lr.py

# Copyright (c) 2020 PaddlePaddle 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
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import math
import warnings

import numpy

import paddle
from paddle import Tensor
from paddle.base import core
from paddle.base.data_feeder import check_type
from paddle.base.framework import (
    Variable,
    default_main_program,
    in_dygraph_mode,
)
from paddle.base.layer_helper import LayerHelper

__all__ = [
    'LRScheduler',
    'NoamDecay',
    'PiecewiseDecay',
    'NaturalExpDecay',
    'InverseTimeDecay',
    'PolynomialDecay',
    'LinearWarmup',
    'ExponentialDecay',
    'MultiStepDecay',
    'StepDecay',
    'LambdaDecay',
    'ReduceOnPlateau',
    'CosineAnnealingDecay',
    'MultiplicativeDecay',
    'OneCycleLR',
    'CyclicLR',
    'LinearLR',
    'CosineAnnealingWarmRestarts',
]


class LRScheduler:
    """

    LRScheduler Base class. Define the common interface of a learning rate scheduler.

    User can import it by ``from paddle.optimizer.lr import LRScheduler`` ,

    then overload it for your subclass and have a custom implementation of ``get_lr()`` .

    Otherwise, an ``NotImplementedError`` exception will be thrown.

    Args:
        learning_rate (float): The initial learning rate. It is a python float number.
        last_epoch (int, optional):  The index of last epoch. Can be set to restart training. Default: -1, means initial learning rate.
        verbose (bool, optional): If ``True``, prints a message to stdout for each update. Default: ``False`` .

    Returns:
        instance to schedule learning rate.

    Examples:
        Here is an example of a simple ``StepDecay`` implementation.

        .. code-block:: python

            >>> import paddle
            >>> from paddle.optimizer.lr import LRScheduler

            >>> class StepDecay(LRScheduler):
            ...     def __init__(self,
            ...                 learning_rate,
            ...                 step_size,
            ...                 gamma=0.1,
            ...                 last_epoch=-1,
            ...                 verbose=False):
            ...         if not isinstance(step_size, int):
            ...             raise TypeError(
            ...                 "The type of 'step_size' must be 'int', but received %s." %
            ...                 type(step_size))
            ...         if gamma >= 1.0:
            ...             raise ValueError('gamma should be < 1.0.')
            ...
            ...         self.step_size = step_size
            ...         self.gamma = gamma
            ...         super().__init__(learning_rate, last_epoch, verbose)
            ...
            ...     def get_lr(self):
            ...         i = self.last_epoch // self.step_size
            ...         return self.base_lr * (self.gamma**i)
            ...
    """

    def __init__(self, learning_rate=0.1, last_epoch=-1, verbose=False):
        if not isinstance(learning_rate, (float, int)):
            raise TypeError(
                "The type of learning rate must be float, but received {}".format(
                    type(learning_rate)
                )
            )
        if learning_rate < 0:
            raise ValueError(f"Invalid learning rate: {learning_rate}")
        self.base_lr = float(learning_rate)
        self.last_lr = float(learning_rate)
        self.last_epoch = last_epoch
        self.verbose = verbose
        self._var_name = None

        self.step()

    def __call__(self):
        """
        Return latest computed learning rate on current epoch.
        """
        return self.last_lr

    def step(self, epoch=None):
        """

        ``step`` should be called after ``optimizer.step`` . It will update the learning rate in optimizer according to current ``epoch`` .
        The new learning rate will take effect on next ``optimizer.step`` .

        Args:
            epoch (int, None): specify current epoch. Default: None. Auto-increment from last_epoch=-1.

        Returns:
            None
        Examples:
            .. code-block:: python

                >>> import paddle
                >>> value = paddle.arange(26, dtype='float32')
                >>> a = paddle.reshape(value, [2, 13])
                >>> linear = paddle.nn.Linear(13, 5)
                >>> adadelta = paddle.optimizer.Adadelta(learning_rate=0.0003, epsilon=1e-06, rho=0.95,
                ...                             parameters = linear.parameters())
                >>> out = linear(a)
                >>> out.backward()
                >>> adadelta.step()
                >>> adadelta.clear_grad()

            .. code-block:: python

                >>> import paddle
                >>> value = paddle.arange(26, dtype='float32')
                >>> a = paddle.reshape(value, [2, 13])
                >>> linear = paddle.nn.Linear(13, 5)
                >>> adadelta = paddle.optimizer.Adadelta(learning_rate=0.0003, epsilon=1e-06, rho=0.95,
                ...                             parameters = linear.parameters())
                >>> out = linear(a)
                >>> out.backward()
                >>> adadelta.step()
                >>> adadelta.clear_grad()
        """
        if epoch is None:
            self.last_epoch += 1
            self.last_lr = self.get_lr()
        else:
            self.last_epoch = epoch
            if hasattr(self, "_get_closed_form_lr"):
                self.last_lr = self._get_closed_form_lr()
            else:
                self.last_lr = self.get_lr()

        if self.verbose:
            print(
                'Epoch {}: {} set learning rate to {}.'.format(
                    self.last_epoch, self.__class__.__name__, self.last_lr
                )
            )

    def state_dict(self):
        """

        Returns the state of the scheduler as a :class:`dict`.

        It is a subset of ``self.__dict__`` .
        """
        self.state_keys()
        state_dict = {}
        for key in self.keys:
            if key not in self.__dict__:
                continue
            value = self.__dict__[key]
            if isinstance(value, Tensor):
                assert (
                    value.size == 1
                ), "numel of Tensor in state_dict must be 1"
                value = float(value)
            state_dict[key] = value

        return state_dict

    # For those subclass who overload LRScheduler, "last_epoch, last_lr" will be saved by default.
    # (Note): you can change it for your subclass.
    def state_keys(self):
        """

        For those subclass who overload ``LRScheduler`` (Base Class). Acquiescently, "last_epoch, last_lr" will be saved by ``self.keys = ['last_epoch', 'last_lr']`` .

        ``last_epoch`` is the current epoch num, and ``last_lr`` is the current learning rate.

        If you want to change the default behavior, you should have a custom implementation of ``_state_keys()`` to redefine ``self.keys`` .

        """
        self.keys = ['last_epoch', 'last_lr']

    def set_state_dict(self, state_dict):
        """

        Loads the schedulers state.
        """
        self.state_keys()
        for key in self.keys:
            if key in state_dict:
                self.__dict__[key] = state_dict[key]
            else:
                raise RuntimeError(
                    f"Please check whether state_dict is correct for optimizer. Can't find [ {key} ] in state_dict"
                )
        if len(state_dict) > len(self.keys):
            warnings.warn(
                "There are some unused values in state_dict. Maybe the optimizer have different 'LearningRateDecay' when invoking state_dict and set_dict"
            )

    # alias for set_state_dict
    set_dict = set_state_dict

    def get_lr(self):
        """

        For those subclass who overload ``LRScheduler`` (Base Class), User should have a custom implementation of ``get_lr()`` .

        Otherwise, an ``NotImplementedError`` exception will be thrown.
        """
        # calculate by python float
        raise NotImplementedError


class NoamDecay(LRScheduler):
    r"""

    Applies Noam Decay to the initial learning rate.

    The algorithm can be described as following.

    .. math::

        new\_learning\_rate = learning\_rate * d_{model}^{-0.5} * min(epoch^{-0.5}, epoch * warmup\_steps^{-1.5})

    Please reference `attention is all you need <https://arxiv.org/pdf/1706.03762.pdf>`_


    Args:
        d$_{model}$(int): The dimensionality of input and output feature vector of model. It is a python int number.
        warmup_steps(int): The number of warmup steps. A super parameter. It is a python int number
        learning_rate (float): The initial learning rate. It is a python float number. Default: 1.0.
        last_epoch (int, optional):  The index of last epoch. Can be set to restart training. Default: -1, means initial learning rate.
        verbose (bool, optional): If ``True``, prints a message to stdout for each update. Default: ``False`` .

    Returns:
        ``NoamDecay`` instance to schedule learning rate.

    Examples:
        .. code-block:: python
            :name: code-example1

            >>> # Example1: train on default dynamic graph mode
            >>> import paddle
            >>> import numpy as np

            >>> # train on default dynamic graph mode
            >>> linear = paddle.nn.Linear(10, 10)
            >>> scheduler = paddle.optimizer.lr.NoamDecay(d_model=0.01, warmup_steps=100, verbose=True)
            >>> sgd = paddle.optimizer.SGD(learning_rate=scheduler, parameters=linear.parameters())
            >>> for epoch in range(20):
            ...     for batch_id in range(5):
            ...         x = paddle.uniform([10, 10])
            ...         out = linear(x)
            ...         loss = paddle.mean(out)
            ...         loss.backward()
            ...         sgd.step()
            ...         sgd.clear_gradients()
            ...         scheduler.step()    # If you update learning rate each step
            ...     # scheduler.step()        # If you update learning rate each epoch

        .. code-block:: python
            :name: code-example2

            >>> # Example2: train on static graph mode
            >>> import paddle
            >>> import numpy as np
            >>> paddle.enable_static()
            >>> main_prog = paddle.static.Program()
            >>> start_prog = paddle.static.Program()
            >>> with paddle.static.program_guard(main_prog, start_prog):
            ...     x = paddle.static.data(name='x', shape=[None, 4, 5])
            ...     y = paddle.static.data(name='y', shape=[None, 4, 5])
            ...     z = paddle.static.nn.fc(x, 100)
            ...     loss = paddle.mean(z)
            ...     scheduler = paddle.optimizer.lr.NoamDecay(d_model=0.01, warmup_steps=100, verbose=True)
            ...     sgd = paddle.optimizer.SGD(learning_rate=scheduler)
            ...     sgd.minimize(loss)
            ...
            >>> exe = paddle.static.Executor()
            >>> exe.run(start_prog)
            >>> for epoch in range(20):
            ...     for batch_id in range(5):
            ...         out = exe.run(
            ...             main_prog,
            ...             feed={
            ...                 'x': np.random.randn(3, 4, 5).astype('float32'),
            ...                 'y': np.random.randn(3, 4, 5).astype('float32')
            ...             },
            ...             fetch_list=loss.name)
            ...         scheduler.step()    # If you update learning rate each step
            ...     # scheduler.step()        # If you update learning rate each epoch
            ...
    """

    def __init__(
        self,
        d_model,
        warmup_steps,
        learning_rate=1.0,
        last_epoch=-1,
        verbose=False,
    ):
        if d_model <= 0:
            raise ValueError("d_model should be grater than 0")

        self.d_model = d_model
        self.warmup_steps = warmup_steps
        super().__init__(learning_rate, last_epoch, verbose)

    def get_lr(self):
        if self.last_epoch == 0:
            a = 1
        else:
            a = self.last_epoch**-0.5
        b = self.warmup_steps**-1.5 * self.last_epoch
        return self.base_lr * (self.d_model**-0.5) * min(a, b)


class PiecewiseDecay(LRScheduler):
    """

    Piecewise learning rate scheduler.

    The algorithm can be described as the code below:

    .. code-block:: text

        boundaries = [100, 200]
        values = [1.0, 0.5, 0.1]
        if epoch < 100:
            learning_rate = 1.0
        elif 100 <= global_step < 200:
            learning_rate = 0.5
        else:
            learning_rate = 0.1

    Args:
        boundaries(list|tuple): A list/tuple of steps numbers. The type of element in the list is python int.
        values(list|tuple): A list/tuple of learning rate values that will be picked during different epoch boundaries.
            The type of element in the list is python float. The ``values`` have one more element than ``boundaries``.
        last_epoch (int, optional):  The index of last epoch. Can be set to restart training. Default: -1, means initial learning rate.
        verbose (bool, optional): If ``True``, prints a message to stdout for each update. Default: ``False`` .

    Returns:
        ``PiecewiseDecay`` instance to schedule learning rate.

    Examples:

        .. code-block:: python
            :name: code-example1

            >>> # Example1: train on default dynamic graph mode
            >>> import paddle
            >>> import numpy as np

            >>> # train on default dynamic graph mode
            >>> linear = paddle.nn.Linear(10, 10)
            >>> scheduler = paddle.optimizer.lr.PiecewiseDecay(boundaries=[3, 6, 9], values=[0.1, 0.2, 0.3, 0.4], verbose=True)
            >>> sgd = paddle.optimizer.SGD(learning_rate=scheduler, parameters=linear.parameters())
            >>> for epoch in range(20):
            ...     for batch_id in range(5):
            ...         x = paddle.uniform([10, 10])
            ...         out = linear(x)
            ...         loss = paddle.mean(out)
            ...         loss.backward()
            ...         sgd.step()
            ...         sgd.clear_gradients()
            ...         scheduler.step()    # If you update learning rate each step
            ...     # scheduler.step()        # If you update learning rate each epoch

        .. code-block:: python
            :name: code-example2

            >>> # Example2: train on static graph mode
            >>> import paddle
            >>> import numpy as np
            >>> paddle.enable_static()
            >>> main_prog = paddle.static.Program()
            >>> start_prog = paddle.static.Program()
            >>> with paddle.static.program_guard(main_prog, start_prog):
            ...     x = paddle.static.data(name='x', shape=[None, 4, 5])
            ...     y = paddle.static.data(name='y', shape=[None, 4, 5])
            ...     z = paddle.static.nn.fc(x, 100)
            ...     loss = paddle.mean(z)
            ...     scheduler = paddle.optimizer.lr.PiecewiseDecay(boundaries=[3, 6, 9], values=[0.1, 0.2, 0.3, 0.4], verbose=True)
            ...     sgd = paddle.optimizer.SGD(learning_rate=scheduler)
            ...     sgd.minimize(loss)
            ...
            >>> exe = paddle.static.Executor()
            >>> exe.run(start_prog)
            >>> for epoch in range(20):
            ...     for batch_id in range(5):
            ...         out = exe.run(
            ...             main_prog,
            ...             feed={
            ...                 'x': np.random.randn(3, 4, 5).astype('float32'),
            ...                 'y': np.random.randn(3, 4, 5).astype('float32')
            ...             },
            ...             fetch_list=loss.name)
            ...         scheduler.step()    # If you update learning rate each step
            ...     # scheduler.step()        # If you update learning rate each epoch
    """

    def __init__(self, boundaries, values, last_epoch=-1, verbose=False):
        if len(boundaries) == 0:
            raise ValueError('The boundaries cannot be empty.')

        if len(values) <= len(boundaries):
            raise ValueError(
                f'The values have one more element than boundaries, but received len(values) [{len(values)}] < len(boundaries) + 1 [{len(boundaries) + 1}].'
            )

        self.boundaries = boundaries
        self.values = values
        super().__init__(last_epoch=last_epoch, verbose=verbose)

    def get_lr(self):
        for i in range(len(self.boundaries)):
            if self.last_epoch < self.boundaries[i]:
                return self.values[i]
        return self.values[len(self.values) - 1]


class NaturalExpDecay(LRScheduler):
    r"""

    Applies natural exponential decay to the initial learning rate.

    The algorithm can be described as following:

    .. math::

        new\_learning\_rate = learning\_rate * e^{- gamma * epoch}

    Args:
        learning_rate (float): The initial learning rate. It is a python float number.
        gamma (float, optional): A Ratio to update the learning rate, should greater than 0.0 to make learning rate decay. Default: 0.1.
        last_epoch (int, optional):  The index of last epoch. Can be set to restart training. Default: -1, means initial learning rate.
        verbose (bool, optional): If ``True``, prints a message to stdout for each update. Default: ``False`` .

    Returns:
        ``NaturalExpDecay`` instance to schedule learning rate.

    Examples:

        .. code-block:: python
            :name: code-example1

            >>> # Example1: train on default dynamic graph mode
            >>> import paddle
            >>> import numpy as np
            >>> linear = paddle.nn.Linear(10, 10)
            >>> scheduler = paddle.optimizer.lr.NaturalExpDecay(learning_rate=0.5, gamma=0.1, verbose=True)
            >>> sgd = paddle.optimizer.SGD(learning_rate=scheduler, parameters=linear.parameters())
            >>> for epoch in range(20):
            ...     for batch_id in range(5):
            ...         x = paddle.uniform([10, 10])
            ...         out = linear(x)
            ...         loss = paddle.mean(out)
            ...         loss.backward()
            ...         sgd.step()
            ...         sgd.clear_gradients()
            ...         scheduler.step()    # If you update learning rate each step
            ...     # scheduler.step()        # If you update learning rate each epoch

        .. code-block:: python
            :name: code-example2

            >>> # Example2: train on static graph mode
            >>> import paddle
            >>> import numpy as np
            >>> paddle.enable_static()
            >>> main_prog = paddle.static.Program()
            >>> start_prog = paddle.static.Program()
            >>> with paddle.static.program_guard(main_prog, start_prog):
            ...     x = paddle.static.data(name='x', shape=[None, 4, 5])
            ...     y = paddle.static.data(name='y', shape=[None, 4, 5])
            ...     z = paddle.static.nn.fc(x, 100)
            ...     loss = paddle.mean(z)
            ...     scheduler = paddle.optimizer.lr.NaturalExpDecay(learning_rate=0.5, gamma=0.1, verbose=True)
            ...     sgd = paddle.optimizer.SGD(learning_rate=scheduler)
            ...     sgd.minimize(loss)
            ...
            >>> exe = paddle.static.Executor()
            >>> exe.run(start_prog)
            >>> for epoch in range(20):
            ...     for batch_id in range(5):
            ...         out = exe.run(
            ...             main_prog,
            ...             feed={
            ...                 'x': np.random.randn(3, 4, 5).astype('float32'),
            ...                 'y': np.random.randn(3, 4, 5).astype('float32')
            ...             },
            ...             fetch_list=loss.name)
            ...         scheduler.step()    # If you update learning rate each step
            ...     # scheduler.step()        # If you update learning rate each epoch
    """

    def __init__(self, learning_rate, gamma, last_epoch=-1, verbose=False):
        assert (
            gamma > 0.0
        ), " 'gamma' must be a positive number so that the learning rate will decay."
        self.gamma = gamma
        super().__init__(learning_rate, last_epoch, verbose)

    def get_lr(self):
        return self.base_lr * math.exp(-1 * self.gamma * self.last_epoch)


class InverseTimeDecay(LRScheduler):
    r"""

    Applies inverse time decay to the initial learning rate.

    The algorithm can be described as following:

    .. math::

        new\_learning\_rate = \frac{learning\_rate}{1 + gamma * epoch}

    Args:
        learning_rate (float): The initial learning rate. It is a python float number.
        gamma (float, optional): The Ratio that the learning rate will be reduced. ``new_lr = origin_lr * gamma`` .
            It should be less than 1.0. Default: 0.1.
        last_epoch (int, optional):  The index of last epoch. Can be set to restart training. Default: -1, means initial learning rate.
        verbose (bool, optional): If ``True``, prints a message to stdout for each update. Default: ``False`` .

    Returns:
        ``InverseTimeDecay`` instance to schedule learning rate.

    Examples:

        .. code-block:: python
            :name: code-example1

            >>> # Example1: train on default dynamic graph mode
            >>> import paddle
            >>> import numpy as np

            >>> # train on default dynamic graph mode
            >>> linear = paddle.nn.Linear(10, 10)
            >>> scheduler = paddle.optimizer.lr.InverseTimeDecay(learning_rate=0.5, gamma=0.1, verbose=True)
            >>> sgd = paddle.optimizer.SGD(learning_rate=scheduler, parameters=linear.parameters())
            >>> for epoch in range(20):
            ...     for batch_id in range(5):
            ...         x = paddle.uniform([10, 10])
            ...         out = linear(x)
            ...         loss = paddle.mean(out)
            ...         loss.backward()
            ...         sgd.step()
            ...         sgd.clear_gradients()
            ...         scheduler.step()    # If you update learning rate each step
            ...     # scheduler.step()        # If you update learning rate each epoch

        .. code-block:: python
            :name: code-example2

            >>> # Example2: train on static graph mode
            >>> import paddle
            >>> import numpy as np
            >>> paddle.enable_static()
            >>> main_prog = paddle.static.Program()
            >>> start_prog = paddle.static.Program()
            >>> with paddle.static.program_guard(main_prog, start_prog):
            ...     x = paddle.static.data(name='x', shape=[None, 4, 5])
            ...     y = paddle.static.data(name='y', shape=[None, 4, 5])
            ...     z = paddle.static.nn.fc(x, 100)
            ...     loss = paddle.mean(z)
            ...     scheduler = paddle.optimizer.lr.InverseTimeDecay(learning_rate=0.5, gamma=0.1, verbose=True)
            ...     sgd = paddle.optimizer.SGD(learning_rate=scheduler)
            ...     sgd.minimize(loss)
            ...
            >>> exe = paddle.static.Executor()
            >>> exe.run(start_prog)
            >>> for epoch in range(20):
            ...     for batch_id in range(5):
            ...         out = exe.run(
            ...             main_prog,
            ...             feed={
            ...                 'x': np.random.randn(3, 4, 5).astype('float32'),
            ...                 'y': np.random.randn(3, 4, 5).astype('float32')
            ...             },
            ...             fetch_list=loss.name)
            ...         scheduler.step()    # If you update learning rate each step
            ...     # scheduler.step()        # If you update learning rate each epoch
            ...
    """

    def __init__(self, learning_rate, gamma, last_epoch=-1, verbose=False):
        self.gamma = gamma
        super().__init__(learning_rate, last_epoch, verbose)

    def get_lr(self):
        return self.base_lr / (1 + self.gamma * self.last_epoch)


class PolynomialDecay(LRScheduler):
    r"""

    Applies polynomial decay to the initial learning rate.

    The algorithm can be described as following.

    If cycle is set to True, then:

    .. math::

        decay\_steps & = decay\_steps * math.ceil(\frac{epoch}{decay\_steps})

        new\_learning\_rate & = (learning\_rate-end\_lr)*(1-\frac{epoch}{decay\_steps})^{power}+end\_lr

    If cycle is set to False, then:

    .. math::

        epoch & = min(epoch, decay\_steps)

        new\_learning\_rate & = (learning\_rate-end\_lr)*(1-\frac{epoch}{decay\_steps})^{power}+end\_lr


    Args:
        learning_rate (float): The initial learning rate. It is a python float number.
        decay_steps(int): The decay step size. It determines the decay cycle. It must be a positive integer.
        end_lr(float, optional): The minimum final learning rate. Default: 0.0001.
        power(float, optional): Power of polynomial, should greater than 0.0 to get learning rate decay. Default: 1.0.
        cycle(bool, optional): Whether the learning rate rises again. If True, then the learning rate will rise when it decrease
            to ``end_lr`` .  If False, the learning rate is monotone decreasing. Default: False.
        last_epoch (int, optional):  The index of last epoch. Can be set to restart training. Default: -1, means initial learning rate.
        verbose (bool, optional): If ``True``, prints a message to stdout for each update. Default: ``False`` .

    Returns:
        ``PolynomialDecay`` instance to schedule learning rate.

    Examples:

        .. code-block:: python
            :name: code-example1

            >>> # Example1: train on default dynamic graph mode
            >>> import paddle
            >>> import numpy as np

            >>> # train on default dynamic graph mode
            >>> linear = paddle.nn.Linear(10, 10)
            >>> scheduler = paddle.optimizer.lr.PolynomialDecay(learning_rate=0.5, decay_steps=20, verbose=True)
            >>> sgd = paddle.optimizer.SGD(learning_rate=scheduler, parameters=linear.parameters())
            >>> for epoch in range(20):
            ...     for batch_id in range(5):
            ...         x = paddle.uniform([10, 10])
            ...         out = linear(x)
            ...         loss = paddle.mean(out)
            ...         loss.backward()
            ...         sgd.step()
            ...         sgd.clear_gradients()
            ...         scheduler.step()    # If you update learning rate each step
            ...     # scheduler.step()        # If you update learning rate each epoch

        .. code-block:: python
            :name: code-example2

            >>> # Example2: train on static graph mode
            >>> import paddle
            >>> import numpy as np
            >>> paddle.enable_static()
            >>> main_prog = paddle.static.Program()
            >>> start_prog = paddle.static.Program()
            >>> with paddle.static.program_guard(main_prog, start_prog):
            ...     x = paddle.static.data(name='x', shape=[None, 4, 5])
            ...     y = paddle.static.data(name='y', shape=[None, 4, 5])
            ...     z = paddle.static.nn.fc(x, 100)
            ...     loss = paddle.mean(z)
            ...     scheduler = paddle.optimizer.lr.PolynomialDecay(learning_rate=0.5, decay_steps=20, verbose=True)
            ...     sgd = paddle.optimizer.SGD(learning_rate=scheduler)
            ...     sgd.minimize(loss)
            ...
            >>> exe = paddle.static.Executor()
            >>> exe.run(start_prog)
            >>> for epoch in range(20):
            ...     for batch_id in range(5):
            ...         out = exe.run(
            ...             main_prog,
            ...             feed={
            ...                 'x': np.random.randn(3, 4, 5).astype('float32'),
            ...                 'y': np.random.randn(3, 4, 5).astype('float32')
            ...             },
            ...             fetch_list=loss.name)
            ...         scheduler.step()    # If you update learning rate each step
            ...     # scheduler.step()        # If you update learning rate each epoch
    """

    def __init__(
        self,
        learning_rate,
        decay_steps,
        end_lr=0.0001,
        power=1.0,
        cycle=False,
        last_epoch=-1,
        verbose=False,
    ):
        assert decay_steps > 0 and isinstance(
            decay_steps, int
        ), " 'decay_steps' must be a positive integer."
        self.decay_steps = decay_steps
        self.end_lr = end_lr
        assert (
            power > 0.0
        ), " 'power' must be greater than 0.0 so that the learning rate will decay."
        self.power = power
        self.cycle = cycle
        super().__init__(learning_rate, last_epoch, verbose)

    def get_lr(self):
        tmp_epoch_num = self.last_epoch
        tmp_decay_steps = self.decay_steps
        if self.cycle:
            div_res = math.ceil(
                float(self.last_epoch) / float(self.decay_steps)
            )

            if self.last_epoch == 0:
                div_res = 1
            tmp_decay_steps = self.decay_steps * div_res
        else:
            tmp_epoch_num = min(self.last_epoch, self.decay_steps)

        return (self.base_lr - self.end_lr) * (
            (1 - float(tmp_epoch_num) / float(tmp_decay_steps)) ** self.power
        ) + self.end_lr


class LinearWarmup(LRScheduler):
    r"""

    Linear learning rate warm up strategy. Update the learning rate preliminarily before the normal learning rate scheduler.
    For more information, please refer to `Bag of Tricks for Image Classification with Convolutional Neural Networks <https://arxiv.org/abs/1812.01187>`_

    When epoch < warmup_steps, learning rate is updated as:

    .. math::

            lr = start\_lr + (end\_lr - start\_lr) * \frac{epoch}{warmup\_steps}

    where start_lr is the initial learning rate, and end_lr is the final learning rate;

    When epoch >= warmup_steps, learning rate is updated as:

    .. math::

            lr = learning_rate

    where ``learning_rate`` is float or any subclass of ``LRScheduler`` .

    Args:
        learning_rate (float|LRScheduler): The learning rate after warm-up. It is a python float number or any subclass of ``LRScheduler`` .
        warmup_steps (int): total steps of warm up. It must be a positive integer.
        start_lr (float): Initial learning rate of warm up.
        end_lr (float): Final learning rate of warm up.
        last_epoch (int, optional):  The index of last epoch. Can be set to restart training. Default: -1, means initial learning rate.
        verbose (bool, optional): If ``True``, prints a message to stdout for each update. Default: ``False`` .

    Returns:
        ``LinearWarmup`` instance to schedule learning rate.

    Examples:

        .. code-block:: python
            :name: code-example1

            >>> # Example1: train on default dynamic graph mode
            >>> import paddle
            >>> import numpy as np

            >>> # train on default dynamic graph mode
            >>> linear = paddle.nn.Linear(10, 10)
            >>> scheduler = paddle.optimizer.lr.LinearWarmup(
            ...         learning_rate=0.5, warmup_steps=20, start_lr=0, end_lr=0.5, verbose=True)
            >>> sgd = paddle.optimizer.SGD(learning_rate=scheduler, parameters=linear.parameters())
            >>> for epoch in range(20):
            ...     for batch_id in range(5):
            ...         x = paddle.uniform([10, 10])
            ...         out = linear(x)
            ...         loss = paddle.mean(out)
            ...         loss.backward()
            ...         sgd.step()
            ...         sgd.clear_gradients()
            ...         scheduler.step()    # If you update learning rate each step
            ...     # scheduler.step()        # If you update learning rate each epoch

        .. code-block:: python
            :name: code-example2

            >>> # Example2: train on static graph mode
            >>> import paddle
            >>> import numpy as np
            >>> paddle.enable_static()
            >>> main_prog = paddle.static.Program()
            >>> start_prog = paddle.static.Program()
            >>> with paddle.static.program_guard(main_prog, start_prog):
            ...     x = paddle.static.data(name='x', shape=[None, 4, 5])
            ...     y = paddle.static.data(name='y', shape=[None, 4, 5])
            ...     z = paddle.static.nn.fc(x, 100)
            ...     loss = paddle.mean(z)
            ...     scheduler = paddle.optimizer.lr.LinearWarmup(
            ...         learning_rate=0.5, warmup_steps=20, start_lr=0, end_lr=0.5, verbose=True)
            ...     sgd = paddle.optimizer.SGD(learning_rate=scheduler)
            ...     sgd.minimize(loss)
            ...
            >>> exe = paddle.static.Executor()
            >>> exe.run(start_prog)
            >>> for epoch in range(20):
            ...     for batch_id in range(5):
            ...         out = exe.run(
            ...             main_prog,
            ...             feed={
            ...                 'x': np.random.randn(3, 4, 5).astype('float32'),
            ...                 'y': np.random.randn(3, 4, 5).astype('float32')
            ...             },
            ...             fetch_list=loss.name)
            ...         scheduler.step()    # If you update learning rate each step
            ...     # scheduler.step()        # If you update learning rate each epoch
    """

    def __init__(
        self,
        learning_rate,
        warmup_steps,
        start_lr,
        end_lr,
        last_epoch=-1,
        verbose=False,
    ):
        type_check = isinstance(learning_rate, (float, int, LRScheduler))
        if not type_check:
            raise TypeError(
                "the type of learning_rate should be [int, float or LRScheduler], the current type is {}".format(
                    learning_rate
                )
            )
        self.learning_rate = learning_rate
        assert warmup_steps > 0 and isinstance(
            warmup_steps, int
        ), " 'warmup_steps' must be a positive integer."
        self.warmup_steps = warmup_steps
        self.start_lr = start_lr
        self.end_lr = end_lr
        assert (
            end_lr > start_lr
        ), f"end_lr {end_lr} must be greater than start_lr {start_lr}"
        super().__init__(start_lr, last_epoch, verbose)

    def state_dict(self):
        """
        Returns the state of the LinearWarmup scheduler as a :class:`dict`.

        It is a subset of ``self.__dict__`` .
        """
        state_dict = super().state_dict()
        if isinstance(self.learning_rate, LRScheduler):
            state_dict["LinearWarmup_LR"] = self.learning_rate.state_dict()
        return state_dict

    def set_state_dict(self, state_dict):
        """
        Loads state_dict for LinearWarmup scheduler.
        """
        super().set_state_dict(state_dict)
        if isinstance(self.learning_rate, LRScheduler):
            self.learning_rate.set_state_dict(state_dict["LinearWarmup_LR"])

    def get_lr(self):
        if self.last_epoch < self.warmup_steps:
            return (self.end_lr - self.start_lr) * float(
                self.last_epoch
            ) / float(self.warmup_steps) + self.start_lr
        else:
            if isinstance(self.learning_rate, LRScheduler):
                self.learning_rate.step(self.last_epoch - self.warmup_steps)
                return self.learning_rate()

            return self.learning_rate


class ExponentialDecay(LRScheduler):
    r"""

    Update learning rate by `gamma` each epoch.

    The algorithm can be described as following.

    .. math::

        new\_learning\_rate = last\_learning\_rate * gamma

    Args:
        learning_rate (float): The initial learning rate. It is a python float number.
        gamma (float): The Ratio that the learning rate will be reduced. ``new_lr = origin_lr * gamma`` .
            It should be in interval (0.0, 1.0).
        last_epoch (int, optional):  The index of last epoch. Can be set to restart training. Default: -1, means initial learning rate.
        verbose (bool, optional): If ``True``, prints a message to stdout for each update. Default: ``False`` .

    Returns:
        ``ExponentialDecay`` instance to schedule learning rate.

    Examples:

        .. code-block:: python
            :name: code-example1

            >>> # Example1: train on default dynamic graph mode
            >>> import paddle
            >>> import numpy as np

            >>> # train on default dynamic graph mode
            >>> linear = paddle.nn.Linear(10, 10)
            >>> scheduler = paddle.optimizer.lr.ExponentialDecay(learning_rate=0.5, gamma=0.9, verbose=True)
            >>> sgd = paddle.optimizer.SGD(learning_rate=scheduler, parameters=linear.parameters())
            >>> for epoch in range(20):
            ...     for batch_id in range(5):
            ...         x = paddle.uniform([10, 10])
            ...         out = linear(x)
            ...         loss = paddle.mean(out)
            ...         loss.backward()
            ...         sgd.step()
            ...         sgd.clear_gradients()
            ...         scheduler.step()    # If you update learning rate each step
            ...     # scheduler.step()        # If you update learning rate each epoch

        .. code-block:: python
            :name: code-example2

            >>> # Example2: train on static graph mode
            >>> import paddle
            >>> import numpy as np
            >>> paddle.enable_static()
            >>> main_prog = paddle.static.Program()
            >>> start_prog = paddle.static.Program()
            >>> with paddle.static.program_guard(main_prog, start_prog):
            ...     x = paddle.static.data(name='x', shape=[None, 4, 5])
            ...     y = paddle.static.data(name='y', shape=[None, 4, 5])
            ...     z = paddle.static.nn.fc(x, 100)
            ...     loss = paddle.mean(z)
            ...     scheduler = paddle.optimizer.lr.ExponentialDecay(learning_rate=0.5, gamma=0.9, verbose=True)
            ...     sgd = paddle.optimizer.SGD(learning_rate=scheduler)
            ...     sgd.minimize(loss)
            ...
            >>> exe = paddle.static.Executor()
            >>> exe.run(start_prog)
            >>> for epoch in range(20):
            ...     for batch_id in range(5):
            ...         out = exe.run(
            ...             main_prog,
            ...             feed={
            ...                 'x': np.random.randn(3, 4, 5).astype('float32'),
            ...                 'y': np.random.randn(3, 4, 5).astype('float32')
            ...             },
            ...             fetch_list=loss.name)
            ...         scheduler.step()    # If you update learning rate each step
            ...     # scheduler.step()        # If you update learning rate each epoch
    """

    def __init__(self, learning_rate, gamma, last_epoch=-1, verbose=False):
        assert (
            gamma > 0.0 and gamma < 1.0
        ), " 'gamma' must be in interval (0.0, 1.0) so that the learning rate will decay."
        self.gamma = gamma
        super().__init__(learning_rate, last_epoch, verbose)

    def get_lr(self):
        return self.base_lr * (self.gamma**self.last_epoch)


class MultiStepDecay(LRScheduler):
    """
    Update the learning rate by ``gamma`` once ``epoch`` reaches one of the milestones.

    The algorithm can be described as the code below.

    .. code-block:: text

        learning_rate = 0.5
        milestones = [30, 50]
        gamma = 0.1
        if epoch < 30:
            learning_rate = 0.5
        elif epoch < 50:
            learning_rate = 0.05
        else:
            learning_rate = 0.005

    Args:
        learning_rate (float): The initial learning rate. It is a python float number.
        milestones (tuple|list): List or tuple of each boundaries. Must be increasing.
        gamma (float, optional): The Ratio that the learning rate will be reduced. ``new_lr = origin_lr * gamma`` .
            It should be less than 1.0. Default: 0.1.
        last_epoch (int, optional):  The index of last epoch. Can be set to restart training. Default: -1, means initial learning rate.
        verbose (bool, optional): If ``True``, prints a message to stdout for each update. Default: ``False`` .


    Returns:
        ``MultiStepDecay`` instance to schedule learning rate.

    Examples:

        .. code-block:: python
            :name: code-example1

            >>> # Example1: train on default dynamic graph mode
            >>> import paddle
            >>> import numpy as np

            >>> # train on default dynamic graph mode
            >>> linear = paddle.nn.Linear(10, 10)
            >>> scheduler = paddle.optimizer.lr.MultiStepDecay(learning_rate=0.5, milestones=[2, 4, 6], gamma=0.8, verbose=True)
            >>> sgd = paddle.optimizer.SGD(learning_rate=scheduler, parameters=linear.parameters())
            >>> for epoch in range(20):
            ...     for batch_id in range(5):
            ...         x = paddle.uniform([10, 10])
            ...         out = linear(x)
            ...         loss = paddle.mean(out)
            ...         loss.backward()
            ...         sgd.step()
            ...         sgd.clear_gradients()
            ...         scheduler.step()    # If you update learning rate each step
            ...     # scheduler.step()        # If you update learning rate each epoch

        .. code-block:: python
            :name: code-example2

            >>> # Example2: train on static graph mode
            >>> import paddle
            >>> import numpy as np
            >>> paddle.enable_static()
            >>> main_prog = paddle.static.Program()
            >>> start_prog = paddle.static.Program()
            >>> with paddle.static.program_guard(main_prog, start_prog):
            ...     x = paddle.static.data(name='x', shape=[None, 4, 5])
            ...     y = paddle.static.data(name='y', shape=[None, 4, 5])
            ...     z = paddle.static.nn.fc(x, 100)
            ...     loss = paddle.mean(z)
            ...     scheduler = paddle.optimizer.lr.MultiStepDecay(learning_rate=0.5, milestones=[2, 4, 6], gamma=0.8, verbose=True)
            ...     sgd = paddle.optimizer.SGD(learning_rate=scheduler)
            ...     sgd.minimize(loss)
            ...
            >>> exe = paddle.static.Executor()
            >>> exe.run(start_prog)
            >>> for epoch in range(20):
            ...     for batch_id in range(5):
            ...         out = exe.run(
            ...             main_prog,
            ...             feed={
            ...                 'x': np.random.randn(3, 4, 5).astype('float32'),
            ...                 'y': np.random.randn(3, 4, 5).astype('float32')
            ...             },
            ...             fetch_list=loss.name)
            ...         scheduler.step()    # If you update learning rate each step
            ...     # scheduler.step()        # If you update learning rate each epoch
    """

    def __init__(
        self, learning_rate, milestones, gamma=0.1, last_epoch=-1, verbose=False
    ):
        if not isinstance(milestones, (tuple, list)):
            raise TypeError(
                "The type of 'milestones' in 'MultiStepDecay' must be 'tuple, list', but received %s."
                % type(milestones)
            )

        if not all(
            milestones[i] < milestones[i + 1]
            for i in range(len(milestones) - 1)
        ):
            raise ValueError('The elements of milestones must be incremented')
        if gamma >= 1.0:
            raise ValueError('gamma should be < 1.0.')

        self.milestones = milestones
        self.gamma = gamma
        super().__init__(learning_rate, last_epoch, verbose)

    def get_lr(self):
        for i in range(len(self.milestones)):
            if self.last_epoch < self.milestones[i]:
                return self.base_lr * (self.gamma**i)
        return self.base_lr * (self.gamma ** len(self.milestones))


class StepDecay(LRScheduler):
    """
    Update the learning rate of ``optimizer`` by ``gamma`` every ``step_size`` number of epoch.

    The algorithm can be described as the code below.

    .. code-block:: text

        learning_rate = 0.5
        step_size = 30
        gamma = 0.1

        learning_rate = 0.5     if epoch < 30
        learning_rate = 0.05    if 30 <= epoch < 60
        learning_rate = 0.005   if 60 <= epoch < 90
        ...

    Args:
        learning_rate (float): The initial learning rate. It is a python float number.
        step_size (int): the interval to update. It must be a positive integer.
        gamma (float, optional): The Ratio that the learning rate will be reduced. ``new_lr = origin_lr * gamma`` .
            It should be less than 1.0. Default: 0.1.
        last_epoch (int, optional):  The index of last epoch. Can be set to restart training. Default: -1, means initial learning rate.
        verbose (bool, optional): If ``True``, prints a message to stdout for each update. Default: ``False`` .

    Returns:
        ``StepDecay`` instance to schedule learning rate.


    Examples:

        .. code-block:: python
            :name: code-example1

            >>> # Example1: train on default dynamic graph mode
            >>> import paddle
            >>> import numpy as np

            >>> # train on default dynamic graph mode
            >>> linear = paddle.nn.Linear(10, 10)
            >>> scheduler = paddle.optimizer.lr.StepDecay(learning_rate=0.5, step_size=5, gamma=0.8, verbose=True)
            >>> sgd = paddle.optimizer.SGD(learning_rate=scheduler, parameters=linear.parameters())
            >>> for epoch in range(20):
            ...     for batch_id in range(5):
            ...         x = paddle.uniform([10, 10])
            ...         out = linear(x)
            ...         loss = paddle.mean(out)
            ...         loss.backward()
            ...         sgd.step()
            ...         sgd.clear_gradients()
            ...         scheduler.step()    # If you update learning rate each step
            ...     # scheduler.step()        # If you update learning rate each epoch

        .. code-block:: python
            :name: code-example2

            >>> # Example2: train on static graph mode
            >>> import paddle
            >>> import numpy as np
            >>> paddle.enable_static()
            >>> main_prog = paddle.static.Program()
            >>> start_prog = paddle.static.Program()
            >>> with paddle.static.program_guard(main_prog, start_prog):
            ...     x = paddle.static.data(name='x', shape=[None, 4, 5])
            ...     y = paddle.static.data(name='y', shape=[None, 4, 5])
            ...     z = paddle.static.nn.fc(x, 100)
            ...     loss = paddle.mean(z)
            ...     scheduler = paddle.optimizer.lr.StepDecay(learning_rate=0.5, step_size=5, gamma=0.8, verbose=True)
            ...     sgd = paddle.optimizer.SGD(learning_rate=scheduler)
            ...     sgd.minimize(loss)
            ...
            >>> exe = paddle.static.Executor()
            >>> exe.run(start_prog)
            >>> for epoch in range(20):
            ...     for batch_id in range(5):
            ...         out = exe.run(
            ...             main_prog,
            ...             feed={
            ...                 'x': np.random.randn(3, 4, 5).astype('float32'),
            ...                 'y': np.random.randn(3, 4, 5).astype('float32')
            ...             },
            ...             fetch_list=loss.name)
            ...         scheduler.step()    # If you update learning rate each step
            ...     # scheduler.step()        # If you update learning rate each epoch
    """

    def __init__(
        self, learning_rate, step_size, gamma=0.1, last_epoch=-1, verbose=False
    ):
        if not isinstance(step_size, int):
            raise TypeError(
                "The type of 'step_size' must be 'int', but received %s."
                % type(step_size)
            )
        if gamma >= 1.0:
            raise ValueError('gamma should be < 1.0.')

        assert step_size > 0 and isinstance(
            step_size, int
        ), " 'step_size' must be a positive integer."
        self.step_size = step_size
        self.gamma = gamma
        super().__init__(learning_rate, last_epoch, verbose)

    def get_lr(self):
        i = self.last_epoch // self.step_size
        return self.base_lr * (self.gamma**i)


class LambdaDecay(LRScheduler):
    """
    Sets the learning rate of ``optimizer`` by function ``lr_lambda`` . ``lr_lambda`` is function which receives ``epoch`` .

    The algorithm can be described as the code below.

    .. code-block:: text

        learning_rate = 0.5        # init learning_rate
        lr_lambda = lambda epoch: 0.95 ** epoch

        learning_rate = 0.5        # epoch 0, 0.5*0.95**0
        learning_rate = 0.475      # epoch 1, 0.5*0.95**1
        learning_rate = 0.45125    # epoch 2, 0.5*0.95**2

    Args:
        learning_rate (float): The initial learning rate. It is a python float number.
        lr_lambda (function): A function which computes a factor by ``epoch`` , and then multiply the initial learning rate by this factor.
        last_epoch (int, optional):  The index of last epoch. Can be set to restart training. Default: -1, means initial learning rate.
        verbose (bool, optional): If ``True``, prints a message to stdout for each update. Default: ``False`` .

    Returns:
        ``LambdaDecay`` instance to schedule learning rate.

    Examples:

        .. code-block:: python
            :name: code-example1

            >>> # Example1: train on default dynamic graph mode
            >>> import paddle
            >>> import numpy as np

            >>> # train on default dynamic graph mode
            >>> linear = paddle.nn.Linear(10, 10)
            >>> scheduler = paddle.optimizer.lr.LambdaDecay(learning_rate=0.5, lr_lambda=lambda x:0.95**x, verbose=True)
            >>> sgd = paddle.optimizer.SGD(learning_rate=scheduler, parameters=linear.parameters())
            >>> for epoch in range(20):
            ...     for batch_id in range(5):
            ...         x = paddle.uniform([10, 10])
            ...         out = linear(x)
            ...         loss = paddle.mean(out)
            ...         loss.backward()
            ...         sgd.step()
            ...         sgd.clear_gradients()
            ...         scheduler.step()    # If you update learning rate each step
            ...     # scheduler.step()        # If you update learning rate each epoch

        .. code-block:: python
            :name: code-example2

            >>> # Example2: train on static graph mode
            >>> import paddle
            >>> import numpy as np
            >>> paddle.enable_static()
            >>> main_prog = paddle.static.Program()
            >>> start_prog = paddle.static.Program()
            >>> with paddle.static.program_guard(main_prog, start_prog):
            ...     x = paddle.static.data(name='x', shape=[None, 4, 5])
            ...     y = paddle.static.data(name='y', shape=[None, 4, 5])
            ...     z = paddle.static.nn.fc(x, 100)
            ...     loss = paddle.mean(z)
            ...     scheduler = paddle.optimizer.lr.LambdaDecay(learning_rate=0.5, lr_lambda=lambda x:0.95**x, verbose=True)
            ...     sgd = paddle.optimizer.SGD(learning_rate=scheduler)
            ...     sgd.minimize(loss)
            ...
            >>> exe = paddle.static.Executor()
            >>> exe.run(start_prog)
            >>> for epoch in range(20):
            ...     for batch_id in range(5):
            ...         out = exe.run(
            ...             main_prog,
            ...             feed={
            ...                 'x': np.random.randn(3, 4, 5).astype('float32'),
            ...                 'y': np.random.randn(3, 4, 5).astype('float32')
            ...             },
            ...             fetch_list=loss.name)
            ...         scheduler.step()    # If you update learning rate each step
            ...     # scheduler.step()        # If you update learning rate each epoch
            ...
    """

    def __init__(self, learning_rate, lr_lambda, last_epoch=-1, verbose=False):
        if not callable(lr_lambda):
            raise TypeError(
                "The type of 'lr_lambda' in 'LambdaDecay' must be 'function', but received %s."
                % type(lr_lambda)
            )

        self.lr_lambda = lr_lambda
        super().__init__(learning_rate, last_epoch, verbose)

    def get_lr(self):
        return self.base_lr * self.lr_lambda(self.last_epoch)


class ReduceOnPlateau(LRScheduler):
    """
    Reduce learning rate when ``metrics`` has stopped descending. Models often benefit from reducing the learning rate
    by 2 to 10 times once model performance has no longer improvement.

    The ``metrics`` is the one which has been pass into ``step`` , it's shape must [] or [1]. When ``metrics``
    stop descending for a ``patience`` number of epochs, the learning rate will be reduced to ``learning_rate * factor`` .
    (Specially, ``mode`` can also be set to ``'max`` , in this case, when ``metrics`` stop ascending for a ``patience``
    number of epochs, the learning rate will be reduced.)

    In addition, After each reduction, it will wait a ``cooldown`` number of epochs before resuming above operation.

    Args:
        learning_rate (float): The initial learning rate. It is a python float number.
        mode (str, optional): ``'min'`` or ``'max'`` can be selected. Normally, it is ``'min'`` , which means that the
            learning rate will reduce when ``loss`` stops descending. Specially, if it's set to ``'max'`` ,  the learning
            rate will reduce when ``loss`` stops ascending. Default: ``'min'`` .
        factor (float, optional): The Ratio that the learning rate will be reduced. ``new_lr = origin_lr * factor`` .
            It should be less than 1.0. Default: 0.1.
        patience (int, optional): When ``loss`` doesn't improve for this number of epochs, learing rate will be reduced.
            Default: 10.
        threshold (float, optional): ``threshold`` and ``threshold_mode`` will determine the minimum change of ``loss`` .
            This make tiny changes of ``loss`` will be ignored. Default: 1e-4.
        threshold_mode (str, optional): ``'rel'`` or ``'abs'`` can be selected. In ``'rel'`` mode, the minimum change of ``loss``
            is ``last_loss * threshold`` , where ``last_loss`` is ``loss`` in last epoch. In ``'abs'`` mode, the minimum
            change of ``loss`` is ``threshold`` . Default: ``'rel'`` .
        cooldown (int, optional): The number of epochs to wait before resuming normal operation. Default: 0.
        min_lr (float, optional): The lower bound of the learning rate after reduction. Default: 0.
        epsilon (float, optional): Minimal decay applied to lr. If the difference between new and old lr is smaller than epsilon,
            the update is ignored. Default: 1e-8.
        verbose (bool, optional): If ``True``, prints a message to stdout for each update. Default: ``False``.


    Returns:
        ``ReduceOnPlateau`` instance to schedule learning rate.


    Examples:
        .. code-block:: python
            :name: code-example1

            >>> # Example1: train on default dynamic graph mode
            >>> import paddle
            >>> import numpy as np

            >>> # train on default dynamic graph mode
            >>> linear = paddle.nn.Linear(10, 10)
            >>> scheduler = paddle.optimizer.lr.ReduceOnPlateau(learning_rate=1.0, factor=0.5, patience=5, verbose=True)
            >>> sgd = paddle.optimizer.SGD(learning_rate=scheduler, parameters=linear.parameters())
            >>> for epoch in range(20):
            ...     for batch_id in range(5):
            ...         x = paddle.uniform([10, 10])
            ...         out = linear(x)
            ...         loss = paddle.mean(out)
            ...         loss.backward()
            ...         sgd.step()
            ...         sgd.clear_gradients()
            ...         scheduler.step(loss)    # If you update learning rate each step
            ...     # scheduler.step(loss)        # If you update learning rate each epoch

        .. code-block:: python
            :name: code-example2

            >>> # Example2: train on static graph mode
            >>> import paddle
            >>> import numpy as np
            >>> paddle.enable_static()
            >>> main_prog = paddle.static.Program()
            >>> start_prog = paddle.static.Program()
            >>> with paddle.static.program_guard(main_prog, start_prog):
            ...     x = paddle.static.data(name='x', shape=[None, 4, 5])
            ...     y = paddle.static.data(name='y', shape=[None, 4, 5])
            ...     z = paddle.static.nn.fc(x, 100)
            ...     loss = paddle.mean(z)
            ...     scheduler = paddle.optimizer.lr.ReduceOnPlateau(learning_rate=1.0, factor=0.5, patience=5, verbose=True)
            ...     sgd = paddle.optimizer.SGD(learning_rate=scheduler)
            ...     sgd.minimize(loss)
            ...
            >>> exe = paddle.static.Executor()
            >>> exe.run(start_prog)
            >>> for epoch in range(20):
            ...     for batch_id in range(5):
            ...         out = exe.run(
            ...             main_prog,
            ...             feed={
            ...                 'x': np.random.randn(3, 4, 5).astype('float32'),
            ...                 'y': np.random.randn(3, 4, 5).astype('float32')
            ...             },
            ...             fetch_list=loss.name)
            ...         scheduler.step(out[0])    # If you update learning rate each step
            ...     # scheduler.step(out[0])        # If you update learning rate each epoch
            ...
    """

    def __init__(
        self,
        learning_rate,
        mode='min',
        factor=0.1,
        patience=10,
        threshold=1e-4,
        threshold_mode='rel',
        cooldown=0,
        min_lr=0,
        epsilon=1e-8,
        verbose=False,
    ):
        mode = mode.lower()
        if mode not in ['min', 'max']:
            raise ValueError('mode: ' + mode + ' is unknown!')
        self.mode = mode

        if factor >= 1.0:
            raise ValueError(
                'new_lr = origin_lr * gamma and gamma should be < 1.0.'
            )
        self.factor = factor

        threshold_mode = threshold_mode.lower()
        if threshold_mode not in ['rel', 'abs']:
            raise ValueError(
                'threshold mode: ' + threshold_mode + ' is unknown!'
            )
        self.threshold_mode = threshold_mode
        if not isinstance(learning_rate, (float, int)):
            raise TypeError(
                "The type of 'learning_rate' in 'ReduceOnPlateau' must be 'float', but received %s."
                % type(learning_rate)
            )

        self.patience = patience
        self.threshold = threshold
        self.threshold_mode = threshold_mode
        self.cooldown = cooldown
        self.min_lr = min_lr
        self.epsilon = epsilon

        self.cooldown_counter = 0
        self.best = None
        self.num_bad_epochs = 0

        # Can not call Parent __init__, so implement here.
        self.base_lr = float(learning_rate)
        self.last_lr = float(learning_rate)
        self.last_epoch = 0
        self.verbose = verbose
        self._var_name = None

    # "cooldown_counter / best / num_bad_epochs / last_epoch / last_lr" will be stored.
    def state_keys(self):
        self.keys = [
            'cooldown_counter',
            'best',
            'num_bad_epochs',
            'last_epoch',
            'last_lr',
        ]

    def step(self, metrics, epoch=None):
        """
        step should be called after `optimizer.step()` . It will update the learning rate in optimizer according to ``metrics`` .
        The new learning rate will take effect on next epoch.

        Args:
            metrics (Tensor|numpy.ndarray|float): Which will be monitored to determine whether the learning rate will reduce.
                If it stop descending for a ``patience`` number of epochs, the learning rate will reduce. If it's 'Tensor' or
                'numpy.ndarray', its numel must be 1.
            epoch (int, None): specify current epoch. Default: None. Auto-increment from last_epoch=-1.

        Returns:
            None

        Examples:
            Please refer to the example of current LRScheduler.
        """
        if epoch is None:
            self.last_epoch = self.last_epoch + 1
        else:
            self.last_epoch = epoch

        # loss must be float, numpy.ndarray or 1-D Tensor with numel 1
        if isinstance(metrics, (core.eager.Tensor, numpy.ndarray)):
            assert metrics.size == 1, (
                "the size of metrics must be 1, but the current metrics.size is {}. Maybe that "
                "you should call paddle.mean to process it first.".format(
                    metrics.size
                )
            )
        elif not isinstance(
            metrics, (int, float, numpy.float32, numpy.float64)
        ):
            raise TypeError(
                "metrics must be 'int', 'float', 'np.float64', 'numpy.ndarray' or 'paddle.Tensor', but receive {}".format(
                    type(metrics)
                )
            )

        if self.cooldown_counter > 0:
            self.cooldown_counter -= 1
        else:
            if self.best is None or self._is_better(metrics, self.best):
                self.best = metrics
                self.num_bad_epochs = 0
            else:
                self.num_bad_epochs += 1

            if self.num_bad_epochs > self.patience:
                self.cooldown_counter = self.cooldown
                self.num_bad_epochs = 0
                new_lr = max(self.last_lr * self.factor, self.min_lr)
                if self.last_lr - new_lr > self.epsilon:
                    self.last_lr = new_lr
                    if self.verbose:
                        print(
                            'Epoch {}: {} set learning rate to {}.'.format(
                                self.last_epoch,
                                self.__class__.__name__,
                                self.last_lr,
                            )
                        )

    def _is_better(self, current, best):
        if self.mode == 'min' and self.threshold_mode == 'rel':
            return current < best - best * self.threshold

        elif self.mode == 'min' and self.threshold_mode == 'abs':
            return current < best - self.threshold

        elif self.mode == 'max' and self.threshold_mode == 'rel':
            return current > best + best * self.threshold

        else:
            return current > best + self.threshold


class CosineAnnealingDecay(LRScheduler):
    r"""

    Set the learning rate using a cosine annealing schedule, where :math:`\eta_{max}` is set to
    the initial learning_rate. :math:`T_{cur}` is the number of epochs since the last restart in
    SGDR.

    The algorithm can be described as following.

    .. math::

        \eta_t & = \eta_{min} + \frac{1}{2}(\eta_{max} - \eta_{min})\left(1
        + \cos\left(\frac{T_{cur}}{T_{max}}\pi\right)\right),
        & T_{cur} \neq (2k+1)T_{max};

        \eta_{t+1} & = \eta_{t} + \frac{1}{2}(\eta_{max} - \eta_{min})
        \left(1 - \cos\left(\frac{1}{T_{max}}\pi\right)\right),
        & T_{cur} = (2k+1)T_{max}.

    It has been proposed in `SGDR: Stochastic Gradient Descent with Warm Restarts <https://arxiv.org/abs/1608.03983>`_.
    Note that this only implements the cosine annealing part of SGDR, and not the restarts.

    Args:
        learning_rate (float): The initial learning rate, that is :math:`\eta_{max}` . It can be set to python float or int number.
        T_max (int): Maximum number of iterations. It is half of the decay cycle of learning rate. It must be a positive integer.
        eta_min (float|int, optional): Minimum learning rate, that is :math:`\eta_{min}` . Default: 0.
        last_epoch (int, optional):  The index of last epoch. Can be set to restart training. Default: -1, means initial learning rate.
        verbose (bool, optional): If ``True``, prints a message to stdout for each update. Default: ``False`` .

    Returns:
        ``CosineAnnealingDecay`` instance to schedule learning rate.

    Examples:

        .. code-block:: python
            :name: code-example1

            >>> # Example1: train on default dynamic graph mode
            >>> import paddle
            >>> import numpy as np

            >>> # train on default dynamic graph mode
            >>> linear = paddle.nn.Linear(10, 10)
            >>> scheduler = paddle.optimizer.lr.CosineAnnealingDecay(learning_rate=0.5, T_max=10, verbose=True)
            >>> sgd = paddle.optimizer.SGD(learning_rate=scheduler, parameters=linear.parameters())
            >>> for epoch in range(20):
            ...     for batch_id in range(5):
            ...         x = paddle.uniform([10, 10])
            ...         out = linear(x)
            ...         loss = paddle.mean(out)
            ...         loss.backward()
            ...         sgd.step()
            ...         sgd.clear_gradients()
            ...         scheduler.step()    # If you update learning rate each step
            ...     # scheduler.step()        # If you update learning rate each epoch

        .. code-block:: python
            :name: code-example2

            >>> # Example2: train on static graph mode
            >>> import paddle
            >>> import numpy as np
            >>> paddle.enable_static()
            >>> main_prog = paddle.static.Program()
            >>> start_prog = paddle.static.Program()
            >>> with paddle.static.program_guard(main_prog, start_prog):
            ...     x = paddle.static.data(name='x', shape=[None, 4, 5])
            ...     y = paddle.static.data(name='y', shape=[None, 4, 5])
            ...     z = paddle.static.nn.fc(x, 100)
            ...     loss = paddle.mean(z)
            ...     scheduler = paddle.optimizer.lr.CosineAnnealingDecay(learning_rate=0.5, T_max=10, verbose=True)
            ...     sgd = paddle.optimizer.SGD(learning_rate=scheduler)
            ...     sgd.minimize(loss)
            ...
            >>> exe = paddle.static.Executor()
            >>> exe.run(start_prog)
            >>> for epoch in range(20):
            ...     for batch_id in range(5):
            ...         out = exe.run(
            ...             main_prog,
            ...             feed={
            ...                 'x': np.random.randn(3, 4, 5).astype('float32'),
            ...                 'y': np.random.randn(3, 4, 5).astype('float32')
            ...             },
            ...             fetch_list=loss.name)
            ...         scheduler.step()    # If you update learning rate each step
            ...     # scheduler.step()        # If you update learning rate each epoch
    """

    def __init__(
        self, learning_rate, T_max, eta_min=0, last_epoch=-1, verbose=False
    ):
        if not isinstance(T_max, int):
            raise TypeError(
                "The type of 'T_max' in 'CosineAnnealingDecay' must be 'int', but received %s."
                % type(T_max)
            )
        if not isinstance(eta_min, (float, int)):
            raise TypeError(
                "The type of 'eta_min' in 'CosineAnnealingDecay' must be 'float, int', but received %s."
                % type(eta_min)
            )
        assert T_max > 0 and isinstance(
            T_max, int
        ), " 'T_max' must be a positive integer."
        self.T_max = T_max
        self.eta_min = float(eta_min)
        super().__init__(learning_rate, last_epoch, verbose)

    def get_lr(self):
        if self.last_epoch == 0:
            return self.base_lr
        elif (self.last_epoch - 1 - self.T_max) % (2 * self.T_max) == 0:
            return (
                self.last_lr
                + (self.base_lr - self.eta_min)
                * (1 - math.cos(math.pi / self.T_max))
                / 2
            )

        return (1 + math.cos(math.pi * self.last_epoch / self.T_max)) / (
            1 + math.cos(math.pi * (self.last_epoch - 1) / self.T_max)
        ) * (self.last_lr - self.eta_min) + self.eta_min

    def _get_closed_form_lr(self):
        return (
            self.eta_min
            + (self.base_lr - self.eta_min)
            * (1 + math.cos(math.pi * self.last_epoch / self.T_max))
            / 2
        )


class MultiplicativeDecay(LRScheduler):
    """
    Multiply the learning rate of ``optimizer`` by the factor given in function ``lr_lambda`` .

    The algorithm can be described as the code below.

    .. code-block:: text

        learning_rate = 0.5        # init learning_rate
        lr_lambda = lambda epoch: 0.95

        learning_rate = 0.5        # epoch 0,
        learning_rate = 0.475      # epoch 1, 0.5*0.95
        learning_rate = 0.45125    # epoch 2, 0.475*0.95

    Args:
        learning_rate (float): The initial learning rate. It is a python float number.
        lr_lambda (function): A function which computes a factor by ``epoch`` , and then multiply the last learning rate by this factor.
        last_epoch (int, optional):  The index of last epoch. Can be set to restart training. Default: -1, means initial learning rate.
        verbose (bool, optional): If ``True``, prints a message to stdout for each update. Default: ``False`` .

    Returns:
        ``MultiplicativeDecay`` instance to schedule learning rate.

    Examples:

        .. code-block:: python

            >>> import paddle

            >>> # train on default dynamic graph mode
            >>> linear = paddle.nn.Linear(10, 10)
            >>> scheduler = paddle.optimizer.lr.MultiplicativeDecay(learning_rate=0.5, lr_lambda=lambda x:0.95, verbose=True)
            >>> sgd = paddle.optimizer.SGD(learning_rate=scheduler, parameters=linear.parameters())
            >>> for epoch in range(20):
            ...     for batch_id in range(5):
            ...         x = paddle.uniform([10, 10])
            ...         out = linear(x)
            ...         loss = paddle.mean(out)
            ...         loss.backward()
            ...         sgd.step()
            ...         sgd.clear_gradients()
            ...         scheduler.step()    # If you update learning rate each step
            ...     # scheduler.step()        # If you update learning rate each epoch
            ...
    """

    def __init__(self, learning_rate, lr_lambda, last_epoch=-1, verbose=False):
        if not callable(lr_lambda):
            raise TypeError(
                "The type of 'lr_lambda' in 'MultiplicativeDecay' must be 'function', but received %s."
                % type(lr_lambda)
            )

        self.lr_lambda = lr_lambda
        super().__init__(learning_rate, last_epoch, verbose)

    def get_lr(self):
        cur_lr = self.base_lr
        for epoch in range(1, self.last_epoch + 1):
            cur_lr = cur_lr * self.lr_lambda(epoch)
        return cur_lr


class OneCycleLR(LRScheduler):
    r"""

    Sets the learning rate according to the one cycle learning rate scheduler.
    The scheduler adjusts the learning rate from an initial learning rate to the maximum learning rate and then
    from that maximum learning rate to the minimum learning rate, which is much less than the initial learning rate.

    It has been proposed in `Super-Convergence: Very Fast Training of Neural Networks Using Large Learning Rates <https://arxiv.org/abs/1708.07120>`_.

    Please note that the default behaviour of this scheduler follows the fastai implementation of one cycle,
    which claims that “unpublished work has shown even better results by using only two phases”.
    If you want the behaviour of this scheduler to be consistent with the paper, please set ``three_phase=True`` .

    Also note that you should update learning rate each step.

    Args:
        max_learning_rate (float): The maximum learning rate. It is a python float number. Functionally, it defines the initial learning rate by ``divide_factor`` .
        total_steps (int): Number of total training steps.
        divide_factor (float, optional): Initial learning rate will be determined by initial_learning_rate = max_learning_rate / divide_factor. Default: 25.
        end_learning_rate (float, optional): The minimum learning rate during training, it should be much less than initial learning rate.
        phase_pct (float): The percentage of total steps which used to increasing learning rate. Default: 0.3.
        anneal_strategy (str, optional): Strategy of adjusting learning rate.'cos' for cosine annealing, 'linear' for linear annealing. Default: 'cos'.
        three_phase (bool, optional): Whether to use three phase.

            If ``True``:

                1. The learning rate will first increase from initial learning rate to maximum learning rate.
                2. Then it will decrease to initial learning rate. Number of step in this phase is the same as the one in first phase.
                3. Finally, it will decrease to minimum learning rate which is much less than initial learning rate.

            If ``False``:

                1. The learning rate will increase to maximum learning rate.
                2. Then it will directly decrease to minimum learning rate.

        last_epoch (int, optional):  The index of last epoch. Can be set to restart training. Default: -1, means initial learning rate.
        verbose (bool, optional): If ``True``, prints a message to stdout for each update. Default: ``False`` .

    Returns:
        ``OneCycleLR`` instance to schedule learning rate.

    Examples:
        .. code-block:: python
            :name: code-example1

            >>> # Example1: train on default dynamic graph mode
            >>> import paddle
            >>> import numpy as np

            >>> # train on default dynamic graph mode
            >>> linear = paddle.nn.Linear(10, 10)
            >>> scheduler = paddle.optimizer.lr.OneCycleLR(max_learning_rate=1.0, total_steps=100, verbose=True)
            >>> sgd = paddle.optimizer.SGD(learning_rate=scheduler, parameters=linear.parameters())
            >>> for epoch in range(5):
            ...     for batch_id in range(20):
            ...         x = paddle.uniform([10, 10])
            ...         out = linear(x)
            ...         loss = paddle.mean(out)
            ...         loss.backward()
            ...         sgd.step()
            ...         sgd.clear_gradients()
            ...         scheduler.step()        # You should update learning rate each step

        .. code-block:: python
            :name: code-example2

            >>> # Example2: train on static graph mode
            >>> import paddle
            >>> import numpy as np
            >>> paddle.enable_static()
            >>> main_prog = paddle.static.Program()
            >>> start_prog = paddle.static.Program()
            >>> with paddle.static.program_guard(main_prog, start_prog):
            ...     x = paddle.static.data(name='x', shape=[None, 4, 5])
            ...     y = paddle.static.data(name='y', shape=[None, 4, 5])
            ...     z = paddle.static.nn.fc(x, 100)
            ...     loss = paddle.mean(z)
            ...     scheduler = paddle.optimizer.lr.OneCycleLR(max_learning_rate=1.0, total_steps=100, verbose=True)
            ...     sgd = paddle.optimizer.SGD(learning_rate=scheduler)
            ...     sgd.minimize(loss)
            ...
            >>> exe = paddle.static.Executor()
            >>> exe.run(start_prog)
            >>> for epoch in range(5):
            ...     for batch_id in range(20):
            ...         out = exe.run(
            ...             main_prog,
            ...             feed={
            ...                 'x': np.random.randn(3, 4, 5).astype('float32'),
            ...                 'y': np.random.randn(3, 4, 5).astype('float32')
            ...             },
            ...             fetch_list=loss.name)
            ...         scheduler.step()    # You should update learning rate each step
            ...
    """

    def __init__(
        self,
        max_learning_rate,
        total_steps,
        divide_factor=25.0,
        end_learning_rate=0.0001,
        phase_pct=0.3,
        anneal_strategy='cos',
        three_phase=False,
        last_epoch=-1,
        verbose=False,
    ):
        # Check type and value of max_learning_rate
        if not isinstance(max_learning_rate, (float, int)):
            raise TypeError(
                "'max_learning_rate' must be 'float' or 'int', but received {}".format(
                    type(max_learning_rate)
                )
            )
        if max_learning_rate < 0:
            raise ValueError("'max_learning_rate' must be a positive integer.")

        # Check type and value of end_learning_rate
        if not isinstance(end_learning_rate, (float, int)):
            raise TypeError(
                "'end_learning_rate' must be 'float' or 'int', but received {}".format(
                    type(end_learning_rate)
                )
            )
        if end_learning_rate < 0:
            raise ValueError("'end_learning_rate' must be a positive integer.")

        # Check type and value of total_steps
        if not isinstance(total_steps, int):
            raise TypeError(
                f"'total_step' must be 'int', but received {type(total_steps)}"
            )
        if total_steps <= 0:
            raise ValueError("'total_step' must be a positive integer.")
        self.total_steps = total_steps

        # Check type and value of pac_start
        if not isinstance(phase_pct, float):
            raise TypeError(
                f"'phase_pct' must be 'float', but received {type(phase_pct)}"
            )
        if phase_pct < 0 or phase_pct > 1:
            raise ValueError(
                f"'phase_pct' must be between 0 and 1, but received {phase_pct}"
            )

        # Check type and value of divide_factor
        if not isinstance(divide_factor, (float, int)):
            raise TypeError(
                "'divide_factor' must be 'float' or 'int', but received {}".format(
                    type(divide_factor)
                )
            )

        initial_lr = max_learning_rate / float(divide_factor)
        min_lr = float(end_learning_rate)

        if three_phase:
            if phase_pct >= 0.5:
                raise ValueError(
                    "When three_phase is True, 'phase_pct' must be less than 0.5"
                )
            # start step and end step of each phase.
            self._step_config = [
                0,
                phase_pct * self.total_steps - 1,
                2 * phase_pct * self.total_steps - 2,
                self.total_steps - 1,
                self.total_steps - 1,  # for the last step.
            ]
            # step size of each phase.
            self._steps_size = [
                self._step_config[1] - self._step_config[0],
                self._step_config[2] - self._step_config[1],
                self._step_config[3] - self._step_config[2],
                self._step_config[3]
                - self._step_config[2],  # for the last step.
            ]
            # start lr and end lr of each phase.
            self._lr_config = [
                initial_lr,
                max_learning_rate,
                initial_lr,
                min_lr,
            ]
        else:
            self._step_config = [
                0,
                phase_pct * self.total_steps - 1,
                self.total_steps - 1,
                self.total_steps - 1,
            ]
            self._steps_size = [
                self._step_config[1] - self._step_config[0],
                self._step_config[2] - self._step_config[1],
                self._step_config[2] - self._step_config[1],
            ]
            self._lr_config = [initial_lr, max_learning_rate, min_lr]

        # Check anneal_strategy
        if anneal_strategy == 'cos':
            self.anneal_func = self._cos_annealing
        elif anneal_strategy == 'linear':
            self.anneal_func = self._linear_annealing
        else:
            raise ValueError(
                "'anneal_strategy' must by one of 'cos' or 'linear', but received {}".format(
                    anneal_strategy
                )
            )
        super().__init__(initial_lr, last_epoch, verbose)

    def _cos_annealing(self, start_lr, end_lr, pct):
        cos_out = math.cos(math.pi * pct) + 1
        return end_lr + (start_lr - end_lr) / 2.0 * cos_out

    def _linear_annealing(self, start_lr, end_lr, pct):
        return (end_lr - start_lr) * pct + start_lr

    def get_lr(self):
        current_step = self.last_epoch

        if current_step > self.total_steps:
            raise ValueError(
                "Tried to step {} times. However the number of total steps is {}".format(
                    current_step, self.total_steps
                )
            )

        for i, (end_step, step_size) in enumerate(
            zip(self._step_config[1:], self._steps_size)
        ):
            # i == len(self._lr_config) - 2 catch the last step, otherwise it will return None.
            if current_step <= end_step or i == len(self._lr_config) - 2:
                # self._step_config[i] means start step of a phase.
                percentage = (current_step - self._step_config[i]) / step_size
                return self.anneal_func(
                    self._lr_config[i], self._lr_config[i + 1], percentage
                )


class CyclicLR(LRScheduler):
    r"""
    Set the learning rate according to the cyclic learning rate (CLR) scheduler.
    The scheduler regards the process of learning rate adjustment as one cycle after another.
    It cycles the learning rate between two boundaries with a constant frequency.
    The distance between the two boundaries can be scaled on a per-iteration or per-cycle basis.

    It has been proposed in `Cyclic Learning Rates for Training Neural Networks <https://arxiv.org/abs/1506.01186>`_.

    According to the paper, the cyclic learning rate schedule has three build-in scale methods:

    * "triangular": A basic triangular cycle without any amplitude scaling.
    * "triangular2": A basic triangular cycle that reduce initial amplitude by half each cycle.
    * "exp_range": A cycle that scales initial amplitude by scale function which is defined as :math:`gamma^{iterations}` .

    The initial amplitude is defined as max_learning_rate - base_learning_rate.
    Also note that you should update learning rate each step.

    Args:
        base_learning_rate (float): Initial learning rate, which is the lower boundary in the cycle. The paper recommends
            that set the base_learning_rate to 1/3 or 1/4 of max_learning_rate.
        max_learning_rate (float): Maximum learning rate in the cycle. It defines the cycle amplitude as above.
            Since there is some scaling operation during process of learning rate adjustment,
            max_learning_rate may not actually be reached.
        step_size_up (int): Number of training steps, which is used to increase learning rate in a cycle.
            The step size of one cycle will be defined by step_size_up + step_size_down. According to the paper, step
            size should be set as at least 3 or 4 times steps in one epoch.
        step_size_down (int, optional): Number of training steps, which is used to decrease learning rate in a cycle.
            If not specified, it's value will initialize to `` step_size_up `` . Default: None
        mode (str, optional): one of 'triangular', 'triangular2' or 'exp_range'.
            If scale_fn is specified, this argument will be ignored. Default: 'triangular'
        exp_gamma (float): Constant in 'exp_range' scaling function: exp_gamma**iterations. Used only when mode = 'exp_range'. Default: 1.0
        scale_fn (function, optional): A custom scaling function, which is used to replace three build-in methods.
            It should only have one argument. For all x >= 0, 0 <= scale_fn(x) <= 1.
            If specified, then 'mode' will be ignored. Default: None
        scale_mode (str, optional): One of 'cycle' or 'iterations'. Defines whether scale_fn is evaluated on cycle
            number or cycle iterations (total iterations since start of training). Default: 'cycle'
        last_epoch (int, optional): The index of last epoch. Can be set to restart training.Default: -1, means initial learning rate.
        verbose: (bool, optional): If ``True``, prints a message to stdout for each update. Default: ``False`` .

    Returns:
        ``CyclicLR`` instance to schedule learning rate.

    Examples:
        .. code-block:: python
            :name: code-example1

            >>> # Example1: train on default dynamic graph mode
            >>> import paddle
            >>> import numpy as np

            >>> # train on default dynamic graph mode
            >>> linear = paddle.nn.Linear(10, 10)
            >>> scheduler = paddle.optimizer.lr.CyclicLR(base_learning_rate=0.5, max_learning_rate=1.0, step_size_up=15, step_size_down=5, verbose=True)
            >>> sgd = paddle.optimizer.SGD(learning_rate=scheduler, parameters=linear.parameters())
            >>> for epoch in range(5):
            ...     for batch_id in range(20):
            ...         x = paddle.uniform([10, 10])
            ...         out = linear(x)
            ...         loss = paddle.mean(out)
            ...         loss.backward()
            ...         sgd.step()
            ...         sgd.clear_gradients()
            ...         scheduler.step()        # You should update learning rate each step

        .. code-block:: python
            :name: code-example2

            >>> # Example2: train on static graph mode
            >>> import paddle
            >>> import numpy as np
            >>> paddle.enable_static()
            >>> main_prog = paddle.static.Program()
            >>> start_prog = paddle.static.Program()
            >>> with paddle.static.program_guard(main_prog, start_prog):
            ...     x = paddle.static.data(name='x', shape=[None, 4, 5])
            ...     y = paddle.static.data(name='y', shape=[None, 4, 5])
            ...     z = paddle.static.nn.fc(x, 100)
            ...     loss = paddle.mean(z)
            ...     scheduler = paddle.optimizer.lr.CyclicLR(base_learning_rate=0.5,
            ...         max_learning_rate=1.0, step_size_up=15, step_size_down=5, verbose=True)
            ...     sgd = paddle.optimizer.SGD(learning_rate=scheduler)
            ...     sgd.minimize(loss)
            ...
            >>> exe = paddle.static.Executor()
            >>> exe.run(start_prog)
            >>> for epoch in range(5):
            ...     for batch_id in range(20):
            ...         out = exe.run(
            ...             main_prog,
            ...             feed={
            ...                 'x': np.random.randn(3, 4, 5).astype('float32'),
            ...                 'y': np.random.randn(3, 4, 5).astype('float32')
            ...             },
            ...             fetch_list=loss.name)
            ...         scheduler.step()    # You should update learning rate each step
    """

    def __init__(
        self,
        base_learning_rate,
        max_learning_rate,
        step_size_up,
        step_size_down=None,
        mode='triangular',
        exp_gamma=1.0,
        scale_fn=None,
        scale_mode='cycle',
        last_epoch=-1,
        verbose=False,
    ):
        # check type and value of max_learning_rate
        if not isinstance(max_learning_rate, (float, int)):
            raise TypeError(
                "'max_learning_rate' must be 'float' or 'int', but received {}".format(
                    type(max_learning_rate)
                )
            )
        if max_learning_rate < 0:
            raise ValueError(
                "'max_learning_rate' must be a positive integer, but received {}".format(
                    max_learning_rate
                )
            )

        # check type and value of step_size_up
        if not isinstance(step_size_up, int):
            raise TypeError(
                "The type of 'step_size_up' must be int, but received {}".format(
                    type(step_size_up)
                )
            )
        if step_size_up <= 0:
            raise ValueError(
                "'step_size_up' must be a positive integer, but received {}".format(
                    step_size_up
                )
            )

        # check type and value of step_size_down
        if step_size_down is not None:
            if not isinstance(step_size_down, int):
                raise TypeError(
                    "The type of 'step_size_down' must be int, but received {}".format(
                        type(step_size_down)
                    )
                )
            if step_size_down <= 0:
                raise ValueError(
                    "'step_size_down' must be a positive integer, but received {}".format(
                        step_size_down
                    )
                )

        # check type of exp_gamma
        if not isinstance(exp_gamma, float):
            raise TypeError(
                f"The type of 'exp_gamma' must be float, but received {type(exp_gamma)}"
            )

        step_size_up = float(step_size_up)
        step_size_down = (
            float(step_size_down)
            if step_size_down is not None
            else step_size_up
        )

        self.cycle_size = step_size_up + step_size_down
        self.step_up_pct = step_size_up / self.cycle_size
        self.max_lr = float(max_learning_rate)
        self.amplitude = self.max_lr - base_learning_rate

        if (
            mode not in ['triangular', 'triangular2', 'exp_range']
            and scale_fn is None
        ):
            raise ValueError(
                "'mode' is invalid and 'scale_fn' is not specified, make sure one of 'mode' or 'scale_fn' is valid"
            )
        if scale_mode not in ['cycle', 'iterations']:
            raise ValueError(
                "'scale_mode' must be one of 'cycle' or 'iterations"
            )

        self.mode = mode
        self.gamma = exp_gamma  # only for exp_range mode

        if scale_fn is None:
            if self.mode == 'triangular':
                self.scale_fn = self._triangular_scale_fn
                self.scale_mode = 'cycle'
            elif self.mode == 'triangular2':
                self.scale_fn = self._triangular2_scale_fn
                self.scale_mode = 'cycle'
            elif self.mode == 'exp_range':
                self.scale_fn = self._exp_range_scale_fn
                self.scale_mode = 'iterations'
        else:
            self.scale_fn = scale_fn
            self.scale_mode = scale_mode
        super().__init__(base_learning_rate, last_epoch, verbose)

    def _triangular_scale_fn(self, x):
        return 1.0

    def _triangular2_scale_fn(self, x):
        return 1 / (2.0 ** (x - 1))

    def _exp_range_scale_fn(self, x):
        return self.gamma**x

    def get_lr(self):
        iterations = self.last_epoch

        cycle = 1 + iterations // self.cycle_size
        pct_per_cycle = 1.0 + iterations / self.cycle_size - cycle

        if pct_per_cycle <= self.step_up_pct:
            scale_factor = pct_per_cycle / self.step_up_pct
        else:
            scale_factor = (1 - pct_per_cycle) / (1 - self.step_up_pct)

        base_height = self.amplitude * scale_factor

        lr = self.base_lr + base_height * self.scale_fn(eval(self.scale_mode))

        return lr


class LinearLR(LRScheduler):
    r"""
    Set the learning rate according to linear scheduler.
    The learning rate will be firstly multiplied by start_factor and linearly increase to end learning rate.

    Args:
        learning_rate (float): The initial learning rate. It is a python float number.
        total_steps (int): Number of iterations that the learning_rate reaches end learning_rate.
        start_factor (float): Start learning rate is defined by `start_factor * learning_rate` . Default: 1./3.
        end_factor (float) End learning rate is defined by `end_factor * learning_rate`. Default: 1.0.
        last_epoch (int, optional): The index of last epoch. Can be set to restart training.Default: -1, means initial learning rate.
        verbose: (bool, optional): If ``True``, prints a message to stdout for each update. Default: ``False`` .

    Returns:
        ``LinearLR`` instance to schedule learning rate.

    Examples:
        .. code-block:: python
            :name: code-dynamic

            >>> # Example1: train on default dynamic graph mode
            >>> import paddle
            >>> import numpy as np

            >>> # train on default dynamic graph mode
            >>> linear = paddle.nn.Linear(10, 10)
            >>> scheduler = paddle.optimizer.lr.LinearLR(learning_rate=0.5, total_steps=5, verbose=True)
            >>> sgd = paddle.optimizer.SGD(learning_rate=scheduler, parameters=linear.parameters())
            >>> for epoch in range(5):
            ...     for batch_id in range(20):
            ...         x = paddle.uniform([10, 10])
            ...         out = linear(x)
            ...         loss = paddle.mean(out)
            ...         loss.backward()
            ...         sgd.step()
            ...         sgd.clear_gradients()
            ...         scheduler.step()

        .. code-block:: python
            :name: code-static

            >>> # Example2: train on static graph mode
            >>> import paddle
            >>> import numpy as np
            >>> paddle.enable_static()
            >>> main_prog = paddle.static.Program()
            >>> start_prog = paddle.static.Program()
            >>> with paddle.static.program_guard(main_prog, start_prog):
            ...     x = paddle.static.data(name='x', shape=[None, 4, 5])
            ...     y = paddle.static.data(name='y', shape=[None, 4, 5])
            ...     z = paddle.static.nn.fc(x, 100)
            ...     loss = paddle.mean(z)
            ...     scheduler = paddle.optimizer.lr.LinearLR(learning_rate=0.5,
            ...        total_steps=5, verbose=True)
            ...     sgd = paddle.optimizer.SGD(learning_rate=scheduler)
            ...     sgd.minimize(loss)
            ...
            >>> exe = paddle.static.Executor()
            >>> exe.run(start_prog)
            >>> for epoch in range(5):
            ...     for batch_id in range(20):
            ...         out = exe.run(
            ...             main_prog,
            ...             feed={
            ...                 'x': np.random.randn(3, 4, 5).astype('float32'),
            ...                 'y': np.random.randn(3, 4, 5).astype('float32')
            ...             },
            ...             fetch_list=loss.name)
            ...         scheduler.step()
    """

    def __init__(
        self,
        learning_rate,
        total_steps,
        start_factor=1.0 / 3,
        end_factor=1.0,
        last_epoch=-1,
        verbose=False,
    ):
        if start_factor > 1.0 or start_factor <= 0:
            raise ValueError(
                "`start_factor` must be greater than 0 and less or equal to 1, but got {}".format(
                    start_factor
                )
            )

        if end_factor > 1.0 or end_factor < 0:
            raise ValueError(
                "`end_factor` must be greater than 0 and less than 1, but got {}".format(
                    end_factor
                )
            )

        if total_steps <= 0:
            raise ValueError(
                f"`total_steps` must be greater than 0, but got {total_steps}"
            )

        self.start_factor = start_factor
        self.end_factor = end_factor
        self.total_steps = total_steps

        super().__init__(learning_rate, last_epoch, verbose)

    def get_lr(self):
        if self.last_epoch == 0:
            return self.base_lr * self.start_factor
        elif self.last_epoch > self.total_steps:
            return self.last_lr
        else:
            base_lr = self.total_steps * self.start_factor
            cur_factor = self.end_factor - self.start_factor
            factor = 1.0 + cur_factor / (
                base_lr + (self.last_epoch - 1) * cur_factor
            )
            return self.last_lr * factor


class CosineAnnealingWarmRestarts(LRScheduler):
    r"""
    Set the learning rate of each parameter group using a cosine annealing
    schedule, where :math:`\eta_{max}` is set to the initial lr, :math:`T_{cur}`
    is the number of epochs since the last restart and :math:`T_{i}` is the number
    of epochs between two warm restarts in SGDR:

    .. math::
        \eta_t = \eta_{min} + \frac{1}{2}(\eta_{max} - \eta_{min})\left(1 +
        \cos\left(\frac{T_{cur}}{T_{i}}\pi\right)\right)

    When :math:`T_{cur}=T_{i}`, set :math:`\eta_t = \eta_{min}`.
    When :math:`T_{cur}=0` after restart, set :math:`\eta_t=\eta_{max}`.

    It has been proposed in `SGDR: Stochastic Gradient Descent with Warm Restarts <https://arxiv.org/abs/1608.03983>`_.

    Args:
        learning_rate (float): Initial learning rate.
        T_0 (int): Number of iterations for the first restart.
        T_mult (int, optional): A factor increases :math:`T_{i}` after a restart. Default: 1.
        eta_min (float, optional): Minimum learning rate. Default: 0.
        last_epoch (int, optional): The index of last epoch. Default: -1, means initial learning rate.
        verbose (bool, optional): If ``True``, prints a message to stdout for
            each update. Default: ``False``.

    Returns:
        ``CosineAnnealingWarmRestarts`` instance to schedule learning rate.

    Examples:
        .. code-block:: python
            :name: code-example1

            >>> import paddle
            >>> import numpy as np
            >>> # train on default dynamic graph mode
            >>> linear = paddle.nn.Linear(10, 10)
            >>> scheduler = paddle.optimizer.lr.CosineAnnealingWarmRestarts(learning_rate=0.5, T_0=1, T_mult=2, verbose=True)
            >>> adam = paddle.optimizer.Adam(learning_rate=scheduler, parameters=linear.parameters())
            >>> for epoch in range(10):
            ...    for batch_id in range(10):
            ...        x = paddle.uniform([10, 10])
            ...        out = linear(x)
            ...        loss = paddle.mean(out)
            ...        loss.backward()
            ...        adam.step()
            ...        adam.clear_grad()
            ...    scheduler.step(epoch)        # You should update learning rate each step

        .. code-block:: python
            :name: code-example2

            >>> import paddle
            >>> import numpy as np
            >>> paddle.enable_static()
            >>> main_prog = paddle.static.Program()
            >>> start_prog = paddle.static.Program()
            >>> with paddle.static.program_guard(main_prog, start_prog):
            ...    x = paddle.static.data(name='x', shape=[None, 4, 5])
            ...    y = paddle.static.data(name='y', shape=[None, 4, 5])
            ...    z = paddle.static.nn.fc(x, 100)
            ...    loss = paddle.mean(z)
            ...    scheduler = paddle.optimizer.lr.CosineAnnealingWarmRestarts(learning_rate=0.5, T_0=1, T_mult=2,verbose=True)
            ...    sgd = paddle.optimizer.SGD(learning_rate=scheduler)
            ...    sgd.minimize(loss)
            >>> exe = paddle.static.Executor()
            >>> exe.run(start_prog)
            >>> for epoch in range(10):
            ...    for batch_id in range(10):
            ...        out = exe.run(
            ...            main_prog,
            ...            feed={
            ...                'x': np.random.randn(3, 4, 5).astype('float32'),
            ...                'y': np.random.randn(3, 4, 5).astype('float32')
            ...            },
            ...            fetch_list=loss.name)
            ...    scheduler.step(epoch)    # You should update learning rate each step
    """

    def __init__(
        self,
        learning_rate,
        T_0,
        T_mult=1,
        eta_min=0,
        last_epoch=-1,
        verbose=False,
    ):
        if T_0 <= 0 or not isinstance(T_0, int):
            raise ValueError(f"Expected positive integer T_0, but got {T_0}")
        if T_mult < 1 or not isinstance(T_mult, int):
            raise ValueError(f"Expected integer T_mult >= 1, but got {T_mult}")
        self.T_0 = T_0
        self.T_i = T_0
        self.T_mult = T_mult
        self.eta_min = eta_min
        self.T_cur = last_epoch
        super().__init__(learning_rate, last_epoch, verbose)

    def get_lr(self):
        return (
            self.eta_min
            + (self.base_lr - self.eta_min)
            * (1 + math.cos(math.pi * self.T_cur / self.T_i))
            / 2
        )

    def step(self, epoch=None):
        """
        step should be called after `optimizer.step()` . It will update the learning rate in optimizer.
        The new learning rate will take effect on next epoch.

        Args:
            epoch (int, None): specify current epoch. Default: None. Auto-increment from last_epoch=-1.

        Returns:
            None

        Examples:
            Please refer to the example of current LRScheduler.
        """

        if epoch is None and self.last_epoch < 0:
            epoch = 0

        if epoch is None:
            epoch = self.last_epoch + 1
            self.T_cur = self.T_cur + 1
            if self.T_cur >= self.T_i:
                self.T_cur = self.T_cur - self.T_i
                self.T_i = self.T_i * self.T_mult
        else:
            if epoch < 0:
                raise ValueError(
                    f"Expected non-negative epoch, but got {epoch}"
                )
            if epoch >= self.T_0:
                if self.T_mult == 1:
                    self.T_cur = epoch % self.T_0
                else:
                    n = int(
                        math.log(
                            (epoch / self.T_0 * (self.T_mult - 1) + 1),
                            self.T_mult,
                        )
                    )
                    self.T_cur = epoch - self.T_0 * (self.T_mult**n - 1) / (
                        self.T_mult - 1
                    )
                    self.T_i = self.T_0 * self.T_mult ** (n)
            else:
                self.T_i = self.T_0
                self.T_cur = epoch
        self.last_epoch = math.floor(epoch)
        self.last_lr = self.get_lr()
        if self.verbose:
            print(
                'Epoch {}: {} set learning rate to {}.'.format(
                    self.last_epoch, self.__class__.__name__, self.last_lr
                )
            )


def autoincreased_step_counter(counter_name=None, begin=1, step=1):
    """
    :api_attr: Static Graph

    Create an auto-increase variable. which will be automatically increased
    by 1 in every iteration. By default, the first return of this counter is 1,
    and the step size is 1.

    Args:
        counter_name(str, optional): The counter name. Default '@STEP_COUNTER@'.
        begin(int, optional): The first return value of this counter. Default 1.
        step(int, optional): The step size. Default 1.

    Returns:
        Variable: The auto-increased Variable with data type int64.

    Examples:
        .. code-block:: python

            >>> import paddle
            >>> paddle.enable_static()
            >>> global_step = paddle.optimizer.lr.autoincreased_step_counter(
            ...     counter_name='@LR_DECAY_COUNTER@', begin=0, step=1)
    """
    helper = LayerHelper('global_step_counter')
    if counter_name is None:
        counter_name = '@STEP_COUNTER@'
    counter, is_new_var = helper.create_or_get_global_variable(
        name=counter_name,
        dtype='int64',
        shape=[1],
        persistable=True,
        belong_to_optimizer=True,
    )
    if is_new_var:
        helper.set_variable_initializer(
            counter,
            initializer=paddle.nn.initializer.ConstantInitializer(
                value=begin - 1, force_cpu=True
            ),
        )
        helper.main_program.global_block()._prepend_op(
            type='increment',
            inputs={'X': [counter]},
            outputs={'Out': [counter]},
            attrs={'step': float(step)},
        )
        counter.stop_gradient = True

    return counter


def _decay_step_counter(begin=0):
    # the first global step is zero in learning rate decay
    global_step = autoincreased_step_counter(
        counter_name='@LR_DECAY_COUNTER@', begin=begin, step=1
    )
    global_step = paddle.cast(global_step, 'float32')
    return global_step


def noam_decay(d_model, warmup_steps, learning_rate=1.0):
    """

    Noam decay method. The numpy implementation of noam decay as follows.

    .. code-block:: python

        >>> import numpy as np
        >>> # set hyper parameters
        >>> base_lr = 0.01
        >>> d_model = 2
        >>> current_steps = 20
        >>> warmup_steps = 200
        >>> # compute
        >>> lr_value = base_lr * np.power(d_model, -0.5) * np.min([
        ...                         np.power(current_steps, -0.5),
        ...                         np.power(warmup_steps, -1.5) * current_steps])

    Please reference `attention is all you need <https://arxiv.org/pdf/1706.03762.pdf>`_.

    Args:
        d_model(Variable): The dimensionality of input and output of model.
        warmup_steps(Variable): A super parameter.
        learning_rate(Variable|float|int): The initial learning rate. If the type
            is Variable, it's a tensor with shape [1], the data type can be
            float32 or float64. It also can be set to python int number. Default 1.0

    Returns:
        The decayed learning rate.

    Examples:
        .. code-block:: python

            >>> import paddle
            >>> warmup_steps = 100
            >>> learning_rate = 0.01
            >>> lr = paddle.optimizer.lr.noam_decay(
            ...                 1/(warmup_steps *(learning_rate ** 2)),
            ...                 warmup_steps,
            ...                 learning_rate)
    """
    with default_main_program()._lr_schedule_guard():
        if in_dygraph_mode():
            decay = paddle.optimizer.lr.NoamDecay(
                d_model, warmup_steps, learning_rate=learning_rate
            )
            return decay
        else:
            global_step = _decay_step_counter(1)

            a = global_step**-0.5
            b = (warmup_steps**-1.5) * global_step
            lr_value = learning_rate * (d_model**-0.5) * paddle.minimum(a, b)

            return lr_value


def exponential_decay(learning_rate, decay_steps, decay_rate, staircase=False):
    """

    Applies exponential decay to the learning rate.

    When training a model, it is often recommended to lower the learning rate as the
    training progresses. By using this function, the learning rate will be decayed by
    'decay_rate' every 'decay_steps' steps.

    Decayed learning rate calculates as follows:

    .. code-block:: text

        >>> if staircase == True:
        >>>     decayed_learning_rate = learning_rate * decay_rate ^ floor(global_step / decay_steps)
        >>> else:
        >>>     decayed_learning_rate = learning_rate * decay_rate ^ (global_step / decay_steps)

    Args:
        learning_rate(Variable|float): The initial learning rate. It should be a Variable
            or a float
        decay_steps(int): The learning rate decay steps. See the decay computation above.
        decay_rate(float): The learning rate decay rate. See the decay computation above.
        staircase(bool): If True, decay the learning rate at discrete intervals, which
            means the learning rate will be decayed by `decay_rate` every
            `decay_steps`. If False, learning rate will be decayed continuously
            and following the formula above. Default: False

    Returns:
        Variable: The decayed learning rate. The data type is float32.

    Examples:
        .. code-block:: python

            >>> import paddle

            >>> paddle.enable_static()
            >>> base_lr = 0.1
            >>> lr = paddle.optimizer.lr.exponential_decay(
            ...     learning_rate=base_lr,
            ...     decay_steps=10000,
            ...     decay_rate=0.5,
            ...     staircase=True
            ... )
    """
    with default_main_program()._lr_schedule_guard():
        if in_dygraph_mode():
            decay = ExponentialDecay(learning_rate, decay_rate)
            return decay
        else:
            global_step = _decay_step_counter()

            div_res = global_step / decay_steps
            if staircase:
                div_res = paddle.floor(div_res)
            decayed_lr = learning_rate * (decay_rate**div_res)

            return decayed_lr


def natural_exp_decay(learning_rate, decay_steps, decay_rate, staircase=False):
    """

    Applies natural exponential decay to the initial learning rate.

    When training a model, it is often recommended to lower the learning rate as the
    training progresses. By using this function, the learning rate will be decayed by
    natural exponential power 'decay_rate' every 'decay_steps' steps.

    Decayed learning rate calculates as follows:

    .. code-block:: text

        >>> if not staircase:
        >>>     decayed_learning_rate = learning_rate * exp(- decay_rate * (global_step / decay_steps))
        >>> else:
        >>>     decayed_learning_rate = learning_rate * exp(- decay_rate * floor(global_step / decay_steps))

    Args:
        learning_rate(Variable|float): The initial learning rate. It should be a Variable
            or a float
        decay_steps(int): The learning rate decay steps. See the decay computation above.
        decay_rate(float): The learning rate decay rate. See the decay computation above.
        staircase(bool): If True, decay the learning rate at discrete intervals, which
            means the learning rate will be decayed by natural exponential power
            `decay_rate` every `decay_steps`. If False, learning rate will be
            decayed continuously and following the formula above. Default: False

    Returns:
        The decayed learning rate. The data type is float32.

    Examples:
        .. code-block:: python

            >>> import paddle

            >>> paddle.enable_static()
            >>> base_lr = 0.1
            >>> lr = paddle.optimizer.lr.natural_exp_decay(
            ...     learning_rate=base_lr,
            ...     decay_steps=10000,
            ...     decay_rate=0.5,
            ...     staircase=True
            ... )
    """
    with default_main_program()._lr_schedule_guard():
        if in_dygraph_mode():
            decay = NaturalExpDecay(learning_rate, decay_rate)
            return decay
        else:
            global_step = _decay_step_counter()

            div_res = global_step / decay_steps
            if staircase:
                div_res = paddle.floor(div_res)
            decayed_lr = learning_rate * paddle.exp(-1 * decay_rate * div_res)

            return decayed_lr


def inverse_time_decay(learning_rate, decay_steps, decay_rate, staircase=False):
    """
    Applies inverse time decay to the initial learning rate.

    When training a model, it is often recommended to lower the learning rate as the
    training progresses. By using this function, an inverse decay function will be
    applied to the initial learning rate.

    Decayed learning rate calculates as follows:

    .. code-block:: text

        >>> if staircase == True:
        >>>     decayed_learning_rate = learning_rate / (1 + decay_rate * floor(global_step / decay_step))
        >>> else:
        >>>     decayed_learning_rate = learning_rate / (1 + decay_rate * global_step / decay_step)

    Args:
        learning_rate(Variable|float): The initial learning rate. It should be a Variable
            or a float
        decay_steps(int): The learning rate decay steps. See the decay computation above.
        decay_rate(float): The learning rate decay rate. See the decay computation above.
        staircase(bool): If True, decay the learning rate at discrete intervals, which
            means the learning rate will be decayed by `decay_rate` times
            every `decay_steps`. If False, learning rate will be decayed
            continuously and following the formula above. Default: False

    Returns:
        Variable: The decayed learning rate. The data type is float32.

    Examples:
        .. code-block:: python

            >>> import paddle
            >>> paddle.enable_static()
            >>> base_lr = 0.1
            >>> lr = paddle.optimizer.lr.inverse_time_decay(
            ...     learning_rate=base_lr,
            ...     decay_steps=10000,
            ...     decay_rate=0.5,
            ...     staircase=True
            ... )
    """
    with default_main_program()._lr_schedule_guard():
        if in_dygraph_mode():
            decay = InverseTimeDecay(learning_rate, decay_rate)
            return decay
        else:
            global_step = _decay_step_counter()

            div_res = global_step / decay_steps
            if staircase:
                div_res = paddle.floor(div_res)

            decayed_lr = learning_rate / (1 + decay_rate * div_res)

            return decayed_lr


def polynomial_decay(
    learning_rate, decay_steps, end_learning_rate=0.0001, power=1.0, cycle=False
):
    """
    Applies polynomial decay to the initial learning rate.

    .. code-block:: text

        if cycle:
            decay_steps = decay_steps * ceil(global_step / decay_steps)
        else:
            global_step = min(global_step, decay_steps)
            decayed_learning_rate = (learning_rate - end_learning_rate) *
                    (1 - global_step / decay_steps) ^ power + end_learning_rate

    Args:
        learning_rate(Variable|float32): A scalar float32 value or a Variable. This
            will be the initial learning rate during training.
        decay_steps(int32): A Python `int32` number.
        end_learning_rate(float): A Python `float` number.
        power(float): A Python `float` number.
        cycle(bool): If set true, decay the learning rate every decay_steps.

    Returns:
        Variable: The decayed learning rate

    Examples:
        .. code-block:: python

            >>> import paddle
            >>> start_lr = 0.01
            >>> total_step = 5000
            >>> end_lr = 0
            >>> lr = paddle.optimizer.lr.polynomial_decay(
            ...     start_lr,
            ...     total_step,
            ...     end_lr,
            ...     power=1
            ... )
    """
    with default_main_program()._lr_schedule_guard():
        if in_dygraph_mode():
            decay = PolynomialDecay(
                learning_rate, decay_steps, end_learning_rate, power, cycle
            )
            return decay
        else:
            global_step = _decay_step_counter()

            if cycle:
                div_res = paddle.ceil(global_step / decay_steps)
                zero_var = paddle.tensor.fill_constant(
                    shape=[1], dtype='float32', value=0.0
                )
                one_var = paddle.tensor.fill_constant(
                    shape=[1], dtype='float32', value=1.0
                )

                div_val = paddle.static.nn.cond(
                    global_step == zero_var, lambda: one_var, lambda: div_res
                )
                paddle.assign(div_val, output=div_res)

                decay_steps = decay_steps * div_res
            else:
                decay_steps_var = paddle.tensor.fill_constant(
                    shape=[1], dtype='float32', value=float(decay_steps)
                )
                global_step = paddle.minimum(x=global_step, y=decay_steps_var)

            decayed_lr = (learning_rate - end_learning_rate) * (
                (1 - global_step / decay_steps) ** power
            ) + end_learning_rate
            return decayed_lr


def piecewise_decay(boundaries, values):
    """
    Applies piecewise decay to the initial learning rate.

    The algorithm can be described as the code below.

    .. code-block:: text

        boundaries = [10000, 20000]
        values = [1.0, 0.5, 0.1]
        if step < 10000:
            learning_rate = 1.0
        elif 10000 <= step < 20000:
            learning_rate = 0.5
        else:
            learning_rate = 0.1

    Args:
        boundaries: A list of steps numbers.
        values: A list of learning rate values that will be picked during
            different step boundaries.

    Returns:
        The decayed learning rate.

    Examples:
        .. code-block:: python

            >>> import paddle
            >>> paddle.enable_static()
            >>> boundaries = [10000, 20000]
            >>> values = [1.0, 0.5, 0.1]
            >>> optimizer = paddle.optimizer.Momentum(
            ...     momentum=0.9,
            ...     learning_rate=paddle.optimizer.lr.PiecewiseDecay(boundaries, values),
            ...     weight_decay=paddle.regularizer.L2Decay(1e-4)
            ... )
    """
    with default_main_program()._lr_schedule_guard():
        if len(values) - len(boundaries) != 1:
            raise ValueError("len(values) - len(boundaries) should be 1")

        if in_dygraph_mode():
            decay = PiecewiseDecay(boundaries, values)
            return decay
        else:
            global_step = _decay_step_counter()

            lr = paddle.static.create_global_var(
                shape=[1],
                value=0.0,
                dtype='float32',
                persistable=True,
                name="learning_rate",
            )
            with paddle.static.nn.control_flow.Switch() as switch:
                for i in range(len(boundaries)):
                    boundary_val = paddle.tensor.fill_constant(
                        shape=[1],
                        dtype='float32',
                        value=float(boundaries[i]),
                        force_cpu=True,
                    )
                    with switch.case(global_step < boundary_val):
                        paddle.tensor.fill_constant(
                            shape=[1],
                            dtype="float32",
                            value=float(values[i]),
                            out=lr,
                        )
                with switch.default():
                    paddle.tensor.fill_constant(
                        shape=[1],
                        dtype="float32",
                        value=float(values[len(values) - 1]),
                        out=lr,
                    )
            return lr


def cosine_decay(learning_rate, step_each_epoch, epochs):
    r"""

    Applies cosine decay to the learning rate.

    when training a model, it is often recommended to lower the learning rate as the
    training progresses. By using this function, the learning rate will be decayed by
    following cosine decay strategy.

    .. math::

        decayed\_lr = learning\_rate * 0.5 * (math.cos * (epoch * \\frac{math.pi}{epochs} ) + 1)

    Args:
        learning_rate(Variable|float): The initial learning rate.
        step_each_epoch(int): the number of steps in an epoch.
        epochs(int): the number of epochs.

    Returns:
        Variable: The decayed learning rate.

    Examples:
        .. code-block:: python

            >>> import paddle
            >>> base_lr = 0.1
            >>> lr = paddle.optimizer.lr.cosine_decay(
            >>> learning_rate = base_lr, step_each_epoch=10000, epochs=120)
    """
    check_type(
        learning_rate, 'learning_rate', (float, Variable), 'cosine_decay'
    )

    with default_main_program()._lr_schedule_guard():
        if in_dygraph_mode():
            decay = CosineAnnealingDecay(learning_rate, epochs)
            return decay
        else:
            global_step = _decay_step_counter()

            cur_epoch = paddle.floor(global_step / step_each_epoch)
            decayed_lr = (
                learning_rate
                * 0.5
                * (paddle.cos(cur_epoch * math.pi / epochs) + 1)
            )
            return decayed_lr


def linear_lr_warmup(learning_rate, warmup_steps, start_lr, end_lr):
    """

    This operator use the linear learning rate warm up strategy to adjust the learning rate preliminarily before the normal learning rate scheduling.
    For more information, please refer to `Bag of Tricks for Image Classification with Convolutional Neural Networks <https://arxiv.org/abs/1812.01187>`_

    When global_step < warmup_steps, learning rate is updated as:

    .. code-block:: text

            linear_step = end_lr - start_lr
            lr = start_lr + linear_step * (global_step / warmup_steps)

    where start_lr is the initial learning rate, and end_lr is the final learning rate;

    When global_step >= warmup_steps, learning rate is updated as:

    .. code-block:: text

        lr = learning_rate

    where lr is the learning_rate after warm-up.

    Args:
        learning_rate (Variable|float): Learning_rate after warm-up, it could be 1D-Tensor or single value with the data type of float32.
        warmup_steps (int): Steps for warm up.
        start_lr (float): Initial learning rate of warm up.
        end_lr (float): Final learning rate of warm up.

    Returns:
        Variable: Warm-up learning rate with the same data type as learning_rate.

    Examples:

        .. code-block:: python

            >>> import paddle
            >>> paddle.enable_static()
            >>> boundaries = [100, 200]
            >>> lr_steps = [0.1, 0.01, 0.001]
            >>> learning_rate = paddle.optimizer.lr.piecewise_decay(boundaries, lr_steps) # case1, 1D-Tensor
            >>> # learning_rate = 0.1  # case2, single-value
            >>> warmup_steps = 50
            >>> start_lr = 0.1
            >>> end_lr = 1. / 3.
            >>> decayed_lr = paddle.optimizer.lr.linear_lr_warmup(
            ...     learning_rate,
            ...     warmup_steps,
            ...     start_lr,
            ...     end_lr
            ... )
            >>> place = paddle.CPUPlace()
            >>> exe = paddle.static.Executor(place)
            >>> exe.run(paddle.static.default_startup_program())
            >>> out, = exe.run(fetch_list=[decayed_lr.name])
            >>> print(out)
            [0.1]
    """
    dtype = 'float32'
    if isinstance(learning_rate, Variable):
        dtype = learning_rate.dtype

    linear_step = float(end_lr) - float(start_lr)
    with default_main_program()._lr_schedule_guard():
        if in_dygraph_mode():
            lr = LinearWarmup(learning_rate, warmup_steps, start_lr, end_lr)
            return lr
        else:
            lr = paddle.static.create_global_var(
                shape=[1],
                value=0.0,
                dtype=dtype,
                persistable=True,
                name="learning_rate_warmup",
            )

            global_step = _decay_step_counter()
            if not isinstance(learning_rate, Variable):
                learning_rate = paddle.tensor.fill_constant(
                    shape=[1], dtype=dtype, value=float(learning_rate)
                )
            lr_val = paddle.static.nn.case(
                pred_fn_pairs=[
                    (
                        global_step < warmup_steps,
                        lambda: start_lr
                        + linear_step * (global_step / float(warmup_steps)),
                    )
                ],
                default=lambda: learning_rate,
            )
            paddle.assign(lr_val, lr)
            return lr