This
    assumption allows more efficient computation as compared to
    `GradientTape.jacobian`. The output, as well as intermediate activations,
    are lower dimensional and avoid a bunch of redundant zeros which would
    result in the jacobian computation given the independence assumption.

    Note: Unless you set `persistent=True` a GradientTape can only be used to
    compute one set of gradients (or jacobians).

    Note: By default the batch_jacobian implementation uses parallel for (pfor),
    which creates a tf.function under the hood for each batch_jacobian call.
    For better performance, and to avoid recompilation and vectorization
    rewrites on each call, enclose GradientTape code in @tf.function.


    Example usage:

    ```python
    with tf.GradientTape() as g:
      x = tf.constant([[1., 2.], [3., 4.]], dtype=tf.float32)
      g.watch(x)
      y = x * x
    batch_jacobian = g.batch_jacobian(y, x)
    # batch_jacobian is [[[2,  0], [0,  4]], [[6,  0], [0,  8]]]
    ```

    Args:
      target: A tensor with rank 2 or higher and with shape [b, y1, ..., y_n].
        `target[i,...]` should only depend on `source[i,...]`.
      source: A tensor with rank 2 or higher and with shape [b, x1, ..., x_m].
      unconnected_gradients: a value which can either hold 'none' or 'zero' and
        alters the value which will be returned if the target and sources are
        unconnected. The possible values and effects are detailed in
        'UnconnectedGradients' and it defaults to 'none'.
      parallel_iterations: A knob to control how many iterations are dispatched
        in parallel. This knob can be used to control the total memory usage.
      experimental_use_pfor: If true, uses pfor for computing the Jacobian. Else
        uses a tf.while_loop.

    Returns:
      A tensor `t` with shape [b, y_1, ..., y_n, x1, ..., x_m] where `t[i, ...]`
      is the jacobian of `target[i, ...]` w.r.t. `source[i, ...]`, i.e. stacked
      per-example jacobians.

    Raises:
      RuntimeError: If called on a used, non-persistent tape.
      RuntimeError: If called on a non-persistent tape with eager execution
        enabled and without enabling experimental_use_pfor.
      ValueError: If vectorization of jacobian computation fails or if first
        dimension of `target` and `source` do not match.
    """
    if self._tape is None:
      raise RuntimeError("A non-persistent GradientTape can only be used to"
                         "compute one set of gradients (or jacobians)")
    target_shape = target.shape
    if target_shape.rank is None:
      dim = tensor_shape.Dimension(None)
    else:
      dim = target_shape.dims[0]
    if not (target_shape.with_rank_at_least(2) and
            source.shape.with_rank_at_least(2) and
            dim.is_compatible_with(source.shape[0])):
      raise ValueError(
          "Need first dimension of target shape (%s) and "
          "source shape (%s) to match." % (target.shape, source.shape))
    if target_shape.is_fully_defined():
      batch_size = int(target_shape[0])
      target_row_size = target_shape.num_elements() // batch_size
    else:
      target_shape = array_ops.shape(target)
      batch_size = target_shape[0]
      target_row_size = array_ops.size(target) // batch_size
    source_shape = array_ops.shape(source)
    # Flatten target to 2-D.
    # Note that we push and pop the tape here and below. This is needed since we
    # need gradients through the enclosed operations.
    with self._ensure_recording():
      with ops.control_dependencies(
          [check_ops.assert_equal(batch_size, source_shape[0])]):
        target = array_ops.reshape(target, [batch_size, target_row_size])

    run_once = False

    def loop_fn(i):
      nonlocal run_once
      if run_once and not self._persistent:
        if parallel_iterations is not None:
          raise RuntimeError(
              "GradientTape must be created with persistent=True"
              " to compute the batch_jacobian with parallel_iterations.")
        else:
          raise RuntimeError(
              "GradientTape must be created with persistent=True"
              " to compute the batch_jacobian.")
      run_once = True

      with self._ensure_recording():
        y = array_ops.gather(target, i, axis=1)
      return self.gradient(y, source,
                           unconnected_gradients=unconnected_gradients)

    if experimental_use_pfor:
      try:
        output = pfor_ops.pfor(loop_fn, target_row_size,
                               parallel_iterations=parallel_iterations)
      except ValueError as err:
        raise ValueError(
            "Encountered an exception while vectorizing the "
            "batch_jacobian computation. Vectorization can be disabled by "
            "setting experimental_use_pfor to False.") from err
    else:
      if context.executing_eagerly() and not self._persistent:
        raise RuntimeError(
            "GradientTape must be created with persistent=True"
            " to compute the batch_jacobian with eager execution enabled and "
            " with experimental_use_pfor set to False.")
      output = pfor_ops.for_loop(loop_fn, target.dtype, target_row_size,
                                 parallel_iterations=parallel_iterations)
    new_shape = array_ops.concat([target_shape, source_shape[1:]], axis=0)
    if output is None:
      # Note that this block is returning zeros when it could use `None` to
      # represent unconnected gradients. This is to maintain compatibility with
      # the previous behavior, which ignored `unconnected_gradients`.
      output = array_ops.zeros(new_shape, target.dtype)
      return output
    else:
      output = array_ops.reshape(output,
                                 [target_row_size, batch_size, -1])
      output = array_ops.transpose(output, [1, 0, 2])

      output = array_ops.reshape(output, new_shape)
      return output