python/training/learning_rate_decay.py

# Copyright 2015 The TensorFlow Authors. All Rights Reserved.
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
#     http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
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# ==============================================================================
"""Various learning rate decay functions."""
from __future__ import absolute_import
from __future__ import division
from __future__ import print_function

import math

from tensorflow.python.framework import constant_op
from tensorflow.python.framework import dtypes
from tensorflow.python.framework import ops
from tensorflow.python.ops import control_flow_ops
from tensorflow.python.ops import math_ops
from tensorflow.python.ops import random_ops
from tensorflow.python.util.tf_export import tf_export


@tf_export("train.exponential_decay")
def exponential_decay(learning_rate,
                      global_step,
                      decay_steps,
                      decay_rate,
                      staircase=False,
                      name=None):
  """Applies exponential decay to the learning rate.

  When training a model, it is often recommended to lower the learning rate as
  the training progresses.  This function applies an exponential decay function
  to a provided initial learning rate.  It requires a `global_step` value to
  compute the decayed learning rate.  You can just pass a TensorFlow variable
  that you increment at each training step.

  The function returns the decayed learning rate.  It is computed as:

  ```python
  decayed_learning_rate = learning_rate *
                          decay_rate ^ (global_step / decay_steps)
  ```

  If the argument `staircase` is `True`, then `global_step / decay_steps` is an
  integer division and the decayed learning rate follows a staircase function.

  Example: decay every 100000 steps with a base of 0.96:

  ```python
  ...
  global_step = tf.Variable(0, trainable=False)
  starter_learning_rate = 0.1
  learning_rate = tf.train.exponential_decay(starter_learning_rate, global_step,
                                             100000, 0.96, staircase=True)
  # Passing global_step to minimize() will increment it at each step.
  learning_step = (
      tf.train.GradientDescentOptimizer(learning_rate)
      .minimize(...my loss..., global_step=global_step)
  )
  ```

  Args:
    learning_rate: A scalar `float32` or `float64` `Tensor` or a
      Python number.  The initial learning rate.
    global_step: A scalar `int32` or `int64` `Tensor` or a Python number.
      Global step to use for the decay computation.  Must not be negative.
    decay_steps: A scalar `int32` or `int64` `Tensor` or a Python number.
      Must be positive.  See the decay computation above.
    decay_rate: A scalar `float32` or `float64` `Tensor` or a
      Python number.  The decay rate.
    staircase: Boolean.  If `True` decay the learning rate at discrete intervals
    name: String.  Optional name of the operation.  Defaults to
      'ExponentialDecay'.

  Returns:
    A scalar `Tensor` of the same type as `learning_rate`.  The decayed
    learning rate.

  Raises:
    ValueError: if `global_step` is not supplied.
  """
  if global_step is None:
    raise ValueError("global_step is required for exponential_decay.")
  with ops.name_scope(
      name, "ExponentialDecay",
      [learning_rate, global_step, decay_steps, decay_rate]) as name:
    learning_rate = ops.convert_to_tensor(learning_rate, name="learning_rate")
    dtype = learning_rate.dtype
    global_step = math_ops.cast(global_step, dtype)
    decay_steps = math_ops.cast(decay_steps, dtype)
    decay_rate = math_ops.cast(decay_rate, dtype)
    p = global_step / decay_steps
    if staircase:
      p = math_ops.floor(p)
    return math_ops.multiply(
        learning_rate, math_ops.pow(decay_rate, p), name=name)


@tf_export("train.piecewise_constant")
def piecewise_constant(x, boundaries, values, name=None):
  """Piecewise constant from boundaries and interval values.

  Example: use a learning rate that's 1.0 for the first 100001 steps, 0.5
    for the next 10000 steps, and 0.1 for any additional steps.

  ```python
  global_step = tf.Variable(0, trainable=False)
  boundaries = [100000, 110000]
  values = [1.0, 0.5, 0.1]
  learning_rate = tf.train.piecewise_constant(global_step, boundaries, values)

  # Later, whenever we perform an optimization step, we increment global_step.
  ```

  Args:
    x: A 0-D scalar `Tensor`. Must be one of the following types: `float32`,
      `float64`, `uint8`, `int8`, `int16`, `int32`, `int64`.
    boundaries: A list of `Tensor`s or `int`s or `float`s with strictly
      increasing entries, and with all elements having the same type as `x`.
    values: A list of `Tensor`s or `float`s or `int`s that specifies the values
      for the intervals defined by `boundaries`. It should have one more element
      than `boundaries`, and all elements should have the same type.
    name: A string. Optional name of the operation. Defaults to
      'PiecewiseConstant'.

  Returns:
    A 0-D Tensor. Its value is `values[0]` when `x <= boundaries[0]`,
    `values[1]` when `x > boundaries[0]` and `x <= boundaries[1]`, ...,
    and values[-1] when `x > boundaries[-1]`.

  Raises:
    ValueError: if types of `x` and `boundaries` do not match, or types of all
        `values` do not match or
        the number of elements in the lists does not match.
  """
  if len(boundaries) != len(values) - 1:
    raise ValueError(
        "The length of boundaries should be 1 less than the length of values")
  with ops.name_scope(name, "PiecewiseConstant",
                      [x, boundaries, values, name]) as name:
    x = ops.convert_to_tensor(x)
    # Avoid explicit conversion to x's dtype. This could result in faulty
    # comparisons, for example if floats are converted to integers.
    boundaries = ops.convert_n_to_tensor(boundaries)
    for i, b in enumerate(boundaries):
      if b.dtype.base_dtype != x.dtype.base_dtype:
        # We can promote int32 boundaries to int64 without loss of precision.
        # This covers the most common case where the user passes in boundaries
        # as an array of Python integers.
        if (b.dtype.base_dtype == dtypes.int32 and
            x.dtype.base_dtype == dtypes.int64):
          b = math_ops.cast(b, x.dtype.base_dtype)
          boundaries[i] = b
        else:
          raise ValueError(
              "Boundaries (%s) must have the same dtype as x (%s)." %
              (b.dtype.base_dtype, x.dtype.base_dtype))
    # TODO(rdipietro): Ensure that boundaries' elements are strictly increasing.
    values = ops.convert_n_to_tensor(values)
    for v in values[1:]:
      if v.dtype.base_dtype != values[0].dtype.base_dtype:
        raise ValueError(
            "Values must have elements all with the same dtype (%s vs %s)." %
            (values[0].dtype.base_dtype, v.dtype.base_dtype))
    pred_fn_pairs = []
    pred_fn_pairs.append((x <= boundaries[0], lambda: values[0]))
    pred_fn_pairs.append((x > boundaries[-1], lambda: values[-1]))
    for low, high, v in zip(boundaries[:-1], boundaries[1:], values[1:-1]):
      # Need to bind v here; can do this with lambda v=v: ...
      pred = (x > low) & (x <= high)
      pred_fn_pairs.append((pred, lambda v=v: v))

    # The default isn't needed here because our conditions are mutually
    # exclusive and exhaustive, but tf.case requires it.
    default = lambda: values[0]
    return control_flow_ops.case(pred_fn_pairs, default, exclusive=True)


@tf_export("train.polynomial_decay")
def polynomial_decay(learning_rate,
                     global_step,
                     decay_steps,
                     end_learning_rate=0.0001,
                     power=1.0,
                     cycle=False,
                     name=None):
  """Applies a polynomial decay to the learning rate.

  It is commonly observed that a monotonically decreasing learning rate, whose
  degree of change is carefully chosen, results in a better performing model.
  This function applies a polynomial decay function to a provided initial
  `learning_rate` to reach an `end_learning_rate` in the given `decay_steps`.

  It requires a `global_step` value to compute the decayed learning rate.  You
  can just pass a TensorFlow variable that you increment at each training step.

  The function returns the decayed learning rate.  It is computed as:

  ```python
  global_step = min(global_step, decay_steps)
  decayed_learning_rate = (learning_rate - end_learning_rate) *
                          (1 - global_step / decay_steps) ^ (power) +
                          end_learning_rate

  ```

  If `cycle` is True then a multiple of `decay_steps` is used, the first one
  that is bigger than `global_steps`.

  ```python
  decay_steps = decay_steps * ceil(global_step / decay_steps)
  decayed_learning_rate = (learning_rate - end_learning_rate) *
                          (1 - global_step / decay_steps) ^ (power) +
                          end_learning_rate

  ```

  Example: decay from 0.1 to 0.01 in 10000 steps using sqrt (i.e. power=0.5):

  ```python
  ...
  global_step = tf.Variable(0, trainable=False)
  starter_learning_rate = 0.1
  end_learning_rate = 0.01
  decay_steps = 10000
  learning_rate = tf.train.polynomial_decay(starter_learning_rate, global_step,
                                            decay_steps, end_learning_rate,
                                            power=0.5)
  # Passing global_step to minimize() will increment it at each step.
  learning_step = (
      tf.train.GradientDescentOptimizer(learning_rate)
      .minimize(...my loss..., global_step=global_step)
  )
  ```

  Args:
    learning_rate: A scalar `float32` or `float64` `Tensor` or a
      Python number.  The initial learning rate.
    global_step: A scalar `int32` or `int64` `Tensor` or a Python number.
      Global step to use for the decay computation.  Must not be negative.
    decay_steps: A scalar `int32` or `int64` `Tensor` or a Python number.
      Must be positive.  See the decay computation above.
    end_learning_rate: A scalar `float32` or `float64` `Tensor` or a
      Python number.  The minimal end learning rate.
    power: A scalar `float32` or `float64` `Tensor` or a
      Python number.  The power of the polynomial. Defaults to linear, 1.0.
    cycle: A boolean, whether or not it should cycle beyond decay_steps.
    name: String.  Optional name of the operation. Defaults to
      'PolynomialDecay'.

  Returns:
    A scalar `Tensor` of the same type as `learning_rate`.  The decayed
    learning rate.

  Raises:
    ValueError: if `global_step` is not supplied.
  """
  if global_step is None:
    raise ValueError("global_step is required for polynomial_decay.")
  with ops.name_scope(
      name, "PolynomialDecay",
      [learning_rate, global_step, decay_steps, end_learning_rate, power
      ]) as name:
    learning_rate = ops.convert_to_tensor(learning_rate, name="learning_rate")
    dtype = learning_rate.dtype
    global_step = math_ops.cast(global_step, dtype)
    decay_steps = math_ops.cast(decay_steps, dtype)
    end_learning_rate = math_ops.cast(end_learning_rate, dtype)
    power = math_ops.cast(power, dtype)
    if cycle:
      # Find the first multiple of decay_steps that is bigger than global_step.
      # If global_step is zero set the multiplier to 1
      multiplier = control_flow_ops.cond(
          math_ops.equal(global_step, 0), lambda: 1.0,
          lambda: math_ops.ceil(global_step / decay_steps))
      decay_steps = math_ops.multiply(decay_steps, multiplier)
    else:
      # Make sure that the global_step used is not bigger than decay_steps.
      global_step = math_ops.minimum(global_step, decay_steps)

    p = math_ops.div(global_step, decay_steps)
    return math_ops.add(
        math_ops.multiply(learning_rate - end_learning_rate,
                          math_ops.pow(1 - p, power)),
        end_learning_rate,
        name=name)


@tf_export("train.natural_exp_decay")
def natural_exp_decay(learning_rate,
                      global_step,
                      decay_steps,
                      decay_rate,
                      staircase=False,
                      name=None):
  """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.  This function applies an exponential decay function
  to a provided initial learning rate.  It requires an `global_step` value to
  compute the decayed learning rate.  You can just pass a TensorFlow variable
  that you increment at each training step.

  The function returns the decayed learning rate.  It is computed as:

  ```python
  decayed_learning_rate = learning_rate * exp(-decay_rate * global_step)
  ```

  Example: decay exponentially with a base of 0.96:

  ```python
  ...
  global_step = tf.Variable(0, trainable=False)
  learning_rate = 0.1
  k = 0.5
  learning_rate = tf.train.exponential_time_decay(learning_rate, global_step, k)

  # Passing global_step to minimize() will increment it at each step.
  learning_step = (
      tf.train.GradientDescentOptimizer(learning_rate)
      .minimize(...my loss..., global_step=global_step)
  )
  ```

  Args:
    learning_rate: A scalar `float32` or `float64` `Tensor` or a
      Python number.  The initial learning rate.
    global_step: A Python number.
      Global step to use for the decay computation.  Must not be negative.
    decay_steps: How often to apply decay.
    decay_rate: A Python number.  The decay rate.
    staircase: Whether to apply decay in a discrete staircase, as opposed to
      continuous, fashion.
    name: String.  Optional name of the operation.  Defaults to
      'ExponentialTimeDecay'.

  Returns:
    A scalar `Tensor` of the same type as `learning_rate`.  The decayed
    learning rate.

  Raises:
    ValueError: if `global_step` is not supplied.
  """
  if global_step is None:
    raise ValueError("global_step is required for natural_exp_decay.")
  with ops.name_scope(name, "NaturalExpDecay",
                      [learning_rate, global_step, decay_rate]) as name:
    learning_rate = ops.convert_to_tensor(learning_rate, name="learning_rate")
    dtype = learning_rate.dtype
    global_step = math_ops.cast(global_step, dtype)
    decay_steps = math_ops.cast(decay_steps, dtype)
    decay_rate = math_ops.cast(decay_rate, dtype)
    p = global_step / decay_steps
    if staircase:
      p = math_ops.floor(p)
    exponent = math_ops.exp(math_ops.multiply(math_ops.negative(decay_rate), p))
    return math_ops.multiply(learning_rate, exponent, name=name)


@tf_export("train.inverse_time_decay")
def inverse_time_decay(learning_rate,
                       global_step,
                       decay_steps,
                       decay_rate,
                       staircase=False,
                       name=None):
  """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.  This function applies an inverse decay function
  to a provided initial learning rate.  It requires an `global_step` value to
  compute the decayed learning rate.  You can just pass a TensorFlow variable
  that you increment at each training step.

  The function returns the decayed learning rate.  It is computed as:

  ```python
  decayed_learning_rate = learning_rate / (1 + decay_rate * global_step /
  decay_step)
  ```

  or, if `staircase` is `True`, as:

  ```python
  decayed_learning_rate = learning_rate / (1 + decay_rate * floor(global_step /
  decay_step))
  ```

  Example: decay 1/t with a rate of 0.5:

  ```python
  ...
  global_step = tf.Variable(0, trainable=False)
  learning_rate = 0.1
  decay_steps = 1.0
  decay_rate = 0.5
  learning_rate = tf.train.inverse_time_decay(learning_rate, global_step,
  decay_steps, decay_rate)

  # Passing global_step to minimize() will increment it at each step.
  learning_step = (
      tf.train.GradientDescentOptimizer(learning_rate)
      .minimize(...my loss..., global_step=global_step)
  )
  ```

  Args:
    learning_rate: A scalar `float32` or `float64` `Tensor` or a
      Python number.  The initial learning rate.
    global_step: A Python number.
      Global step to use for the decay computation.  Must not be negative.
    decay_steps: How often to apply decay.
    decay_rate: A Python number.  The decay rate.
    staircase: Whether to apply decay in a discrete staircase, as opposed to
      continuous, fashion.
    name: String.  Optional name of the operation.  Defaults to
      'InverseTimeDecay'.

  Returns:
    A scalar `Tensor` of the same type as `learning_rate`.  The decayed
    learning rate.

  Raises:
    ValueError: if `global_step` is not supplied.
  """
  if global_step is None:
    raise ValueError("global_step is required for inverse_time_decay.")
  with ops.name_scope(name, "InverseTimeDecay",
                      [learning_rate, global_step, decay_rate]) as name:
    learning_rate = ops.convert_to_tensor(learning_rate, name="learning_rate")
    dtype = learning_rate.dtype
    global_step = math_ops.cast(global_step, dtype)
    decay_steps = math_ops.cast(decay_steps, dtype)
    decay_rate = math_ops.cast(decay_rate, dtype)
    p = global_step / decay_steps
    if staircase:
      p = math_ops.floor(p)
    const = math_ops.cast(constant_op.constant(1), learning_rate.dtype)
    denom = math_ops.add(const, math_ops.multiply(decay_rate, p))
    return math_ops.div(learning_rate, denom, name=name)


@tf_export("train.cosine_decay")
def cosine_decay(learning_rate, global_step, decay_steps, alpha=0.0, name=None):
  """Applies cosine decay to the learning rate.

  See [Loshchilov & Hutter, ICLR2016], SGDR: Stochastic Gradient Descent
  with Warm Restarts. https://arxiv.org/abs/1608.03983

  When training a model, it is often recommended to lower the learning rate as
  the training progresses.  This function applies a cosine decay function
  to a provided initial learning rate.  It requires a `global_step` value to
  compute the decayed learning rate.  You can just pass a TensorFlow variable
  that you increment at each training step.

  The function returns the decayed learning rate.  It is computed as:
  ```python
  global_step = min(global_step, decay_steps)
  cosine_decay = 0.5 * (1 + cos(pi * global_step / decay_steps))
  decayed = (1 - alpha) * cosine_decay + alpha
  decayed_learning_rate = learning_rate * decayed
  ```

  Example usage:
  ```python
  decay_steps = 1000
  lr_decayed = cosine_decay(learning_rate, global_step, decay_steps)
  ```

  Args:
    learning_rate: A scalar `float32` or `float64` Tensor or a Python number.
      The initial learning rate.
    global_step: A scalar `int32` or `int64` `Tensor` or a Python number.
      Global step to use for the decay computation.
    decay_steps: A scalar `int32` or `int64` `Tensor` or a Python number.
      Number of steps to decay over.
    alpha: A scalar `float32` or `float64` Tensor or a Python number.
      Minimum learning rate value as a fraction of learning_rate.
    name: String. Optional name of the operation.  Defaults to 'CosineDecay'.
  Returns:
    A scalar `Tensor` of the same type as `learning_rate`.  The decayed
    learning rate.
  Raises:
    ValueError: if `global_step` is not supplied.
  """
  if global_step is None:
    raise ValueError("cosine decay requires global_step")
  with ops.name_scope(name, "CosineDecay",
                      [learning_rate, global_step]) as name:
    learning_rate = ops.convert_to_tensor(learning_rate, name="learning_rate")
    dtype = learning_rate.dtype
    global_step = math_ops.cast(global_step, dtype)
    decay_steps = math_ops.cast(decay_steps, dtype)
    global_step = math_ops.minimum(global_step, decay_steps)
    completed_fraction = global_step / decay_steps
    cosine_decayed = 0.5 * (
        1.0 + math_ops.cos(constant_op.constant(math.pi) * completed_fraction))

    decayed = (1 - alpha) * cosine_decayed + alpha
    return math_ops.multiply(learning_rate, decayed)


@tf_export("train.cosine_decay_restarts")
def cosine_decay_restarts(learning_rate,
                          global_step,
                          first_decay_steps,
                          t_mul=2.0,
                          m_mul=1.0,
                          alpha=0.0,
                          name=None):
  """Applies cosine decay with restarts to the learning rate.

  See [Loshchilov & Hutter, ICLR2016], SGDR: Stochastic Gradient Descent
  with Warm Restarts. https://arxiv.org/abs/1608.03983

  When training a model, it is often recommended to lower the learning rate as
  the training progresses.  This function applies a cosine decay function with
  restarts to a provided initial learning rate.  It requires a `global_step`
  value to compute the decayed learning rate.  You can just pass a TensorFlow
  variable that you increment at each training step.

  The function returns the decayed learning rate while taking into account
  possible warm restarts. The learning rate multiplier first decays
  from 1 to `alpha` for `first_decay_steps` steps. Then, a warm
  restart is performed. Each new warm restart runs for `t_mul` times more steps
  and with `m_mul` times smaller initial learning rate.

  Example usage:
  ```python
  first_decay_steps = 1000
  lr_decayed = cosine_decay_restarts(learning_rate, global_step,
                                     first_decay_steps)
  ```

  Args:
    learning_rate: A scalar `float32` or `float64` Tensor or a Python number.
      The initial learning rate.
    global_step: A scalar `int32` or `int64` `Tensor` or a Python number.
      Global step to use for the decay computation.
    first_decay_steps: A scalar `int32` or `int64` `Tensor` or a Python number.
      Number of steps to decay over.
    t_mul: A scalar `float32` or `float64` `Tensor` or a Python number.
      Used to derive the number of iterations in the i-th period
    m_mul: A scalar `float32` or `float64` `Tensor` or a Python number.
      Used to derive the initial learning rate of the i-th period:
    alpha: A scalar `float32` or `float64` Tensor or a Python number.
      Minimum learning rate value as a fraction of the learning_rate.
    name: String. Optional name of the operation.  Defaults to 'SGDRDecay'.
  Returns:
    A scalar `Tensor` of the same type as `learning_rate`.  The decayed
    learning rate.
  Raises:
    ValueError: if `global_step` is not supplied.
  """
  if global_step is None:
    raise ValueError("cosine decay restarts requires global_step")
  with ops.name_scope(name, "SGDRDecay", [learning_rate, global_step]) as name:
    learning_rate = ops.convert_to_tensor(
        learning_rate, name="initial_learning_rate")
    dtype = learning_rate.dtype
    global_step = math_ops.cast(global_step, dtype)
    first_decay_steps = math_ops.cast(first_decay_steps, dtype)
    alpha = math_ops.cast(alpha, dtype)
    t_mul = math_ops.cast(t_mul, dtype)
    m_mul = math_ops.cast(m_mul, dtype)

    completed_fraction = global_step / first_decay_steps

    def compute_step(completed_fraction, geometric=False):
      if geometric:
        i_restart = math_ops.floor(
            math_ops.log(1.0 - completed_fraction * (1.0 - t_mul)) /
            math_ops.log(t_mul))

        sum_r = (1.0 - t_mul**i_restart) / (1.0 - t_mul)
        completed_fraction = (completed_fraction - sum_r) / t_mul**i_restart

      else:
        i_restart = math_ops.floor(completed_fraction)
        completed_fraction = completed_fraction - i_restart

      return i_restart, completed_fraction

    i_restart, completed_fraction = control_flow_ops.cond(
        math_ops.equal(t_mul, 1.0),
        lambda: compute_step(completed_fraction, geometric=False),
        lambda: compute_step(completed_fraction, geometric=True))

    m_fac = m_mul**i_restart
    cosine_decayed = 0.5 * m_fac * (
        1.0 + math_ops.cos(constant_op.constant(math.pi) * completed_fraction))
    decayed = (1 - alpha) * cosine_decayed + alpha

  return math_ops.multiply(learning_rate, decayed, name=name)


@tf_export("train.linear_cosine_decay")
def linear_cosine_decay(learning_rate,
                        global_step,
                        decay_steps,
                        num_periods=0.5,
                        alpha=0.0,
                        beta=0.001,
                        name=None):
  """Applies linear cosine decay to the learning rate.

  See [Bello et al., ICML2017] Neural Optimizer Search with RL.
  https://arxiv.org/abs/1709.07417

  For the idea of warm starts here controlled by `num_periods`,
  see [Loshchilov & Hutter, ICLR2016] SGDR: Stochastic Gradient Descent
  with Warm Restarts. https://arxiv.org/abs/1608.03983

  Note that linear cosine decay is more aggressive than cosine decay and
  larger initial learning rates can typically be used.

  When training a model, it is often recommended to lower the learning rate as
  the training progresses.  This function applies a linear cosine decay function
  to a provided initial learning rate.  It requires a `global_step` value to
  compute the decayed learning rate.  You can just pass a TensorFlow variable
  that you increment at each training step.

  The function returns the decayed learning rate.  It is computed as:
  ```python
  global_step = min(global_step, decay_steps)
  linear_decay = (decay_steps - global_step) / decay_steps)
  cosine_decay = 0.5 * (
      1 + cos(pi * 2 * num_periods * global_step / decay_steps))
  decayed = (alpha + linear_decay) * cosine_decay + beta
  decayed_learning_rate = learning_rate * decayed
  ```

  Example usage:
  ```python
  decay_steps = 1000
  lr_decayed = linear_cosine_decay(learning_rate, global_step, decay_steps)
  ```

  Args:
    learning_rate: A scalar `float32` or `float64` Tensor or a Python number.
      The initial learning rate.
    global_step: A scalar `int32` or `int64` `Tensor` or a Python number.
      Global step to use for the decay computation.
    decay_steps: A scalar `int32` or `int64` `Tensor` or a Python number.
      Number of steps to decay over.
    num_periods: Number of periods in the cosine part of the decay.
      See computation above.
    alpha: See computation above.
    beta: See computation above.
    name: String.  Optional name of the operation.  Defaults to
      'LinearCosineDecay'.
  Returns:
    A scalar `Tensor` of the same type as `learning_rate`.  The decayed
    learning rate.
  Raises:
    ValueError: if `global_step` is not supplied.
  """
  if global_step is None:
    raise ValueError("linear cosine decay requires global_step")
  with ops.name_scope(name, "LinearCosineDecay",
                      [learning_rate, global_step]) as name:
    learning_rate = ops.convert_to_tensor(learning_rate, name="learning_rate")
    dtype = learning_rate.dtype
    global_step = math_ops.cast(global_step, dtype)
    decay_steps = math_ops.cast(decay_steps, dtype)
    num_periods = math_ops.cast(num_periods, dtype)
    global_step = math_ops.minimum(global_step, decay_steps)
    alpha = math_ops.cast(alpha, dtype)
    beta = math_ops.cast(beta, dtype)

    linear_decayed = (decay_steps - global_step) / decay_steps
    completed_fraction = global_step / decay_steps
    fraction = 2.0 * num_periods * completed_fraction
    cosine_decayed = 0.5 * (
        1.0 + math_ops.cos(constant_op.constant(math.pi) * fraction))

    linear_cosine_decayed = (alpha + linear_decayed) * cosine_decayed + beta
    return math_ops.multiply(learning_rate, linear_cosine_decayed, name=name)


@tf_export("train.noisy_linear_cosine_decay")
def noisy_linear_cosine_decay(learning_rate,
                              global_step,
                              decay_steps,
                              initial_variance=1.0,
                              variance_decay=0.55,
                              num_periods=0.5,
                              alpha=0.0,
                              beta=0.001,
                              name=None):
  """Applies noisy linear cosine decay to the learning rate.

  See [Bello et al., ICML2017] Neural Optimizer Search with RL.
  https://arxiv.org/abs/1709.07417

  For the idea of warm starts here controlled by `num_periods`,
  see [Loshchilov & Hutter, ICLR2016] SGDR: Stochastic Gradient Descent
  with Warm Restarts. https://arxiv.org/abs/1608.03983

  Note that linear cosine decay is more aggressive than cosine decay and
  larger initial learning rates can typically be used.

  When training a model, it is often recommended to lower the learning rate as
  the training progresses.  This function applies a noisy linear
  cosine decay function to a provided initial learning rate.
  It requires a `global_step` value to compute the decayed learning rate.
  You can just pass a TensorFlow variable that you increment at each
  training step.

  The function returns the decayed learning rate.  It is computed as:
  ```python
  global_step = min(global_step, decay_steps)
  linear_decay = (decay_steps - global_step) / decay_steps)
  cosine_decay = 0.5 * (
      1 + cos(pi * 2 * num_periods * global_step / decay_steps))
  decayed = (alpha + linear_decay + eps_t) * cosine_decay + beta
  decayed_learning_rate = learning_rate * decayed
  ```
  where eps_t is 0-centered gaussian noise with variance
  initial_variance / (1 + global_step) ** variance_decay

  Example usage:
  ```python
  decay_steps = 1000
  lr_decayed = noisy_linear_cosine_decay(
    learning_rate, global_step, decay_steps)
  ```

  Args:
    learning_rate: A scalar `float32` or `float64` Tensor or a Python number.
      The initial learning rate.
    global_step: A scalar `int32` or `int64` `Tensor` or a Python number.
      Global step to use for the decay computation.
    decay_steps: A scalar `int32` or `int64` `Tensor` or a Python number.
      Number of steps to decay over.
    initial_variance: initial variance for the noise. See computation above.
    variance_decay: decay for the noise's variance. See computation above.
    num_periods: Number of periods in the cosine part of the decay.
      See computation above.
    alpha: See computation above.
    beta: See computation above.
    name: String.  Optional name of the operation.  Defaults to
      'NoisyLinearCosineDecay'.
  Returns:
    A scalar `Tensor` of the same type as `learning_rate`.  The decayed
    learning rate.
  Raises:
    ValueError: if `global_step` is not supplied.
  """
  if global_step is None:
    raise ValueError("noisy linear cosine decay requires global_step")
  with ops.name_scope(name, "NoisyLinearCosineDecay",
                      [learning_rate, global_step]) as name:
    learning_rate = ops.convert_to_tensor(learning_rate, name="learning_rate")
    dtype = learning_rate.dtype
    global_step = math_ops.cast(global_step, dtype)
    decay_steps = math_ops.cast(decay_steps, dtype)
    global_step = math_ops.minimum(global_step, decay_steps)
    initial_variance = math_ops.cast(initial_variance, dtype)
    variance_decay = math_ops.cast(variance_decay, dtype)
    num_periods = math_ops.cast(num_periods, dtype)
    alpha = math_ops.cast(alpha, dtype)
    beta = math_ops.cast(beta, dtype)

    linear_decayed = (decay_steps - global_step) / decay_steps
    variance = initial_variance / (
        math_ops.pow(1.0 + global_step, variance_decay))
    std = math_ops.sqrt(variance)
    noisy_linear_decayed = (
        linear_decayed +
        random_ops.random_normal(linear_decayed.shape, stddev=std))

    completed_fraction = global_step / decay_steps
    fraction = 2.0 * num_periods * completed_fraction
    cosine_decayed = 0.5 * (
        1.0 + math_ops.cos(constant_op.constant(math.pi) * fraction))
    noisy_linear_cosine_decayed = (
        (alpha + noisy_linear_decayed) * cosine_decayed + beta)

    return math_ops.multiply(
        learning_rate, noisy_linear_cosine_decayed, name=name)