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

This library contains classes for estimating blocks in a model's Fisher
Information matrix. Suppose one has a model that parameterizes a posterior
distribution over 'y' given 'x' with parameters 'params', p(y | x, params). Its
Fisher Information matrix is given by,

  $$F(params) = E[ v(x, y, params) v(x, y, params)^T ]$$

where,

  $$v(x, y, params) = (d / d params) log p(y | x, params)$$

and the expectation is taken with respect to the data's distribution for 'x' and
the model's posterior distribution for 'y',

  x ~ p(x)
  y ~ p(y | x, params)

"""

from __future__ import absolute_import
from __future__ import division
from __future__ import print_function

import abc
import enum  # pylint: disable=g-bad-import-order

import numpy as np
import six

from tensorflow.contrib.kfac.python.ops import fisher_factors
from tensorflow.contrib.kfac.python.ops import utils
from tensorflow.python.framework import ops
from tensorflow.python.ops import array_ops
from tensorflow.python.ops import math_ops
from tensorflow.python.util import nest

# For blocks corresponding to convolutional layers, or any type of block where
# the parameters can be thought of as being replicated in time or space,
# we want to adjust the scale of the damping by
#   damping /= num_replications ** NORMALIZE_DAMPING_POWER
NORMALIZE_DAMPING_POWER = 1.0

# Methods for adjusting damping for FisherBlocks. See
# compute_pi_adjusted_damping() for details.
PI_OFF_NAME = "off"
PI_TRACENORM_NAME = "tracenorm"
PI_TYPE = PI_TRACENORM_NAME


def set_global_constants(normalize_damping_power=None, pi_type=None):
  """Sets various global constants used by the classes in this module."""
  global NORMALIZE_DAMPING_POWER
  global PI_TYPE

  if normalize_damping_power is not None:
    NORMALIZE_DAMPING_POWER = normalize_damping_power

  if pi_type is not None:
    PI_TYPE = pi_type


def normalize_damping(damping, num_replications):
  """Normalize damping after adjusting scale by NORMALIZE_DAMPING_POWER."""
  if NORMALIZE_DAMPING_POWER:
    return damping / (num_replications ** NORMALIZE_DAMPING_POWER)
  return damping


def compute_pi_tracenorm(left_cov, right_cov):
  """Computes the scalar constant pi for Tikhonov regularization/damping.

  $$\pi = \sqrt{ (trace(A) / dim(A)) / (trace(B) / dim(B)) }$$
  See section 6.3 of https://arxiv.org/pdf/1503.05671.pdf for details.

  Args:
    left_cov: The left Kronecker factor "covariance".
    right_cov: The right Kronecker factor "covariance".

  Returns:
    The computed scalar constant pi for these Kronecker Factors (as a Tensor).
  """

  def _trace(cov):
    if len(cov.shape) == 1:
      # Diagonal matrix.
      return math_ops.reduce_sum(cov)
    elif len(cov.shape) == 2:
      # Full matrix.
      return math_ops.trace(cov)
    else:
      raise ValueError(
          "What's the trace of a Tensor of rank %d?" % len(cov.shape))

  # Instead of dividing by the dim of the norm, we multiply by the dim of the
  # other norm. This works out the same in the ratio.
  left_norm = _trace(left_cov) * right_cov.shape.as_list()[0]
  right_norm = _trace(right_cov) * left_cov.shape.as_list()[0]
  return math_ops.sqrt(left_norm / right_norm)


def compute_pi_adjusted_damping(left_cov, right_cov, damping):

  if PI_TYPE == PI_TRACENORM_NAME:
    pi = compute_pi_tracenorm(left_cov, right_cov)
    return (damping * pi, damping / pi)

  elif PI_TYPE == PI_OFF_NAME:
    return (damping, damping)


class PackagedFunc(object):
  """A Python thunk with a stable ID.

  Enables stable names for lambdas.
  """

  def __init__(self, func, func_id):
    """Initializes PackagedFunc.

    Args:
      func: a zero-arg Python function.
      func_id: a hashable, function that produces a hashable, or a list/tuple
        thereof.
    """
    self._func = func
    func_id = func_id if isinstance(func_id, (tuple, list)) else (func_id,)
    self._func_id = func_id

  def __call__(self):
    return self._func()

  @property
  def func_id(self):
    """A hashable identifier for this function."""
    return tuple(elt() if callable(elt) else elt for elt in self._func_id)


def _package_func(func, func_id):
  return PackagedFunc(func, func_id)


@six.add_metaclass(abc.ABCMeta)
class FisherBlock(object):
  """Abstract base class for objects modeling approximate Fisher matrix blocks.

  Subclasses must implement register_matpower, multiply_matpower,
  instantiate_factors, tensors_to_compute_grads, and num_registered_towers
  methods.
  """

  def __init__(self, layer_collection):
    self._layer_collection = layer_collection

  @abc.abstractmethod
  def instantiate_factors(self, grads_list, damping):
    """Creates and registers the component factors of this Fisher block.

    Args:
      grads_list: A list gradients (each a Tensor or tuple of Tensors) with
          respect to the tensors returned by tensors_to_compute_grads() that
          are to be used to estimate the block.
      damping: The damping factor (float or Tensor).
    """
    pass

  @abc.abstractmethod
  def register_matpower(self, exp):
    """Registers a matrix power to be computed by the block.

    Args:
      exp: A float representing the power to raise the block by.
    """
    pass

  def register_inverse(self):
    """Registers a matrix inverse to be computed by the block."""
    self.register_matpower(-1)

  @abc.abstractmethod
  def multiply_matpower(self, vector, exp):
    """Multiplies the vector by the (damped) matrix-power of the block.

    Args:
      vector: The vector (a Tensor or tuple of Tensors) to be multiplied.
      exp: A float representing the power to raise the block by before
        multiplying it by the vector.

    Returns:
      The vector left-multiplied by the (damped) matrix-power of the block.
    """
    pass

  def multiply_inverse(self, vector):
    """Multiplies the vector by the (damped) inverse of the block.

    Args:
      vector: The vector (a Tensor or tuple of Tensors) to be multiplied.

    Returns:
      The vector left-multiplied by the (damped) inverse of the block.
    """
    return self.multiply_matpower(vector, -1)

  def multiply(self, vector):
    """Multiplies the vector by the (damped) block.

    Args:
      vector: The vector (a Tensor or tuple of Tensors) to be multiplied.

    Returns:
      The vector left-multiplied by the (damped) block.
    """
    return self.multiply_matpower(vector, 1)

  @abc.abstractmethod
  def tensors_to_compute_grads(self):
    """Returns the Tensor(s) with respect to which this FisherBlock needs grads.
    """
    pass

  @abc.abstractproperty
  def num_registered_towers(self):
    """Number of towers registered for this FisherBlock.

    Typically equal to the number of towers in a multi-tower setup.
    """
    pass


class FullFB(FisherBlock):
  """FisherBlock using a full matrix estimate (no approximations).

  FullFB uses a full matrix estimate (no approximations), and should only ever
  be used for very low dimensional parameters.

  Note that this uses the naive "square the sum estimator", and so is applicable
  to any type of parameter in principle, but has very high variance.
  """

  def __init__(self, layer_collection, params):
    """Creates a FullFB block.

    Args:
      layer_collection: The collection of all layers in the K-FAC approximate
          Fisher information matrix to which this FisherBlock belongs.
      params: The parameters of this layer (Tensor or tuple of Tensors).
    """
    self._batch_sizes = []
    self._params = params

    super(FullFB, self).__init__(layer_collection)

  def instantiate_factors(self, grads_list, damping):
    self._damping_func = _package_func(lambda: damping, (damping,))

    self._factor = self._layer_collection.make_or_get_factor(
        fisher_factors.FullFactor, (grads_list, self._batch_size))

  def register_matpower(self, exp):
    self._factor.register_matpower(exp, self._damping_func)

  def multiply_matpower(self, vector, exp):
    vector_flat = utils.tensors_to_column(vector)
    out_flat = self._factor.left_multiply_matpower(
        vector_flat, exp, self._damping_func)
    return utils.column_to_tensors(vector, out_flat)

  def full_fisher_block(self):
    """Explicitly constructs the full Fisher block."""
    return self._factor.get_cov()

  def tensors_to_compute_grads(self):
    return self._params

  def register_additional_tower(self, batch_size):
    """Register an additional tower.

    Args:
      batch_size: The batch size, used in the covariance estimator.
    """
    self._batch_sizes.append(batch_size)

  @property
  def num_registered_towers(self):
    return len(self._batch_sizes)

  @property
  def _batch_size(self):
    return math_ops.reduce_sum(self._batch_sizes)


class NaiveDiagonalFB(FisherBlock):
  """FisherBlock using a diagonal matrix approximation.

  This type of approximation is generically applicable but quite primitive.

  Note that this uses the naive "square the sum estimator", and so is applicable
  to any type of parameter in principle, but has very high variance.
  """

  def __init__(self, layer_collection, params):
    """Creates a NaiveDiagonalFB block.

    Args:
      layer_collection: The collection of all layers in the K-FAC approximate
          Fisher information matrix to which this FisherBlock belongs.
      params: The parameters of this layer (Tensor or tuple of Tensors).
    """
    self._params = params
    self._batch_sizes = []

    super(NaiveDiagonalFB, self).__init__(layer_collection)

  def instantiate_factors(self, grads_list, damping):
    self._damping_func = _package_func(lambda: damping, (damping,))

    self._factor = self._layer_collection.make_or_get_factor(
        fisher_factors.NaiveDiagonalFactor, (grads_list, self._batch_size))

  def register_matpower(self, exp):
    # Not needed for this.  Matrix powers are computed on demand in the
    # diagonal case
    pass

  def multiply_matpower(self, vector, exp):
    vector_flat = utils.tensors_to_column(vector)
    out_flat = self._factor.left_multiply_matpower(
        vector_flat, exp, self._damping_func)
    return utils.column_to_tensors(vector, out_flat)

  def full_fisher_block(self):
    return self._factor.get_cov()

  def tensors_to_compute_grads(self):
    return self._params

  def register_additional_tower(self, batch_size):
    """Register an additional tower.

    Args:
      batch_size: The batch size, used in the covariance estimator.
    """
    self._batch_sizes.append(batch_size)

  @property
  def num_registered_towers(self):
    return len(self._batch_sizes)

  @property
  def _batch_size(self):
    return math_ops.reduce_sum(self._batch_sizes)


class InputOutputMultiTower(object):
  """Mix-in class for blocks with inputs & outputs and multiple mini-batches."""

  def __init__(self, *args, **kwargs):
    self.__inputs = []
    self.__outputs = []
    super(InputOutputMultiTower, self).__init__(*args, **kwargs)

  def _process_data(self, grads_list):
    """Process data into the format used by the factors.

    This function takes inputs and grads_lists data and processes it into
    one of the formats expected by the FisherFactor classes (depending on
    the value of the global configuration variable TOWER_STRATEGY).

    The initial format of self._inputs is expected to be a list of Tensors
    over towers. Similarly grads_lists is expected to be a list over sources
    of such lists.

    If TOWER_STRATEGY is "concat", 'inputs' becomes a tuple containing a single
    tensor (represented as a PartitionedTensor object) equal to the
    concatenation (across towers) of all of the elements of self._inputs. And
    similarly grads_list is formatted into a tuple (over sources) of such
    tensors (also represented as PartitionedTensors).

    If TOWER_STRATEGY is "separate", formatting of inputs and grads_list
    remains unchanged from the initial format (although possibly converting
    from lists into tuples).

    Args:
      grads_list: grads_list in its initial format (see above).

    Returns:
      inputs: self._inputs transformed into the appropriate format (see
        above).
      grads_list: grads_list transformed into the appropriate format (see
        above).

    Raises:
      ValueError: if TOWER_STRATEGY is not one of "separate" or "concat".
    """
    inputs = self._inputs
    # inputs is a list over towers of Tensors
    # grads_list is a list of list with the first index being sources and the
    # second being towers.
    if fisher_factors.TOWER_STRATEGY == "concat":
      # Merge towers together into a PartitionedTensor. We package it in
      # a singleton tuple since the factors will expect a list over towers
      inputs = (utils.PartitionedTensor(inputs),)
      # Do the same for grads_list but preserve leading sources dimension
      grads_list = tuple((utils.PartitionedTensor(grads),)
                         for grads in grads_list)
    elif fisher_factors.TOWER_STRATEGY == "separate":
      inputs = tuple(inputs)
      grads_list = tuple(grads_list)

    else:
      raise ValueError("Global config variable TOWER_STRATEGY must be one of "
                       "'concat' or 'separate'.")

    return inputs, grads_list

  def tensors_to_compute_grads(self):
    """Tensors to compute derivative of loss with respect to."""
    return tuple(self._outputs)

  def register_additional_tower(self, inputs, outputs):
    self._inputs.append(inputs)
    self._outputs.append(outputs)

  @property
  def num_registered_towers(self):
    result = len(self._inputs)
    assert result == len(self._outputs)
    return result

  @property
  def _inputs(self):
    return self.__inputs

  @property
  def _outputs(self):
    return self.__outputs


class FullyConnectedDiagonalFB(InputOutputMultiTower, FisherBlock):
  """FisherBlock for fully-connected (dense) layers using a diagonal approx.

  Estimates the Fisher Information matrix's diagonal entries for a fully
  connected layer. Unlike NaiveDiagonalFB this uses the low-variance "sum of
  squares" estimator.

  Let 'params' be a vector parameterizing a model and 'i' an arbitrary index
  into it. We are interested in Fisher(params)[i, i]. This is,

    $$Fisher(params)[i, i] = E[ v(x, y, params) v(x, y, params)^T ][i, i]
                         = E[ v(x, y, params)[i] ^ 2 ]$$

  Consider fully connected layer in this model with (unshared) weight matrix
  'w'. For an example 'x' that produces layer inputs 'a' and output
  preactivations 's',

    $$v(x, y, w) = vec( a (d loss / d s)^T )$$

  This FisherBlock tracks Fisher(params)[i, i] for all indices 'i' corresponding
  to the layer's parameters 'w'.
  """

  def __init__(self, layer_collection, has_bias=False):
    """Creates a FullyConnectedDiagonalFB block.

    Args:
      layer_collection: The collection of all layers in the K-FAC approximate
          Fisher information matrix to which this FisherBlock belongs.
      has_bias: Whether the component Kronecker factors have an additive bias.
          (Default: False)
    """
    self._has_bias = has_bias

    super(FullyConnectedDiagonalFB, self).__init__(layer_collection)

  def instantiate_factors(self, grads_list, damping):
    inputs, grads_list = self._process_data(grads_list)

    self._factor = self._layer_collection.make_or_get_factor(
        fisher_factors.FullyConnectedDiagonalFactor,
        (inputs, grads_list, self._has_bias))

    self._damping_func = _package_func(lambda: damping, (damping,))

  def register_matpower(self, exp):
    # Not needed for this.  Matrix powers are computed on demand in the
    # diagonal case
    pass

  def multiply_matpower(self, vector, exp):
    """Multiplies the vector by the (damped) matrix-power of the block.

    Args:
      vector: Tensor or 2-tuple of Tensors. if self._has_bias, Tensor of shape
        [input_size, output_size] corresponding to layer's weights. If not, a
        2-tuple of the former and a Tensor of shape [output_size] corresponding
        to the layer's bias.
      exp: A scalar representing the power to raise the block before multiplying
           it by the vector.

    Returns:
      The vector left-multiplied by the (damped) matrix-power of the block.
    """
    reshaped_vec = utils.layer_params_to_mat2d(vector)
    reshaped_out = self._factor.left_multiply_matpower(
        reshaped_vec, exp, self._damping_func)
    return utils.mat2d_to_layer_params(vector, reshaped_out)


class ConvDiagonalFB(InputOutputMultiTower, FisherBlock):
  """FisherBlock for 2-D convolutional layers using a diagonal approx.

  Estimates the Fisher Information matrix's diagonal entries for a convolutional
  layer. Unlike NaiveDiagonalFB this uses the low-variance "sum of squares"
  estimator.

  Let 'params' be a vector parameterizing a model and 'i' an arbitrary index
  into it. We are interested in Fisher(params)[i, i]. This is,

    $$Fisher(params)[i, i] = E[ v(x, y, params) v(x, y, params)^T ][i, i]
                         = E[ v(x, y, params)[i] ^ 2 ]$$

  Consider a convoluational layer in this model with (unshared) filter matrix
  'w'. For an example image 'x' that produces layer inputs 'a' and output
  preactivations 's',

    $$v(x, y, w) = vec( sum_{loc} a_{loc} (d loss / d s_{loc})^T )$$

  where 'loc' is a single (x, y) location in an image.

  This FisherBlock tracks Fisher(params)[i, i] for all indices 'i' corresponding
  to the layer's parameters 'w'.
  """

  def __init__(self,
               layer_collection,
               params,
               strides,
               padding,
               data_format=None,
               dilations=None):
    """Creates a ConvDiagonalFB block.

    Args:
      layer_collection: The collection of all layers in the K-FAC approximate
          Fisher information matrix to which this FisherBlock belongs.
      params: The parameters (Tensor or tuple of Tensors) of this layer. If
        kernel alone, a Tensor of shape [kernel_height, kernel_width,
        in_channels, out_channels]. If kernel and bias, a tuple of 2 elements
        containing the previous and a Tensor of shape [out_channels].
      strides: The stride size in this layer (1-D Tensor of length 4).
      padding: The padding in this layer (e.g. "SAME").
      data_format: str or None. Format of input data.
      dilations: List of 4 ints or None. Rate for dilation along all dimensions.

    Raises:
      ValueError: if strides is not length-4.
      ValueError: if dilations is not length-4.
      ValueError: if channel is not last dimension.
    """
    if len(strides) != 4:
      raise ValueError("strides must contain 4 numbers.")

    if dilations is None:
      dilations = [1, 1, 1, 1]

    if len(dilations) != 4:
      raise ValueError("dilations must contain 4 numbers.")

    if not utils.is_data_format_channel_last(data_format):
      raise ValueError("data_format must be channels-last.")

    self._strides = maybe_tuple(strides)
    self._padding = padding
    self._data_format = data_format
    self._dilations = maybe_tuple(dilations)
    self._has_bias = isinstance(params, (tuple, list))

    fltr = params[0] if self._has_bias else params
    self._filter_shape = tuple(fltr.shape.as_list())

    if len(self._filter_shape) != 4:
      raise ValueError(
          "Convolution filter must be of shape"
          " [filter_height, filter_width, in_channels, out_channels].")

    super(ConvDiagonalFB, self).__init__(layer_collection)

  def instantiate_factors(self, grads_list, damping):
    inputs, grads_list = self._process_data(grads_list)

    # Infer number of locations upon which convolution is applied.
    self._num_locations = num_conv_locations(inputs[0].shape.as_list(),
                                             self._strides)

    self._factor = self._layer_collection.make_or_get_factor(
        fisher_factors.ConvDiagonalFactor,
        (inputs, grads_list, self._filter_shape, self._strides, self._padding,
         self._data_format, self._dilations, self._has_bias))

    def damping_func():
      return self._num_locations * normalize_damping(damping,
                                                     self._num_locations)

    damping_id = (self._num_locations, "mult", "normalize_damping", damping,
                  self._num_locations)
    self._damping_func = _package_func(damping_func, damping_id)

  def register_matpower(self, exp):
    # Not needed for this.  Matrix powers are computed on demand in the
    # diagonal case
    pass

  def multiply_matpower(self, vector, exp):
    reshaped_vect = utils.layer_params_to_mat2d(vector)
    reshaped_out = self._factor.left_multiply_matpower(
        reshaped_vect, exp, self._damping_func)
    return utils.mat2d_to_layer_params(vector, reshaped_out)


class KroneckerProductFB(FisherBlock):
  """A base class for blocks with separate input and output Kronecker factors.

  The Fisher block is approximated as a Kronecker product of the input and
  output factors.
  """

  def __init__(self, layer_collection):
    super(KroneckerProductFB, self).__init__(layer_collection)

  def _setup_damping(self, damping, normalization=None):
    """Makes functions that compute the damping values for both factors."""
    def compute_damping():
      if normalization is not None:
        maybe_normalized_damping = normalize_damping(damping, normalization)
      else:
        maybe_normalized_damping = damping

      return compute_pi_adjusted_damping(self._input_factor.get_cov(),
                                         self._output_factor.get_cov(),
                                         maybe_normalized_damping**0.5)

    if normalization is not None:
      damping_id = ("compute_pi_adjusted_damping",
                    "cov", self._input_factor.name,
                    "cov", self._output_factor.name,
                    "normalize_damping", damping, normalization, "power", 0.5)
    else:
      damping_id = ("compute_pi_adjusted_damping",
                    "cov", self._input_factor.name,
                    "cov", self._output_factor.name,
                    damping, "power", 0.5)

    self._input_damping_func = _package_func(lambda: compute_damping()[0],
                                             damping_id + ("ref", 0))
    self._output_damping_func = _package_func(lambda: compute_damping()[1],
                                              damping_id + ("ref", 1))

  def register_matpower(self, exp):
    self._input_factor.register_matpower(exp, self._input_damping_func)
    self._output_factor.register_matpower(exp, self._output_damping_func)

  @property
  def _renorm_coeff(self):
    """Kronecker factor multiplier coefficient.

    If this FisherBlock is represented as 'FB = c * kron(left, right)', then
    this is 'c'.

    Returns:
      0-D Tensor.
    """
    return 1.0

  def multiply_matpower(self, vector, exp):
    reshaped_vector = utils.layer_params_to_mat2d(vector)
    reshaped_out = self._output_factor.right_multiply_matpower(
        reshaped_vector, exp, self._output_damping_func)
    reshaped_out = self._input_factor.left_multiply_matpower(
        reshaped_out, exp, self._input_damping_func)
    if self._renorm_coeff != 1.0:
      renorm_coeff = math_ops.cast(self._renorm_coeff, dtype=reshaped_out.dtype)
      reshaped_out *= math_ops.cast(renorm_coeff**exp, dtype=reshaped_out.dtype)
    return utils.mat2d_to_layer_params(vector, reshaped_out)

  def full_fisher_block(self):
    """Explicitly constructs the full Fisher block.

    Used for testing purposes. (In general, the result may be very large.)

    Returns:
      The full Fisher block.
    """
    left_factor = self._input_factor.get_cov()
    right_factor = self._output_factor.get_cov()
    return self._renorm_coeff * utils.kronecker_product(left_factor,
                                                        right_factor)


class EmbeddingKFACFB(InputOutputMultiTower, KroneckerProductFB):
  """K-FAC FisherBlock for embedding layers.

  This FisherBlock is similar to FullyConnectedKFACBasicFB, except that its
  input factor is approximated by a diagonal matrix. In the case that each
  example references exactly one embedding, this approximation is exact.

  Does not support bias parameters.
  """

  def __init__(self, layer_collection, vocab_size):
    """Creates a EmbeddingKFACFB block.

    Args:
      layer_collection: The collection of all layers in the K-FAC approximate
          Fisher information matrix to which this FisherBlock belongs.
      vocab_size: int. Size of vocabulary for this embedding layer.
    """
    self._vocab_size = vocab_size

    super(EmbeddingKFACFB, self).__init__(layer_collection)

  def instantiate_factors(self, grads_list, damping):
    """Instantiate Kronecker Factors for this FisherBlock.

    Args:
      grads_list: List of list of Tensors. grads_list[i][j] is the
        gradient of the loss with respect to 'outputs' from source 'i' and
        tower 'j'. Each Tensor has shape [tower_minibatch_size, output_size].
      damping: 0-D Tensor or float. 'damping' * identity is approximately added
        to this FisherBlock's Fisher approximation.
    """
    inputs, grads_list = self._process_data(grads_list)

    self._input_factor = self._layer_collection.make_or_get_factor(
        fisher_factors.EmbeddingInputKroneckerFactor,
        (inputs, self._vocab_size))
    self._output_factor = self._layer_collection.make_or_get_factor(
        fisher_factors.FullyConnectedKroneckerFactor, (grads_list,))
    self._setup_damping(damping)


class FullyConnectedKFACBasicFB(InputOutputMultiTower, KroneckerProductFB):
  """K-FAC FisherBlock for fully-connected (dense) layers.

  This uses the Kronecker-factorized approximation from the original
  K-FAC paper (https://arxiv.org/abs/1503.05671)
  """

  def __init__(self, layer_collection, has_bias=False):
    """Creates a FullyConnectedKFACBasicFB block.

    Args:
      layer_collection: The collection of all layers in the K-FAC approximate
          Fisher information matrix to which this FisherBlock belongs.
      has_bias: Whether the component Kronecker factors have an additive bias.
          (Default: False)
    """
    self._has_bias = has_bias

    super(FullyConnectedKFACBasicFB, self).__init__(layer_collection)

  def instantiate_factors(self, grads_list, damping):
    """Instantiate Kronecker Factors for this FisherBlock.

    Args:
      grads_list: List of list of Tensors. grads_list[i][j] is the
        gradient of the loss with respect to 'outputs' from source 'i' and
        tower 'j'. Each Tensor has shape [tower_minibatch_size, output_size].
      damping: 0-D Tensor or float. 'damping' * identity is approximately added
        to this FisherBlock's Fisher approximation.
    """
    inputs, grads_list = self._process_data(grads_list)

    self._input_factor = self._layer_collection.make_or_get_factor(
        fisher_factors.FullyConnectedKroneckerFactor,
        ((inputs,), self._has_bias))
    self._output_factor = self._layer_collection.make_or_get_factor(
        fisher_factors.FullyConnectedKroneckerFactor,
        (grads_list,))
    self._setup_damping(damping)


class ConvKFCBasicFB(InputOutputMultiTower, KroneckerProductFB):
  """FisherBlock for convolutional layers using the basic KFC approx.

  Estimates the Fisher Information matrix's blog for a convolutional
  layer.

  Consider a convoluational layer in this model with (unshared) filter matrix
  'w'. For a minibatch that produces inputs 'a' and output preactivations 's',
  this FisherBlock estimates,

    $$F(w) = \#locations * kronecker(E[flat(a) flat(a)^T],
                                  E[flat(ds) flat(ds)^T])$$

  where

    $$ds = (d / ds) log p(y | x, w)$$
    #locations = number of (x, y) locations where 'w' is applied.

  where the expectation is taken over all examples and locations and flat()
  concatenates an array's leading dimensions.

  See equation 23 in https://arxiv.org/abs/1602.01407 for details.
  """

  def __init__(self,
               layer_collection,
               params,
               padding,
               strides=None,
               dilation_rate=None,
               data_format=None,
               extract_patches_fn=None):
    """Creates a ConvKFCBasicFB block.

    Args:
      layer_collection: The collection of all layers in the K-FAC approximate
          Fisher information matrix to which this FisherBlock belongs.
      params: The parameters (Tensor or tuple of Tensors) of this layer. If
        kernel alone, a Tensor of shape [..spatial_filter_shape..,
        in_channels, out_channels]. If kernel and bias, a tuple of 2 elements
        containing the previous and a Tensor of shape [out_channels].
      padding: str. Padding method.
      strides: List of ints or None. Contains [..spatial_filter_strides..] if
        'extract_patches_fn' is compatible with tf.nn.convolution(), else
        [1, ..spatial_filter_strides, 1].
      dilation_rate: List of ints or None. Rate for dilation along each spatial
        dimension if 'extract_patches_fn' is compatible with
        tf.nn.convolution(), else [1, ..spatial_dilation_rates.., 1].
      data_format: str or None. Format of input data.
      extract_patches_fn: str or None. Name of function that extracts image
        patches. One of "extract_convolution_patches", "extract_image_patches",
        "extract_pointwise_conv2d_patches".
    """
    self._padding = padding
    self._strides = maybe_tuple(strides)
    self._dilation_rate = maybe_tuple(dilation_rate)
    self._data_format = data_format
    self._extract_patches_fn = extract_patches_fn
    self._has_bias = isinstance(params, (tuple, list))

    fltr = params[0] if self._has_bias else params
    self._filter_shape = tuple(fltr.shape.as_list())

    super(ConvKFCBasicFB, self).__init__(layer_collection)

  def instantiate_factors(self, grads_list, damping):
    inputs, grads_list = self._process_data(grads_list)

    # Infer number of locations upon which convolution is applied.
    self._num_locations = num_conv_locations(inputs[0].shape.as_list(),
                                             self._strides)

    self._input_factor = self._layer_collection.make_or_get_factor(
        fisher_factors.ConvInputKroneckerFactor,
        (inputs, self._filter_shape, self._padding, self._strides,
         self._dilation_rate, self._data_format, self._extract_patches_fn,
         self._has_bias))
    self._output_factor = self._layer_collection.make_or_get_factor(
        fisher_factors.ConvOutputKroneckerFactor, (grads_list,))

    self._setup_damping(damping, normalization=self._num_locations)

  @property
  def _renorm_coeff(self):
    return self._num_locations


class DepthwiseConvDiagonalFB(ConvDiagonalFB):
  """FisherBlock for depthwise_conv2d().

  Equivalent to ConvDiagonalFB applied to each input channel in isolation.
  """

  def __init__(self,
               layer_collection,
               params,
               strides,
               padding,
               rate=None,
               data_format=None):
    """Creates a DepthwiseConvKFCBasicFB block.

    Args:
      layer_collection: The collection of all layers in the K-FAC approximate
          Fisher information matrix to which this FisherBlock belongs.
      params: Tensor of shape [filter_height, filter_width, in_channels,
        channel_multiplier].
      strides: List of 4 ints. Strides along all dimensions.
      padding: str. Padding method.
      rate: List of 4 ints or None. Rate for dilation along all dimensions.
      data_format: str or None. Format of input data.

    Raises:
      NotImplementedError: If parameters contains bias.
      ValueError: If filter is not 4-D.
      ValueError: If strides is not length-4.
      ValueError: If rates is not length-2.
      ValueError: If channels are not last dimension.
    """
    if isinstance(params, (tuple, list)):
      raise NotImplementedError("Bias not yet supported.")

    if params.shape.ndims != 4:
      raise ValueError("Filter must be 4-D.")

    if len(strides) != 4:
      raise ValueError("strides must account for 4 dimensions.")

    if rate is not None:
      if len(rate) != 2:
        raise ValueError("rate must only account for spatial dimensions.")
      rate = [1, rate[0], rate[1], 1]  # conv2d expects 4-element rate.

    if not utils.is_data_format_channel_last(data_format):
      raise ValueError("data_format must be channels-last.")

    super(DepthwiseConvDiagonalFB, self).__init__(
        layer_collection=layer_collection,
        params=params,
        strides=strides,
        padding=padding,
        dilations=rate,
        data_format=data_format)

    # This is a hack to overwrite the same setting in ConvKFCBasicFB.__init__().
    filter_height, filter_width, in_channels, channel_multiplier = (
        params.shape.as_list())
    self._filter_shape = (filter_height, filter_width, in_channels,
                          in_channels * channel_multiplier)

  def multiply_matpower(self, vector, exp):
    conv2d_vector = depthwise_conv2d_filter_to_conv2d_filter(vector)
    conv2d_result = super(DepthwiseConvDiagonalFB, self).multiply_matpower(
        conv2d_vector, exp)
    return conv2d_filter_to_depthwise_conv2d_filter(conv2d_result)


class DepthwiseConvKFCBasicFB(ConvKFCBasicFB):
  """FisherBlock for depthwise_conv2d().

  Equivalent to ConvKFCBasicFB applied to each input channel in isolation.
  """

  def __init__(self,
               layer_collection,
               params,
               strides,
               padding,
               rate=None,
               data_format=None):
    """Creates a DepthwiseConvKFCBasicFB block.

    Args:
      layer_collection: The collection of all layers in the K-FAC approximate
          Fisher information matrix to which this FisherBlock belongs.
      params: Tensor of shape [filter_height, filter_width, in_channels,
        channel_multiplier].
      strides: List of 4 ints. Strides along all dimensions.
      padding: str. Padding method.
      rate: List of 4 ints or None. Rate for dilation along all dimensions.
      data_format: str or None. Format of input data.

    Raises:
      NotImplementedError: If parameters contains bias.
      ValueError: If filter is not 4-D.
      ValueError: If strides is not length-4.
      ValueError: If rates is not length-2.
      ValueError: If channels are not last dimension.
    """
    if isinstance(params, (tuple, list)):
      raise NotImplementedError("Bias not yet supported.")

    if params.shape.ndims != 4:
      raise ValueError("Filter must be 4-D.")

    if len(strides) != 4:
      raise ValueError("strides must account for 4 dimensions.")

    if rate is not None:
      if len(rate) != 2:
        raise ValueError("rate must only account for spatial dimensions.")
      rate = [1, rate[0], rate[1], 1]  # conv2d expects 4-element rate.

    if not utils.is_data_format_channel_last(data_format):
      raise ValueError("data_format must be channels-last.")

    super(DepthwiseConvKFCBasicFB, self).__init__(
        layer_collection=layer_collection,
        params=params,
        padding=padding,
        strides=strides,
        dilation_rate=rate,
        data_format=data_format,
        extract_patches_fn="extract_image_patches")

    # This is a hack to overwrite the same setting in ConvKFCBasicFB.__init__().
    filter_height, filter_width, in_channels, channel_multiplier = (
        params.shape.as_list())
    self._filter_shape = (filter_height, filter_width, in_channels,
                          in_channels * channel_multiplier)

  def multiply_matpower(self, vector, exp):
    conv2d_vector = depthwise_conv2d_filter_to_conv2d_filter(vector)
    conv2d_result = super(DepthwiseConvKFCBasicFB, self).multiply_matpower(
        conv2d_vector, exp)
    return conv2d_filter_to_depthwise_conv2d_filter(conv2d_result)


def depthwise_conv2d_filter_to_conv2d_filter(filter, name=None):  # pylint: disable=redefined-builtin
  """Converts a convolution filter for use with conv2d.

  Transforms a filter for use with tf.nn.depthwise_conv2d() to one that's
  compatible with tf.nn.conv2d().

  Args:
    filter: Tensor of shape [height, width, in_channels, channel_multiplier].
    name: None or str. Name of Op.

  Returns:
    Tensor of shape [height, width, in_channels, out_channels].

  """
  with ops.name_scope(name, "depthwise_conv2d_filter_to_conv2d_filter",
                      [filter]):
    filter = ops.convert_to_tensor(filter)
    filter_height, filter_width, in_channels, channel_multiplier = (
        filter.shape.as_list())

    results = []
    for i in range(in_channels):
      # Slice out one in_channel's filter. Insert zeros around it to force it
      # to affect that channel and that channel alone.
      elements = []
      if i > 0:
        elements.append(
            array_ops.zeros(
                [filter_height, filter_width, i, channel_multiplier]))
      elements.append(filter[:, :, i:(i + 1), :])
      if i + 1 < in_channels:
        elements.append(
            array_ops.zeros([
                filter_height, filter_width, in_channels - (i + 1),
                channel_multiplier
            ]))

      # Concat along in_channel.
      results.append(
          array_ops.concat(elements, axis=-2, name="in_channel_%d" % i))

    # Concat along out_channel.
    return array_ops.concat(results, axis=-1, name="out_channel")


def conv2d_filter_to_depthwise_conv2d_filter(filter, name=None):  # pylint: disable=redefined-builtin
  """Converts a convolution filter for use with depthwise_conv2d.

  Transforms a filter for use with tf.nn.conv2d() to one that's
  compatible with tf.nn.depthwise_conv2d(). Ignores all filters but those along
  the diagonal.

  Args:
    filter: Tensor of shape [height, width, in_channels, out_channels].
    name: None or str. Name of Op.

  Returns:
    Tensor of shape,
      [height, width, in_channels, channel_multiplier]

  Raises:
    ValueError: if out_channels is not evenly divisible by in_channels.
  """
  with ops.name_scope(name, "conv2d_filter_to_depthwise_conv2d_filter",
                      [filter]):
    filter = ops.convert_to_tensor(filter)
    filter_height, filter_width, in_channels, out_channels = (
        filter.shape.as_list())

    if out_channels % in_channels != 0:
      raise ValueError("out_channels must be evenly divisible by in_channels.")
    channel_multiplier = out_channels // in_channels

    results = []
    filter = array_ops.reshape(filter, [
        filter_height, filter_width, in_channels, in_channels,
        channel_multiplier
    ])
    for i in range(in_channels):
      # Slice out output corresponding to the correct filter.
      filter_slice = array_ops.reshape(
          filter[:, :, i, i, :],
          [filter_height, filter_width, 1, channel_multiplier])
      results.append(filter_slice)

    # Concat along out_channel.
    return array_ops.concat(results, axis=-2, name="in_channels")


def maybe_tuple(obj):
  if not isinstance(obj, list):
    return obj
  return tuple(obj)


def num_conv_locations(input_shape, strides):
  """Returns the number of spatial locations a 2D Conv kernel is applied to.

  Args:
    input_shape: List of ints representing shape of inputs to
      tf.nn.convolution().
    strides: List of ints representing strides along spatial dimensions as
      passed in to tf.nn.convolution().

  Returns:
    A scalar |T| denoting the number of spatial locations for the Conv layer.
  """
  spatial_input_locations = np.prod(input_shape[1:-1])

  if strides is None:
    spatial_strides_divisor = 1
  else:
    spatial_strides_divisor = np.prod(strides)

  return spatial_input_locations // spatial_strides_divisor


class InputOutputMultiTowerMultiUse(InputOutputMultiTower):
  """Adds methods for multi-use/time-step case to InputOutputMultiTower."""

  def __init__(self, num_uses=None, *args, **kwargs):
    self._num_uses = num_uses
    super(InputOutputMultiTowerMultiUse, self).__init__(*args, **kwargs)

  def _process_data(self, grads_list):
    """Process temporal/multi-use data into the format used by the factors.

    This function takes inputs and grads_lists data and processes it into
    one of the formats expected by the FisherFactor classes (depending on
    the value of the global configuration variable TOWER_STRATEGY).

    It accepts the data in one of two initial formats. The first possible
    format is where self._inputs is a list of list of Tensors. The first index
    is tower, the second is use/time-step. grads_list, meanwhile, is a list
    over sources of such lists of lists.

    The second possible data format is where self._inputs is a Tensor with
    uses/times-steps folded into the batch dimension.  i.e. it is a Tensor
    of shape [num_uses * size_batch, ...] which represents a reshape of a
    Tensor of shape [num_uses, size_batch, ...].  And similarly grads_list is
    a list over sources of such Tensors.

    There are two possible formats which inputs and grads_list are transformed
    into.

    If TOWER_STRATEGY is "concat", 'inputs' becomes a tuple containing
    a single tensor (represented as a PartitionedTensor object) with all of
    the data from the towers, as well as the uses/time-steps, concatenated
    together. In this tensor the leading dimension is the batch and
    use/time-step dimensions folded together (with 'use' being the major of
    these two, so that the tensors can be thought of as reshapes of ones of
    shape [num_uses, batch_size, ...]). grads_list is similarly formatted as a
    tuple over sources of such tensors.

    If TOWER_STRATEGY is "separate" the inputs are formatted into lists of
    tensors over towers. Each of these tensors has a similar format to
    the tensor produced by the "concat" option, except that each contains
    only the data from a single tower.  grads_list is similarly formatted
    into a tuple over sources of such tuples.

    Args:
      grads_list: grads_list in its initial format (see above).

    Returns:
      inputs: self._inputs transformed into the appropriate format (see
        above).
      grads_list: grads_list transformed into the appropriate format (see
        above).

    Raises:
      ValueError: If TOWER_STRATEGY is not one of "separate" or "concat".
      ValueError: If the given/initial format of self._inputs and grads_list
        isn't recognized, or doesn't agree with self._num_uses.
    """

    inputs = self._inputs

    if isinstance(inputs[0], (list, tuple)):
      num_uses = len(inputs[0])
      if self._num_uses is not None and self._num_uses != num_uses:
        raise ValueError("num_uses argument doesn't match length of inputs.")
      else:
        self._num_uses = num_uses

      # Check that all mini-batches/towers have the same number of uses
      if not all(len(input_) == num_uses for input_ in inputs):
        raise ValueError("Length of inputs argument is inconsistent across "
                         "towers.")

      if fisher_factors.TOWER_STRATEGY == "concat":
        # Reverse the tower and use/time-step indices, so that use is now first,
        # and towers is second
        inputs = tuple(zip(*inputs))

        # Flatten the two dimensions
        inputs = nest.flatten(inputs)

        # Merge everything together into a PartitionedTensor. We package it in
        # a singleton tuple since the factors will expect a list over towers
        inputs = (utils.PartitionedTensor(inputs),)

      elif fisher_factors.TOWER_STRATEGY == "separate":
        # Merge together the uses/time-step dimension into PartitionedTensors,
        # but keep the leading dimension (towers) intact for the factors to
        # process individually.
        inputs = tuple(utils.PartitionedTensor(input_) for input_ in inputs)

      else:
        raise ValueError("Global config variable TOWER_STRATEGY must be one of "
                         "'concat' or 'separate'.")

    # Now we perform the analogous processing for grads_list
    if isinstance(grads_list[0][0], (list, tuple)):
      num_uses = len(grads_list[0][0])
      if self._num_uses is not None and self._num_uses != num_uses:
        raise ValueError("num_uses argument doesn't match length of outputs, "
                         "or length of outputs is inconsistent with length of "
                         "inputs.")
      else:
        self._num_uses = num_uses

      if not all(len(grad) == num_uses for grads in grads_list
                 for grad in grads):
        raise ValueError("Length of outputs argument is inconsistent across "
                         "towers.")

      if fisher_factors.TOWER_STRATEGY == "concat":
        # Reverse the tower and use/time-step indices, so that use is now first,
        # and towers is second
        grads_list = tuple(tuple(zip(*grads)) for grads in grads_list)

        # Flatten the two dimensions, leaving the leading dimension (source)
        # intact
        grads_list = tuple(nest.flatten(grads) for grads in grads_list)

        # Merge inner dimensions together into PartitionedTensors. We package
        # them in a singleton tuple since the factors will expect a list over
        # towers
        grads_list = tuple((utils.PartitionedTensor(grads),)
                           for grads in grads_list)

      elif fisher_factors.TOWER_STRATEGY == "separate":
        # Merge together the uses/time-step dimension into PartitionedTensors,
        # but keep the leading dimension (towers) intact for the factors to
        # process individually.
        grads_list = tuple(tuple(utils.PartitionedTensor(grad)
                                 for grad in grads)
                           for grads in grads_list)

      else:
        raise ValueError("Global config variable TOWER_STRATEGY must be one of "
                         "'concat' or 'separate'.")

    if self._num_uses is None:
      raise ValueError("You must supply a value for the num_uses argument if "
                       "the number of uses cannot be inferred from inputs or "
                       "outputs arguments (e.g. if they are both given in the "
                       "single Tensor format, instead of as lists of Tensors.")

    return inputs, grads_list


class FullyConnectedMultiIndepFB(InputOutputMultiTowerMultiUse,
                                 KroneckerProductFB):
  """FisherBlock for fully-connected layers that share parameters.

  This class implements the "independence across time" approximation from the
  following paper:
    https://openreview.net/pdf?id=HyMTkQZAb
  """

  def __init__(self, layer_collection, has_bias=False, num_uses=None):
    """Creates a FullyConnectedMultiIndepFB block.

    Args:
      layer_collection: LayerCollection instance.
      has_bias: bool. If True, estimates Fisher with respect to a bias
        parameter as well as the layer's parameters.
      num_uses: int or None. Number of uses of the layer in the model's graph.
        Only required if the data is formatted with uses/time folded into the
        batch dimension (instead of uses/time being a list dimension).
        (Default: None)
    """
    self._has_bias = has_bias

    super(FullyConnectedMultiIndepFB, self).__init__(
        layer_collection=layer_collection,
        num_uses=num_uses)

  def instantiate_factors(self, grads_list, damping):
    inputs, grads_list = self._process_data(grads_list)

    self._input_factor = self._layer_collection.make_or_get_factor(
        fisher_factors.FullyConnectedMultiKF,
        ((inputs,), self._num_uses, self._has_bias))

    self._output_factor = self._layer_collection.make_or_get_factor(
        fisher_factors.FullyConnectedMultiKF, (grads_list, self._num_uses))

    self._setup_damping(damping, normalization=self._num_uses)

  @property
  def _renorm_coeff(self):
    return float(self._num_uses)


class ConvKFCBasicMultiIndepFB(InputOutputMultiTowerMultiUse,
                               KroneckerProductFB):
  """FisherBlock for 2D convolutional layers using the basic KFC approx.

  Similar to ConvKFCBasicFB except that this version supports multiple
  uses/time-steps via a standard independence approximation.  Similar to the
  "independence across time" used in FullyConnectedMultiIndepFB but generalized
  in the obvious way to conv layers.
  """

  def __init__(self,
               layer_collection,
               params,
               padding,
               strides=None,
               dilation_rate=None,
               data_format=None,
               extract_patches_fn=None,
               num_uses=None):
    """Creates a ConvKFCBasicMultiIndepFB block.

    Args:
      layer_collection: The collection of all layers in the K-FAC approximate
          Fisher information matrix to which this FisherBlock belongs.
      params: The parameters (Tensor or tuple of Tensors) of this layer. If
        kernel alone, a Tensor of shape [..spatial_filter_shape..,
        in_channels, out_channels]. If kernel and bias, a tuple of 2 elements
        containing the previous and a Tensor of shape [out_channels].
      padding: str. Padding method.
      strides: List of ints or None. Contains [..spatial_filter_strides..] if
        'extract_patches_fn' is compatible with tf.nn.convolution(), else
        [1, ..spatial_filter_strides, 1].
      dilation_rate: List of ints or None. Rate for dilation along each spatial
        dimension if 'extract_patches_fn' is compatible with
        tf.nn.convolution(), else [1, ..spatial_dilation_rates.., 1].
      data_format: str or None. Format of input data.
      extract_patches_fn: str or None. Name of function that extracts image
        patches. One of "extract_convolution_patches", "extract_image_patches",
        "extract_pointwise_conv2d_patches".
      num_uses: int or None. Number of uses of the layer in the model's graph.
        Only required if the data is formatted with uses/time folded into the
        batch dimension (instead of uses/time being a list dimension).
        (Default: None)
    """
    self._padding = padding
    self._strides = maybe_tuple(strides)
    self._dilation_rate = maybe_tuple(dilation_rate)
    self._data_format = data_format
    self._extract_patches_fn = extract_patches_fn
    self._has_bias = isinstance(params, (tuple, list))

    fltr = params[0] if self._has_bias else params
    self._filter_shape = tuple(fltr.shape.as_list())

    super(ConvKFCBasicMultiIndepFB, self).__init__(
        layer_collection=layer_collection,
        num_uses=num_uses)

  def instantiate_factors(self, grads_list, damping):
    inputs, grads_list = self._process_data(grads_list)

    # Infer number of locations upon which convolution is applied.
    self._num_locations = num_conv_locations(inputs[0].shape.as_list(),
                                             self._strides)

    self._input_factor = self._layer_collection.make_or_get_factor(
        fisher_factors.ConvInputKroneckerFactor,
        (inputs, self._filter_shape, self._padding, self._strides,
         self._dilation_rate, self._data_format, self._extract_patches_fn,
         self._has_bias))
    self._output_factor = self._layer_collection.make_or_get_factor(
        fisher_factors.ConvOutputKroneckerFactor, (grads_list,))

    self._setup_damping(damping, normalization=
                        (self._num_locations * self._num_uses))

  @property
  def _renorm_coeff(self):
    return self._num_locations * self._num_uses


class EmbeddingKFACMultiIndepFB(InputOutputMultiTowerMultiUse,
                                KroneckerProductFB):
  """K-FAC FisherBlock for embedding layers used multiple times in the graph.

  Similar to EmbeddingKFACFB except that this version supports multiple uses
  of the parameter within a single model. These uses could correspond to time
  steps in an RNN architecture, but they don't have to.

  Does not support bias parameters.
  """

  def __init__(self, layer_collection, vocab_size, num_uses=None):
    """Creates a EmbeddingKFACMultiIndepFB block.

    Args:
      layer_collection: The collection of all layers in the K-FAC approximate
          Fisher information matrix to which this FisherBlock belongs.
      vocab_size: int. Size of vocabulary for this embedding layer.
      num_uses: int or None. Number of uses of the layer in the model's graph.
        Only required if the data is formatted with time folded into the batch
        dimension (instead of time being a list dimension). (Default: None)
    """
    self._vocab_size = vocab_size

    super(EmbeddingKFACMultiIndepFB, self).__init__(
        layer_collection=layer_collection,
        num_uses=num_uses)

  def instantiate_factors(self, grads_list, damping):
    """Instantiate Kronecker Factors for this FisherBlock.

    Args:
      grads_list: List of list of list of Tensors. grads_list[i][j][k] is the
        gradient of the loss with respect to 'outputs' from source 'i',
        tower/mini-batch 'j', and use/time-step 'k'. Each Tensor has shape
        [tower_minibatch_size, output_size].
      damping: 0-D Tensor or float. 'damping' * identity is approximately added
        to this FisherBlock's Fisher approximation.
    """
    inputs, grads_list = self._process_data(grads_list)

    self._input_factor = self._layer_collection.make_or_get_factor(
        fisher_factors.EmbeddingInputKroneckerFactor,
        (inputs, self._vocab_size))
    self._output_factor = self._layer_collection.make_or_get_factor(
        fisher_factors.FullyConnectedMultiKF, (grads_list, self._num_uses))
    self._setup_damping(damping, normalization=self._num_uses)

  @property
  def _renorm_coeff(self):
    return float(self._num_uses)


class SeriesFBApproximation(enum.IntEnum):
  """See FullyConnectedSeriesFB.__init__ for description and usage."""
  option1 = 1
  option2 = 2


class FullyConnectedSeriesFB(InputOutputMultiTowerMultiUse,
                             KroneckerProductFB):
  """FisherBlock for fully-connected layers that share parameters across time.

  This class implements the "Option 1" and "Option 2" approximation from the
  following paper:
    https://openreview.net/pdf?id=HyMTkQZAb

  See the end of the appendix of the paper for a pseudo-code of the
  algorithm being implemented by multiply_matpower here.  Note that we are
  using pre-computed versions of certain matrix-matrix products to speed
  things up.  This is explicitly explained wherever it is done.
  """

  def __init__(self,
               layer_collection,
               has_bias=False,
               num_uses=None,
               option=SeriesFBApproximation.option2):
    """Constructs a new `FullyConnectedSeriesFB`.

    Args:
      layer_collection: The collection of all layers in the K-FAC approximate
        Fisher information matrix to which this FisherBlock belongs.
      has_bias: Whether the layer includes a bias parameter.
      num_uses: int or None. Number of time-steps over which the layer
        is used. Only required if the data is formatted with time folded into
        the batch dimension (instead of time being a list dimension).
        (Default: None)
      option: A `SeriesFBApproximation` specifying the simplifying assumption
        to be used in this block. `option1` approximates the cross-covariance
        over time as a symmetric matrix, while `option2` makes
        the assumption that training sequences are infinitely long. See section
        3.5 of the paper for more details.
    """

    self._has_bias = has_bias
    self._option = option

    super(FullyConnectedSeriesFB, self).__init__(
        layer_collection=layer_collection,
        num_uses=num_uses)

  @property
  def _num_timesteps(self):
    return self._num_uses

  @property
  def _renorm_coeff(self):
    # This should no longer be used since the multiply_X functions from the base
    # class have been overridden
    assert False

  def instantiate_factors(self, grads_list, damping):
    inputs, grads_list = self._process_data(grads_list)

    self._input_factor = self._layer_collection.make_or_get_factor(
        fisher_factors.FullyConnectedMultiKF,
        ((inputs,), self._num_uses, self._has_bias))
    self._input_factor.register_cov_dt1()

    self._output_factor = self._layer_collection.make_or_get_factor(
        fisher_factors.FullyConnectedMultiKF, (grads_list, self._num_uses))
    self._output_factor.register_cov_dt1()

    self._setup_damping(damping, normalization=self._num_uses)

  def register_matpower(self, exp):
    if exp != -1:
      raise NotImplementedError("FullyConnectedSeriesFB only supports inverse"
                                "multiplications.")

    if self._option == SeriesFBApproximation.option1:
      self._input_factor.register_option1quants(self._input_damping_func)
      self._output_factor.register_option1quants(self._output_damping_func)
    elif self._option == SeriesFBApproximation.option2:
      self._input_factor.register_option2quants(self._input_damping_func)
      self._output_factor.register_option2quants(self._output_damping_func)
    else:
      raise ValueError(
          "Unrecognized FullyConnectedSeriesFB approximation: {}".format(
              self._option))

  def multiply_matpower(self, vector, exp):
    if exp != -1:
      raise NotImplementedError("FullyConnectedSeriesFB only supports inverse"
                                "multiplications.")

    # pylint: disable=invalid-name

    Z = utils.layer_params_to_mat2d(vector)

    # Derivations were done for "batch_dim==1" case so we need to convert to
    # that orientation:
    Z = array_ops.transpose(Z)

    if self._option == SeriesFBApproximation.option1:

      # Note that \\(L_A = A0^{-1/2} * U_A and L_G = G0^{-1/2} * U_G.\\)
      L_A, psi_A = self._input_factor.get_option1quants(
          self._input_damping_func)
      L_G, psi_G = self._output_factor.get_option1quants(
          self._output_damping_func)

      def gamma(x):
        # We are assuming that each case has the same number of time-steps.
        # If this stops being the case one shouldn't simply replace this T
        # with its average value.  Instead, one needs to go back to the
        # definition of the gamma function from the paper.
        T = self._num_timesteps
        return (1 - x)**2 / (T * (1 - x**2) - 2 * x * (1 - x**T))

      # \\(Y = \gamma( psi_G*psi_A^T )\\) (computed element-wise)
      # Even though Y is Z-independent we are recomputing it from the psi's
      # each since Y depends on both A and G quantities, and it is relatively
      # cheap to compute.
      Y = gamma(array_ops.reshape(psi_G, [int(psi_G.shape[0]), -1]) * psi_A)

      # \\(Z = L_G^T * Z * L_A\\)
      # This is equivalent to the following computation from the original
      # pseudo-code:
      # \\(Z = G0^{-1/2} * Z * A0^{-1/2}\\)
      # \\(Z = U_G^T * Z * U_A\\)
      Z = math_ops.matmul(L_G, math_ops.matmul(Z, L_A), transpose_a=True)

      # \\(Z = Z .* Y\\)
      Z *= Y

      # \\(Z = L_G * Z * L_A^T\\)
      # This is equivalent to the following computation from the original
      # pseudo-code:
      # \\(Z = U_G * Z * U_A^T\\)
      # \\(Z = G0^{-1/2} * Z * A0^{-1/2}\\)
      Z = math_ops.matmul(L_G, math_ops.matmul(Z, L_A, transpose_b=True))

    elif self._option == SeriesFBApproximation.option2:

      # Note that \\(P_A = A_1^T * A_0^{-1} and P_G = G_1^T * G_0^{-1}\\),
      # and \\(K_A = A_0^{-1/2} * E_A\ and\ K_G = G_0^{-1/2} * E_G.\\)
      P_A, K_A, mu_A = self._input_factor.get_option2quants(
          self._input_damping_func)
      P_G, K_G, mu_G = self._output_factor.get_option2quants(
          self._output_damping_func)

      # Our approach differs superficially from the pseudo-code in the paper
      # in order to reduce the total number of matrix-matrix multiplies.
      # In particular, the first three computations in the pseudo code are
      # \\(Z = G0^{-1/2} * Z * A0^{-1/2}\\)
      # \\(Z = Z - hPsi_G^T * Z * hPsi_A\\)
      # \\(Z = E_G^T * Z * E_A\\)
      # Noting that hPsi = C0^{-1/2} * C1 * C0^{-1/2}\\), so that
      # \\(C0^{-1/2} * hPsi = C0^{-1} * C1 * C0^{-1/2} = P^T * C0^{-1/2}\\)
      # the entire computation can be written as
      # \\(Z = E_G^T * (G0^{-1/2} * Z * A0^{-1/2}\\)
      # \\(    - hPsi_G^T * G0^{-1/2} * Z * A0^{-1/2} * hPsi_A) * E_A\\)
      # \\(  = E_G^T * (G0^{-1/2} * Z * A0^{-1/2}\\)
      # \\(    - G0^{-1/2} * P_G * Z * P_A^T * A0^{-1/2}) * E_A\\)
      # \\(  = E_G^T * G0^{-1/2} * Z * A0^{-1/2} * E_A\\)
      # \\(    -  E_G^T* G0^{-1/2} * P_G * Z * P_A^T * A0^{-1/2} * E_A\\)
      # \\(  = K_G^T * Z * K_A  -  K_G^T * P_G * Z * P_A^T * K_A\\)
      # This final expression is computed by the following two lines:
      # \\(Z = Z - P_G * Z * P_A^T\\)
      Z -= math_ops.matmul(P_G, math_ops.matmul(Z, P_A, transpose_b=True))
      # \\(Z = K_G^T * Z * K_A\\)
      Z = math_ops.matmul(K_G, math_ops.matmul(Z, K_A), transpose_a=True)

      # \\(Z = Z ./ (1*1^T - mu_G*mu_A^T)\\)
      # Be careful with the outer product.  We don't want to accidentally
      # make it an inner-product instead.
      tmp = 1.0 - array_ops.reshape(mu_G, [int(mu_G.shape[0]), -1]) * mu_A
      # Prevent some numerical issues by setting any 0.0 eigs to 1.0
      tmp += 1.0 * math_ops.cast(math_ops.equal(tmp, 0.0), dtype=tmp.dtype)
      Z /= tmp

      # We now perform the transpose/reverse version of the operations
      # derived above, whose derivation from the original pseudo-code is
      # analgous.
      # \\(Z = K_G * Z * K_A^T\\)
      Z = math_ops.matmul(K_G, math_ops.matmul(Z, K_A, transpose_b=True))

      # \\(Z = Z - P_G^T * Z * P_A\\)
      Z -= math_ops.matmul(P_G, math_ops.matmul(Z, P_A), transpose_a=True)

      # \\(Z = normalize (1/E[T]) * Z\\)
      # Note that this normalization is done because we compute the statistics
      # by averaging, not summing, over time. (And the gradient is presumably
      # summed over time, not averaged, and thus their scales are different.)
      Z /= math_ops.cast(self._num_timesteps, Z.dtype)

    # Convert back to the "batch_dim==0" orientation.
    Z = array_ops.transpose(Z)

    return utils.mat2d_to_layer_params(vector, Z)

    # pylint: enable=invalid-name
