distributed_shampoo.py

#from https://github.com/google-research/google-research/blob/master/scalable_shampoo/optax/distributed_shampoo.py

# coding=utf-8
# Copyright 2022 The Google Research Authors.
#
# 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.

# An implementation of distributed Shampoo optimizer from:
#
#  Scalable Second Order Optimization for Deep Learning
#  Rohan Anil, Vineet Gupta, Tomer Koren, Kevin Regan, Yoram Singer
#  Preprint Paper: https://arxiv.org/abs/2002.09018
#
# This implementation moves computation of inverse pth root back to the
# accelerator (if higher precision is available).
#
# Authors: Rohan Anil (rohananil at google dot com)
#    &     Vineet Gupta (vineet at google dot com)
#

"""Distributed Shampoo Implementation."""

import enum
import functools
import itertools
from typing import Any, List, NamedTuple

import chex
from flax import struct
import jax
from jax import lax
import jax.experimental.pjit as pjit
import jax.numpy as jnp
import numpy as np
import optax

PartitionSpec = pjit.PartitionSpec


# pylint:disable=no-value-for-parameter
@struct.dataclass
class QuantizedValue:
  """State associated with quantized value."""
  quantized: chex.Array
  diagonal: chex.Array  # Diagonal (if extract_diagonal is set)
  bucket_size: chex.Array
  quantized_dtype: jnp.dtype = struct.field(
      pytree_node=False)  # Dtype for the quantized value.
  extract_diagonal: bool = struct.field(
      pytree_node=False)  # In case its centered.
  shape: Any = struct.field(pytree_node=False)  # Shape of the tensor.

  @classmethod
  def from_float_value(cls, fvalue, quantized_dtype, extract_diagonal=False):
    if isinstance(fvalue, list) and not fvalue:
      return QuantizedValue([], [], [], quantized_dtype, extract_diagonal, [])
    quantized, diagonal_fvalue, bucket_size = QuantizedValue.quantize(
        fvalue, quantized_dtype, extract_diagonal)
    return QuantizedValue(quantized, diagonal_fvalue, bucket_size,
                          quantized_dtype, extract_diagonal,
                          list(quantized.shape))

  # Quantization is from Lingvo JAX optimizers.
  # We extend it for int16 quantization of PSD matrices.
  @classmethod
  def quantize(cls, fvalue, quantized_dtype, extract_diagonal=False):
    """Returns quantized value and the bucket."""
    if quantized_dtype == jnp.float32:
      return fvalue, [], []
    elif quantized_dtype == jnp.bfloat16:
      return fvalue.astype(jnp.bfloat16), [], []

    float_dtype = fvalue.dtype
    if quantized_dtype == jnp.int8:
      # value -128 is not used.
      num_buckets = jnp.array(127.0, dtype=float_dtype)
    elif quantized_dtype == jnp.int16:
      # value -32768 is not used.
      num_buckets = jnp.array(32767.0, dtype=float_dtype)
    else:
      raise ValueError(f'Quantized dtype {quantized_dtype} not supported.')
    # max value is mapped to num_buckets

    if extract_diagonal and fvalue.ndim != 2:
      raise ValueError(
          f'Input array {fvalue} must be 2D to work with extract_diagonal.')

    diagonal_fvalue = []
    if extract_diagonal:
      diagonal_fvalue = jnp.diag(fvalue)
      # Remove the diagonal entries.
      fvalue = fvalue - jnp.diag(diagonal_fvalue)

    # TODO(rohananil): Extend this by making use of information about the blocks
    # SM3 style which will be useful for diagonal statistics
    # We first decide the scale.
    if fvalue.ndim < 1:
      raise ValueError(
          f'Input array {fvalue} must have a strictly positive number of '
          'dimensions.')

    max_abs = jnp.max(jnp.abs(fvalue), axis=0)
    bucket_size = max_abs / num_buckets
    bs_expanded = bucket_size[jnp.newaxis, Ellipsis]
    # To avoid divide by 0.0
    bs_nonzero = jnp.where(bs_expanded > 0.0, bs_expanded,
                           jnp.ones_like(bs_expanded))
    ratio = fvalue / bs_nonzero
    # We use rounding to remove bias.
    quantized = jnp.round(ratio)
    return quantized.astype(quantized_dtype), diagonal_fvalue, bucket_size

  def to_float(self):
    """Returns the float value."""
    if isinstance(self.quantized, list) and not self.quantized:
      return self.quantized

    if self.quantized_dtype == jnp.float32:
      return self.quantized

    if self.quantized_dtype == jnp.bfloat16:
      return self.quantized.astype(jnp.float32)

    float_dtype = self.bucket_size.dtype
    bucket_size = self.bucket_size[jnp.newaxis, Ellipsis]
    val = self.quantized.astype(float_dtype) * bucket_size
    if self.extract_diagonal:
      val += jnp.diag(self.diagonal)
    return val


# Per parameter optimizer state used in data-parallel training.
class ParameterStats(NamedTuple):
  """State associated to each parameter of the model being trained."""
  diagonal_statistics: QuantizedValue  # Accumulator for diagonal preconditioner
  statistics: List[Any]  # Statistics (QuantizedValue, chex.Array)
  preconditioners: List[Any]  # Preconditioners (QuantizedValue, chex.Array)
  diagonal_momentum: QuantizedValue  # Momentum for the diagonal preconditioner
  momentum: QuantizedValue  # Momentum for the shampoo preconditioner


# For training extremely large model; We keep a global state with a concatenated
# statistics and preconditioner states for all vars. This is so that we can
# annotate the leading axis to be sharded to save memory at the cost of
# communication.
@struct.dataclass
class GlobalShardedParameterStats:
  statistics: chex.Array  # Statistics
  preconditioners: chex.Array  # Preconditioners
  exponents: chex.Array  # exponents


# These are per-parameter local states; All statistics here mirror the parameter
# Thus the sharding is copied over from the param specification.
@struct.dataclass
class LocalShardedParameterStats:
  """State associated to each parameter of the model being trained."""
  diagonal_statistics: QuantizedValue  # Accumulator for diagonal preconditioner
  diagonal_momentum: QuantizedValue  # Momentum for the diagonal preconditioner
  momentum: QuantizedValue  # Momentum for the shampoo preconditioner
  index_start: np.int32 = struct.field(
      pytree_node=False)  # Index into global statistics array
  sizes: Any = struct.field(pytree_node=False)  # Sizes of the statistics.


class ShardedShampooStats(NamedTuple):
  """Shampoo state in sharded mode."""
  global_stats: Any
  local_stats: Any


class ShampooState(NamedTuple):
  count: chex.Array
  stats: Any


class InitFnState(NamedTuple):
  init_fn: Any
  pspec_fn: Any
  shape_and_dtype_fn: Any


class GraftingType(enum.IntEnum):
  SGD = 1
  ADAGRAD = 2
  RMSPROP = 3
  RMSPROP_NORMALIZED = 4


def power_iteration(
    matrix,
    num_iters=100,
    error_tolerance=1e-6,
    precision=lax.Precision.HIGHEST):
  r"""Power iteration algorithm.

  The power iteration algorithm takes a symmetric PSD matrix `A`, and produces
  a scalar `\lambda` , which is the greatest (in absolute value) eigenvalue
  of `A`, and a vector v, which is the corresponding eigenvector of `A`.

  References:
    [Wikipedia, 2021](https://en.wikipedia.org/wiki/Power_iteration)

  Args:
    matrix: the symmetric PSD matrix.
    num_iters: Number of iterations.
    error_tolerance: Iterative exit condition.
    precision: precision XLA related flag, the available options are:
      a) lax.Precision.DEFAULT (better step time, but not precise)
      b) lax.Precision.HIGH (increased precision, slower)
      c) lax.Precision.HIGHEST (best possible precision, slowest)

  Returns:
    eigen vector, eigen value
  """
  matrix_size = matrix.shape[-1]
  def _iter_condition(state):
    i, unused_v, unused_s, unused_s_v, run_step = state
    return jnp.logical_and(i < num_iters, run_step)

  def _iter_body(state):
    """One step of power iteration."""
    i, new_v, s, s_v, unused_run_step = state
    new_v = new_v / jnp.linalg.norm(new_v)

    s_v = jnp.einsum('ij,j->i', matrix, new_v, precision=precision)
    s_new = jnp.einsum('i,i->', new_v, s_v, precision=precision)
    return (i + 1, s_v, s_new, s_v,
            jnp.greater(jnp.abs(s_new - s), error_tolerance))

  # Figure out how to use step as seed for random.
  v_0 = np.random.RandomState(1729).uniform(-1.0, 1.0,
                                            matrix_size).astype(matrix.dtype)

  init_state = tuple([0, v_0, jnp.zeros([], dtype=matrix.dtype), v_0, True])
  _, v_out, s_out, _, _ = lax.while_loop(
      _iter_condition, _iter_body, init_state)
  v_out = v_out / jnp.linalg.norm(v_out)
  return v_out, s_out


def matrix_inverse_pth_root(
    matrix,
    p,
    num_iters=100,
    ridge_epsilon=1e-6,
    error_tolerance=1e-6,
    precision=lax.Precision.HIGHEST):
  """Computes `matrix^(-1/p)`, where `p` is a positive integer.

  This function uses the Coupled newton iterations algorithm for
  the computation of a matrix's inverse pth root.


  References:
    [Functions of Matrices, Theory and Computation,
     Nicholas J Higham, Pg 184, Eq 7.18](
     https://epubs.siam.org/doi/book/10.1137/1.9780898717778)

  Args:
    matrix: the symmetric PSD matrix whose power it to be computed
    p: exponent, for p a positive integer.
    num_iters: Maximum number of iterations.
    ridge_epsilon: Ridge epsilon added to make the matrix positive definite.
    error_tolerance: Error indicator, useful for early termination.
    precision: precision XLA related flag, the available options are:
      a) lax.Precision.DEFAULT (better step time, but not precise)
      b) lax.Precision.HIGH (increased precision, slower)
      c) lax.Precision.HIGHEST (best possible precision, slowest)

  Returns:
    matrix^(-1/p)
  """

  assert matrix.shape[0] == matrix.shape[1]

  # We use float32 for the matrix inverse pth root.
  # Switch to f64 if you have hardware that supports it.
  matrix_size = matrix.shape[0]
  alpha = jnp.asarray(-1.0 / p, jnp.float32)
  identity = jnp.eye(matrix_size, dtype=jnp.float32)
  _, max_ev = power_iteration(
      matrix=matrix, num_iters=100,
      error_tolerance=1e-6, precision=precision)
  ridge_epsilon = ridge_epsilon * jnp.maximum(max_ev, 1e-16)

  def _unrolled_mat_pow_1(mat_m):
    """Computes mat_m^1."""
    return mat_m

  def _unrolled_mat_pow_2(mat_m):
    """Computes mat_m^2."""
    return jnp.matmul(mat_m, mat_m, precision=precision)

  def _unrolled_mat_pow_4(mat_m):
    """Computes mat_m^4."""
    mat_pow_2 = _unrolled_mat_pow_2(mat_m)
    return jnp.matmul(
        mat_pow_2, mat_pow_2, precision=precision)

  def _unrolled_mat_pow_8(mat_m):
    """Computes mat_m^4."""
    mat_pow_4 = _unrolled_mat_pow_4(mat_m)
    return jnp.matmul(
        mat_pow_4, mat_pow_4, precision=precision)

  def mat_power(mat_m, p):
    """Computes mat_m^p, for p == 1, 2, 4 or 8.

    Args:
      mat_m: a square matrix
      p: a positive integer

    Returns:
      mat_m^p
    """
    # We unrolled the loop for performance reasons.
    exponent = jnp.round(jnp.log2(p))
    return lax.switch(
        jnp.asarray(exponent, jnp.int32), [
            _unrolled_mat_pow_1,
            _unrolled_mat_pow_2,
            _unrolled_mat_pow_4,
            _unrolled_mat_pow_8,
        ], (mat_m))

  def _iter_condition(state):
    (i, unused_mat_m, unused_mat_h, unused_old_mat_h, error,
     run_step) = state
    error_above_threshold = jnp.logical_and(
        error > error_tolerance, run_step)
    return jnp.logical_and(i < num_iters, error_above_threshold)

  def _iter_body(state):
    (i, mat_m, mat_h, unused_old_mat_h, error, unused_run_step) = state
    mat_m_i = (1 - alpha) * identity + alpha * mat_m
    new_mat_m = jnp.matmul(mat_power(mat_m_i, p), mat_m, precision=precision)
    new_mat_h = jnp.matmul(mat_h, mat_m_i, precision=precision)
    new_error = jnp.max(jnp.abs(new_mat_m - identity))
    # sometimes error increases after an iteration before decreasing and
    # converging. 1.2 factor is used to bound the maximal allowed increase.
    return (i + 1, new_mat_m, new_mat_h, mat_h, new_error,
            new_error < error * 1.2)

  if matrix_size == 1:
    resultant_mat_h = (matrix + ridge_epsilon)**alpha
    error = 0
  else:
    damped_matrix = matrix + ridge_epsilon * identity

    z = (1 + p) / (2 * jnp.linalg.norm(damped_matrix))
    new_mat_m_0 = damped_matrix * z
    new_error = jnp.max(jnp.abs(new_mat_m_0 - identity))
    new_mat_h_0 = identity * jnp.power(z, 1.0 / p)
    init_state = tuple(
        [0, new_mat_m_0, new_mat_h_0, new_mat_h_0, new_error, True])
    _, mat_m, mat_h, old_mat_h, error, convergence = lax.while_loop(
        _iter_condition, _iter_body, init_state)
    error = jnp.max(jnp.abs(mat_m - identity))
    is_converged = jnp.asarray(convergence, old_mat_h.dtype)
    resultant_mat_h = is_converged * mat_h + (1 - is_converged) * old_mat_h
    resultant_mat_h = jnp.asarray(resultant_mat_h, matrix.dtype)
  return resultant_mat_h, error


def merge_small_dims(shape_to_merge, max_dim):
  """Merge small dimensions.

  If there are some small dimensions, we collapse them:
  e.g. [1, 2, 512, 1, 2048, 1, 3, 4] --> [1024, 2048, 12] if max_dim = 1024
       [1, 2, 768, 1, 2048] --> [2, 768, 2048]

  Args:
    shape_to_merge: Shape to merge small dimensions.
    max_dim: Maximal dimension of output shape used in merging.

  Returns:
    Merged shape.
  """
  resulting_shape = []
  product = 1
  for d in shape_to_merge:
    if product * d <= max_dim:
      product *= d
    else:
      if product > 1:
        resulting_shape.append(product)
      product = d
  if product > 1:
    resulting_shape.append(product)
  return resulting_shape


def pad_matrix(mat, max_size):
  """Pad a matrix to a max_size.

  Args:
    mat: a matrix to pad.
    max_size: matrix size requested.

  Returns:
    Given M returns [[M, 0], [0, I]]
  """
  size = mat.shape[0]
  assert size <= max_size
  if size == max_size:
    return mat
  pad_size = max_size - size
  zs1 = jnp.zeros([size, pad_size], dtype=mat.dtype)
  zs2 = jnp.zeros([pad_size, size], dtype=mat.dtype)
  eye = jnp.eye(pad_size, dtype=mat.dtype)
  mat = jnp.concatenate([mat, zs1], 1)
  mat = jnp.concatenate([mat, jnp.concatenate([zs2, eye], 1)], 0)
  return mat


def pad_vector(vec, max_size):
  """Pad a vector to a max_size.

  Args:
    vec: a vector to pad.
    max_size: matrix size requested.

  Returns:
    Given V returns [V, 0]
  """
  size = vec.shape[0]
  assert size <= max_size
  if size == max_size:
    return vec
  pad_size = max_size - size
  zs1 = jnp.zeros([pad_size], dtype=vec.dtype)
  return jnp.concatenate([vec, zs1], 0)


def efficient_cond(predicate, compute_fn, init_state, *args, **kwargs):
  """Avoids wasteful buffer allocation with XLA."""

  def _iter_body(unused_state):
    results = compute_fn(*args, **kwargs)
    return tuple([False] + list(results))

  def _iter_condition(state):
    return state[0]

  results = jax.lax.while_loop(_iter_condition, _iter_body,
                               tuple([predicate] + init_state))
  return tuple(results[1:])


class BlockPartitioner:
  """Partitions a tensor into smaller tensors."""

  def __init__(self, param, block_size):
    self._shape = param.shape
    self._splits = []
    split_sizes = []
    # We split params into smaller blocks. Here we store the metadata to make
    # that split.
    for i, d in enumerate(param.shape):
      if 0 < block_size < d:
        # d-1, otherwise split appends a 0-size array.
        nsplit = (d - 1) // block_size
        indices = (np.arange(nsplit, dtype=np.int32) + 1) * block_size
        sizes = np.ones(nsplit + 1, dtype=np.int32) * block_size
        sizes[-1] = d - indices[-1]
        self._splits.append((i, indices))
        split_sizes.append(sizes)
      else:
        split_sizes.append(np.array([d], dtype=np.int32))
    self._num_splits = len(split_sizes)
    self._preconditioner_shapes = []
    for t in itertools.product(*split_sizes):
      self._preconditioner_shapes.extend([[d, d] for d in t])

  def shapes_for_preconditioners(self):
    return self._preconditioner_shapes

  def num_splits(self):
    return self._num_splits

  def partition(self, tensor):
    """Partition tensor into blocks."""

    assert tensor.shape == self._shape
    tensors = [tensor]
    for (i, indices) in self._splits:
      tensors_local = []
      for t in tensors:
        tensors_local.extend(jnp.split(t, indices_or_sections=indices, axis=i))
      tensors = tensors_local
    return tensors

  def merge_partitions(self, partitions):
    """Merge partitions back to original shape."""

    for (i, indices) in reversed(self._splits):
      n = len(indices) + 1
      partial_merged_tensors = []
      ind = 0
      while ind < len(partitions):
        partial_merged_tensors.append(
            jnp.concatenate(partitions[ind:ind + n], axis=i))
        ind += n
      partitions = partial_merged_tensors
    assert len(partitions) == 1
    return partitions[0]


class Preconditioner:
  """Compute statistics/shape from gradients for preconditioning."""

  def __init__(self, param, block_size, best_effort_shape_interpretation):
    self._original_shape = param.shape
    self._transformed_shape = param.shape
    if best_effort_shape_interpretation:
      self._transformed_shape = merge_small_dims(self._original_shape,
                                                 block_size)
    reshaped_param = jnp.reshape(param, self._transformed_shape)
    self._partitioner = BlockPartitioner(reshaped_param, block_size)

  def statistics_from_grad(self, grad):
    """Compute statistics from gradients.

    Args:
      grad: Gradient to compute statistics from.

    Returns:
      A list of gradient statistics for each partition.
    """
    reshaped_grad = jnp.reshape(grad, self._transformed_shape)
    partitioned_grads = self._partitioner.partition(reshaped_grad)
    stats = []
    for g in partitioned_grads:
      g_stats = []
      rank = len(g.shape)
      for i in range(rank):
        axes = list(range(i)) + list(range(i + 1, rank))
        stat = jnp.tensordot(g, g, axes=(axes, axes))
        g_stats.append(stat)
      stats.extend(g_stats)
    return stats

  def shapes_for_preconditioners(self):
    """Returns shape from statistics."""
    return self._partitioner.shapes_for_preconditioners()

  def exponent_for_preconditioner(self):
    """Returns exponent to use for inverse-pth root M^{-1/p}."""
    return 2 * len(self._transformed_shape)

  def preconditioned_grad(self, grad, preconditioners):
    """Precondition the gradient.

    Args:
      grad: A gradient tensor to precondition.
      preconditioners: A list of preconditioners to apply.

    Returns:
      A preconditioned gradient.
    """

    reshaped_grad = jnp.reshape(grad, self._transformed_shape)
    partitioned_grads = self._partitioner.partition(reshaped_grad)
    preconditioned_partitioned_grads = []
    num_splits = self._partitioner.num_splits()
    for i, g in enumerate(partitioned_grads):
      preconditioners_for_grad = preconditioners[i * num_splits:(i + 1) *
                                                 num_splits]
      rank = len(g.shape)
      precond_g = g
      for j in range(rank):
        precond_g = jnp.tensordot(
            precond_g, preconditioners_for_grad[j], axes=[[0], [0]])
      preconditioned_partitioned_grads.append(precond_g)
    merged_grad = self._partitioner.merge_partitions(
        preconditioned_partitioned_grads)
    return jnp.reshape(merged_grad, self._original_shape)


def _convert_to_parameter_stats(global_stats, local_stat):
  """Creates parameter stats from sharded stats."""
  index_start = int(local_stat.index_start)
  index_end = int(len(local_stat.sizes)) + index_start
  statistics = global_stats.statistics[index_start:index_end, :, :]
  preconditioners = global_stats.preconditioners[index_start:index_end, :, :]
  new_statistics = []
  new_preconditioners = []
  for i, size in enumerate(local_stat.sizes):
    new_statistics.append(statistics[i][:size, :size])
    new_preconditioners.append(preconditioners[i][:size, :size])
  return ParameterStats(local_stat.diagonal_statistics, new_statistics,
                        new_preconditioners, local_stat.diagonal_momentum,
                        local_stat.momentum)


def _convert_from_parameter_stats(parameter_stats, local_stats):
  """Creates sharded stats from paramter stats."""
  return LocalShardedParameterStats(parameter_stats.diagonal_statistics,
                                    parameter_stats.diagonal_momentum,
                                    parameter_stats.momentum,
                                    local_stats.index_start, local_stats.sizes)


def batch(x, num_devices):
  """Batch `x` so that so that leading axis is num_devices."""
  n = len(x)
  b = int(n / num_devices)
  return jnp.stack([jnp.stack(x[idx:idx + b]) for idx in range(0, n, b)])


def unbatch(batched_values):
  """Unbatch values across leading axis and return a list of elements."""
  b1, b2 = batched_values.shape[0], batched_values.shape[1]
  results = []
  for v_array in jnp.split(batched_values, indices_or_sections=b1, axis=0):
    v_array = jnp.squeeze(v_array)
    # b2 = batches (number of preconditioner computation) per core.
    if b2 > 1:
      for v in jnp.split(v_array, indices_or_sections=b2, axis=0):
        results.append(jnp.squeeze(v))
    else:
      results.append(v_array)
  return results


def distributed_shampoo(
    learning_rate,
    block_size,
    beta1=0.9,
    beta2=0.999,
    diagonal_epsilon=1e-10,
    matrix_epsilon=1e-6,
    weight_decay=0.0,
    start_preconditioning_step=5,
    preconditioning_compute_steps=1,
    statistics_compute_steps=1,
    best_effort_shape_interpretation=True,
    graft_type=GraftingType.SGD,
    nesterov=True,
    exponent_override=0,
    # Pass pmap 'batch axis name' in pmap mode.
    batch_axis_name=None,
    ### Only set following 3 params in pjit/spmd mode.
    ### WARNING: Experimental
    statistics_partition_spec=None,
    preconditioner_partition_spec=None,
    num_devices_for_pjit=None,
    shard_optimizer_states=False,
    ###
    ### Experimental memory reduction mode
    best_effort_memory_usage_reduction=False,
    ###
    inverse_failure_threshold=0.1,
    moving_average_for_momentum=False,
    skip_preconditioning_dim_size_gt=4096,
    clip_by_scaled_gradient_norm=None,
    precision=lax.Precision.HIGHEST):
  """Distributed Shampoo optimizer.

  Distributed Shampoo is a second-order preconditioned method (concretely, a
  variant of full-matrix Adagrad), that provides significant convergence and
  wall-clock time improvements compared to conventional first-order methods,
  and that has been shown to scale to large state-of-the-art deep learning
  models.

  References:
    Scalable Second Order Optimization for Deep Learning,
    Rohan Anil, Vineet Gupta, Tomer Koren, Kevin Regan, Yoram Singer

    Preprint: https://arxiv.org/abs/2002.09018

  Args:
    learning_rate: the step size used to update the parameters.
    block_size: Block size for large layers (if > 0). Preconditioning compute
      operation is cubic in the dimension of the tensor. Block size allows us to
      chunk the layers into sub-layers of maximal dimension dictated by this
      value. Use 128 as default (increase if you have compute budget).
    beta1: momentum parameter.
    beta2: second moment averaging parameter.
    diagonal_epsilon: epsilon for diagonal adagrad (only if layerwise grafting
      to AdaGrad is enabled).
    matrix_epsilon: epsilon to add to statistics before computing inverse pth
      root. If you are running in f32 precision for inverse pth root
      (recommended today) this can go upto 1e-6. If you have latest hardware
      with native f64 precision, set this upto 1e-12.
    weight_decay: Weight decay for regularization.
    start_preconditioning_step: When to start Shampoo update before which
      diagonal update is used. This is because we dont have enough information
      to do stable inverse.
    preconditioning_compute_steps: How often to compute preconditioner.
      Performance tuning params for controlling memory and compute requirements.
      Ideally set this and statistics_compute_steps params to 1.
    statistics_compute_steps: How often to compute statistics.
    best_effort_shape_interpretation: If there are some small dimensions,
      collapse them e.g. [1, 2, 512, 1, 2048, 1, 3, 4] --> [1024, 2048, 12] if
      block = 1024, [1, 2, 768, 1, 2048] --> [2, 768, 2048]
    graft_type: Grafting is a technique to fix the layerwise scale of Shampoo
      optimizer. This allows us to plugin the Shampoo optimizer into settings
      where SGD/AdaGrad is already well tuned. Available options are:
        GraftingType.SGD and GraftingType.ADAGRAD.
    nesterov: Nesterov momentum.
    exponent_override: Override the exponent used in matrix inverse.
    batch_axis_name: labeled axis over pmap for data-parallel training the
      optimizer used for.
    statistics_partition_spec: PartitionSpec to be used in sharded mode.
    preconditioner_partition_spec: PartitionSpec to be used in sharded mode.
    num_devices_for_pjit: Number of devices to parallelize over when using pjit.
    shard_optimizer_states: Shard optimizer states to save memory in model
      parallel training.
    best_effort_memory_usage_reduction: Best effort memory usage reduction.
      diagonal_statistics -> jnp.bfloat16
      momentum buffers (2x) -> jnp.int8
      statistics, preconditioners -> jnp.int16 + diagonals
    inverse_failure_threshold: numerics are hard and inverses fail sometimes; we
      determine that using this threshold.
    moving_average_for_momentum: Whether to use moving average for momentum
      instead of exponential moving average.
    skip_preconditioning_dim_size_gt: Skip if preconditioning dim size is
        greater than this value.
    clip_by_scaled_gradient_norm: Clip by scaled gradient norm (only useful
      when using RMSProp Grafting).
    precision: precision XLA related flag, the available options are: a)
      lax.Precision.DEFAULT (better step time, but not precise) b)
      lax.Precision.HIGH (increased precision, slower) c) lax.Precision.HIGHEST
      (best possible precision, slowest)

  Returns:
    a GradientTransformation.
  """

  def quantized_dtype_for_momentum_buffers():
    return jnp.int8 if best_effort_memory_usage_reduction else jnp.float32

  # TODO(rohananil): Explore int8-16 quantization with non-linear bucket sizes.
  def quantized_dtype_for_diagonal_statistics_buffers():
    return jnp.bfloat16 if best_effort_memory_usage_reduction else jnp.float32

  # Preconditioner and statistics are both stores as int16 in this mode.
  # We take out the diagonal to make quantization easier.
  def quantized_dtype_for_second_moment_statistics_buffers():
    return jnp.int16 if best_effort_memory_usage_reduction and batch_axis_name else jnp.float32

  # Preconditioner and statistics are both stores as int16 in this mode.
  # We take out the diagonal to make quantization easier.
  def quantized_dtype_for_second_moment_preconditioner_buffers():
    return jnp.int16 if best_effort_memory_usage_reduction and batch_axis_name else jnp.float32

  def _to_float(maybe_quantized):
    if isinstance(maybe_quantized, QuantizedValue):
      return maybe_quantized.to_float()
    else:
      return maybe_quantized

  def _maybe_quantize_statistics(statistics_list):
    return _maybe_quantize_matrices_with_dtype(
        statistics_list, quantized_dtype_for_second_moment_statistics_buffers())

  def _maybe_quantize_preconditioners(statistics_list):
    return _maybe_quantize_matrices_with_dtype(
        statistics_list,
        quantized_dtype_for_second_moment_preconditioner_buffers())

  def _maybe_quantize_matrices_with_dtype(statistics_list, quantized_dtype):
    if quantized_dtype != jnp.float32:
      return ([
          QuantizedValue.from_float_value(
              s, quantized_dtype, extract_diagonal=True)
          for s in statistics_list
      ])
    else:
      return statistics_list

  def _maybe_dequantize_preconditioners(preconditioner_list):
    return _maybe_dequantize_matrices_with_dtype(
        preconditioner_list,
        quantized_dtype_for_second_moment_preconditioner_buffers())

  def _maybe_dequantize_matrices_with_dtype(statistics_list, quantized_dtype):
    if quantized_dtype != jnp.float32:
      return [s.to_float() for s in statistics_list]
    else:
      return statistics_list

  def _quantize_diagonal_statistics(diagonal_statistics):
    return QuantizedValue.from_float_value(
        diagonal_statistics, quantized_dtype_for_diagonal_statistics_buffers())

  def _quantize_momentum(momentum_statistics):
    return QuantizedValue.from_float_value(
        momentum_statistics, quantized_dtype_for_momentum_buffers())

  def sharded_init_fn(params):
    """Returns optimizer state (for PJIT mode).

    Args:
      params: the parameters that should be updated.
    """
    params_flat, treedef = jax.tree_flatten(params)
    # Find max size to pad to.
    max_size = 0
    for param in params_flat:
      preconditioner = Preconditioner(param, block_size,
                                      best_effort_shape_interpretation)
      if not _skip_preconditioning(param):
        shapes = preconditioner.shapes_for_preconditioners()
        sizes = [s[0] for s in shapes]
        max_size = max(max(sizes), max_size)

    padded_statistics = []
    padded_preconditioners = []
    local_stats_flat = []
    exponents = []
    for param in params_flat:
      preconditioner = Preconditioner(param, block_size,
                                      best_effort_shape_interpretation)
      shapes = preconditioner.shapes_for_preconditioners()
      sizes = []

      statistics = []
      preconditioners = []
      index_start = len(padded_statistics)
      if not _skip_preconditioning(param):
        sizes = [s[0] for s in shapes]
        shapes = preconditioner.shapes_for_preconditioners()
        statistics = [matrix_epsilon * jnp.eye(max_size) for s in shapes]
        preconditioners = [jnp.eye(max_size) for s in shapes]
        padded_statistics.extend(statistics)
        padded_preconditioners.extend(preconditioners)
        exponent = (
            preconditioner.exponent_for_preconditioner()
            if exponent_override == 0 else exponent_override)
        exponents.extend([exponent] * len(shapes))

      diagonal_statistics = []
      if graft_type != GraftingType.SGD:
        diagonal_statistics = jnp.zeros_like(param)
      local_stats_flat.append(
          LocalShardedParameterStats(
              _quantize_diagonal_statistics(diagonal_statistics),
              _quantize_momentum(jnp.zeros_like(param)),
              _quantize_momentum(jnp.zeros_like(param)), index_start, sizes))

    local_stats = jax.tree_unflatten(treedef, local_stats_flat)
    # Pad the statistics and preconditioner matrices to be a multiple of
    # num devices.
    # TODO(rohananil): Relax to only the size of the mesh axis where the dim
    # is split on.
    to_pad = -len(padded_statistics) % num_devices_for_pjit
    padded_statistics.extend([
        jnp.eye(max_size, dtype=padded_statistics[0].dtype)
        for _ in range(to_pad)
    ])
    padded_preconditioners.extend([
        jnp.eye(max_size, dtype=padded_statistics[0].dtype)
        for _ in range(to_pad)
    ])
    exponents.extend([1 for _ in range(to_pad)])
    global_stats = GlobalShardedParameterStats(
        jnp.stack(padded_statistics), jnp.stack(padded_preconditioners),
        jnp.stack(exponents))
    return ShampooState(
        count=jnp.zeros([], jnp.int32),
        stats=ShardedShampooStats(global_stats, local_stats))

  def _max_statistics_size_from_params(params):
    max_size = 0
    for param in params:
      param_clone = jnp.zeros(param.shape, dtype=param.dtype)
      preconditioner = Preconditioner(param_clone, block_size,
                                      best_effort_shape_interpretation)
      if not _skip_preconditioning(param):
        shapes = preconditioner.shapes_for_preconditioners()
        sizes = [s[0] for s in shapes]
        max_size = max(max(sizes), max_size)
    return max_size

  def _remove_leading_sharding_annotation(pspec):
    """Mapping from N-d to (N-1)-d, used for quantization, factoring etc."""
    # None and PSpec(None) are valid PSpecs.
    if pspec and len(pspec) > 1:
      return PartitionSpec(*pspec[1:])
    else:
      return pspec

  def sharded_init_partition_spec_fn(params, params_partition_spec,
                                     partition_spec_for_statistics):
    """Returns a parallel state tree with PartitionSpec associated with state.


    Args:
      params: A pytree with params.
      params_partition_spec: A pytree with PartitionSpec for params.
      partition_spec_for_statistics: PartitionSpec for the statistics.
    """
    # Parallel lists of spec, and params.
    param_pspec_flat, _ = jax.tree_flatten(params_partition_spec)
    params_flat, treedef = jax.tree_flatten(params)
    assert param_pspec_flat
    assert params_flat
    # Step is replicated across cores.
    # None means cores.
    local_stats_flat = []
    num_statistics = 0
    for param, param_pspec in zip(params_flat, param_pspec_flat):
      param_clone = jnp.zeros(param.shape, dtype=param.dtype)
      preconditioner = Preconditioner(param_clone, block_size,
                                      best_effort_shape_interpretation)
      shapes = preconditioner.shapes_for_preconditioners()
      sizes = []

      index_start = num_statistics
      if not _skip_preconditioning(param):
        sizes = [s[0] for s in shapes]
        shapes = preconditioner.shapes_for_preconditioners()
        num_statistics += len(shapes)

      diagonal_statistics_pspec = []
      diagonal_statistics_scale_pspec = []
      if graft_type != GraftingType.SGD:
        # Identically shaped param.
        diagonal_statistics_pspec = param_pspec
        if quantized_dtype_for_diagonal_statistics_buffers() != jnp.float32:
          diagonal_statistics_scale_pspec = _remove_leading_sharding_annotation(
              param_pspec)

      m1_pspec = param_pspec
      m2_pspec = param_pspec

      m1_scale_pspec = []
      m2_scale_pspec = []

      if quantized_dtype_for_momentum_buffers() != jnp.float32:
        m1_scale_pspec = _remove_leading_sharding_annotation(m1_pspec)
        m2_scale_pspec = _remove_leading_sharding_annotation(m2_pspec)

      local_stats_flat.append(
          LocalShardedParameterStats(
              QuantizedValue(diagonal_statistics_pspec, [],
                             diagonal_statistics_scale_pspec,
                             quantized_dtype_for_diagonal_statistics_buffers(),
                             False, list(param.shape)),
              QuantizedValue(m1_pspec, [], m1_scale_pspec,
                             quantized_dtype_for_momentum_buffers(), False,
                             list(param.shape)),
              QuantizedValue(m2_pspec, [], m2_scale_pspec,
                             quantized_dtype_for_momentum_buffers(), False,
                             list(param.shape)), index_start, sizes))

    local_stats = jax.tree_unflatten(treedef, local_stats_flat)
    global_stats = GlobalShardedParameterStats(partition_spec_for_statistics,
                                               partition_spec_for_statistics,
                                               PartitionSpec())
    count_pspec = PartitionSpec()
    return ShampooState(
        count=count_pspec, stats=ShardedShampooStats(global_stats, local_stats))

  def sharded_init_shape_and_dtype_fn(params):
    """Returns a parallel state tree with shape, dtype associated with state.


    Args:
      params: A pytree with params.
    """
    # Parallel lists of spec, and params.
    params_flat, treedef = jax.tree_flatten(params)
    assert params_flat
    # Step is replicated across cores.
    # None means cores.
    local_stats_flat = []
    num_statistics = 0
    for param in params_flat:
      param_clone = jnp.zeros(param.shape, dtype=param.dtype)
      preconditioner = Preconditioner(param_clone, block_size,
                                      best_effort_shape_interpretation)
      shapes = preconditioner.shapes_for_preconditioners()
      sizes = []

      index_start = num_statistics
      if not _skip_preconditioning(param):
        sizes = [s[0] for s in shapes]
        shapes = preconditioner.shapes_for_preconditioners()
        num_statistics += len(shapes)

      diagonal_statistics_shape_and_dtype = []
      diagonal_statistics_scale_shape_and_dtype = []
      if graft_type != GraftingType.SGD:
        diagonal_statistics_shape_and_dtype = [list(param.shape), param.dtype]
        qdtype = quantized_dtype_for_diagonal_statistics_buffers()
        if qdtype != jnp.float32:
          diagonal_statistics_shape_and_dtype = [list(param.shape), qdtype]
          diagonal_statistics_scale_shape_and_dtype = [
              list(param.shape)[1:], param.dtype
          ]

      m1_shape_and_dtype = [list(param.shape), param.dtype]
      m2_shape_and_dtype = [list(param.shape), param.dtype]

      m1_scale_shape_and_dtype = []
      m2_scale_shape_and_dtype = []

      qdtype = quantized_dtype_for_momentum_buffers()
      if qdtype != jnp.float32:
        m1_shape_and_dtype = [list(param.shape), qdtype]
        m2_shape_and_dtype = [list(param.shape), qdtype]

        m1_scale_shape_and_dtype = [list(param.shape)[1:], qdtype]
        m2_scale_shape_and_dtype = [list(param.shape)[1:], qdtype]

      local_stats_flat.append(
          LocalShardedParameterStats(
              QuantizedValue(diagonal_statistics_shape_and_dtype, [],
                             diagonal_statistics_scale_shape_and_dtype,
                             quantized_dtype_for_diagonal_statistics_buffers(),
                             False, list(param.shape)),
              QuantizedValue(m1_shape_and_dtype, [], m1_scale_shape_and_dtype,
                             quantized_dtype_for_momentum_buffers(), False,
                             list(param.shape)),
              QuantizedValue(m2_shape_and_dtype, [], m2_scale_shape_and_dtype,
                             quantized_dtype_for_momentum_buffers(), False,
                             list(param.shape)), index_start, sizes))

    local_stats = jax.tree_unflatten(treedef, local_stats_flat)
    max_statistics_size = _max_statistics_size_from_params(params_flat)
    to_pad = -num_statistics % num_devices_for_pjit
    num_statistics += to_pad
    statistics_shape = [
        num_statistics, max_statistics_size, max_statistics_size
    ]
    global_stats = GlobalShardedParameterStats([statistics_shape, jnp.float32],
                                               [statistics_shape, jnp.float32],
                                               [[num_statistics], jnp.int32])
    return ShampooState(
        count=[[], jnp.float32],
        stats=ShardedShampooStats(global_stats, local_stats))

  def sharded_update_fn(grads, state, params):
    """Transform the input gradient and update all statistics in sharded mode.

    Args:
      grads: the gradient tensors for the parameters.
      state: a named tuple containing the state of the optimizer
      params: the parameters that should be updated.

    Returns:
      A tuple containing the new parameters and the new optimizer state.
    """
    params_flat, treedef = jax.tree_flatten(params)
    grads_flat = treedef.flatten_up_to(grads)

    global_stats = state.stats.global_stats
    local_stats_flat = treedef.flatten_up_to(state.stats.local_stats)
    stats_flat = [
        _convert_to_parameter_stats(global_stats, local_stat)
        for local_stat in local_stats_flat
    ]
    new_stats_flat = jax.tree_multimap(
        lambda g, s, p: _compute_stats(g, s, p, state.count), grads_flat,
        stats_flat, params_flat)

    outputs = jax.tree_multimap(
        lambda g, s, p: _transform_grad(g, s, p, state.count), grads_flat,
        new_stats_flat, params_flat)
    updates_flat, new_stats_flat = list(zip(*outputs)) if outputs else ((), ())

    updates = jax.tree_unflatten(treedef, updates_flat)
    # Create new local_stats
    new_local_stats_flat = [
        _convert_from_parameter_stats(new_stat, local_stat)
        for new_stat, local_stat in zip(new_stats_flat, local_stats_flat)
    ]
    new_local_stats = jax.tree_unflatten(treedef, new_local_stats_flat)

    max_size = global_stats.statistics.shape[1]
    new_padded_statistics = []
    for stat in new_stats_flat:
      new_padded_statistics.extend(
          [pad_matrix(stat, max_size) for stat in stat.statistics])

    # Create global stats
    # TODO(rohananil): Preconditioner is not updated every step, so cost of
    # stack/pad can be obviated away.
    # Pad the statistics and preconditioner matrices to be a multiple of
    # num devices.
    # TODO(rohananil): Relax to only the size of the mesh axis where the dim
    # is split on.
    to_pad = -len(new_padded_statistics) % num_devices_for_pjit
    new_padded_statistics.extend([
        jnp.eye(max_size, dtype=new_padded_statistics[0].dtype)
        for _ in range(to_pad)
    ])
    new_stacked_padded_statistics = jnp.stack(new_padded_statistics)
    new_stacked_padded_statistics = pjit.with_sharding_constraint(
        new_stacked_padded_statistics, statistics_partition_spec)
    def _internal_inverse_pth_root_all():
      preconditioners, errors = _matrix_inverse_pth_root_pjit(
          new_stacked_padded_statistics, global_stats.exponents,
          statistics_partition_spec)
      return preconditioners, errors

    if preconditioning_compute_steps == 1:
      new_preconditioners, errors = _internal_inverse_pth_root_all()
    else:
      # Passing statistics instead of preconditioners as they are similarly
      # shaped tensors. Note statistics will be ignored as we are passing in
      # a large init value for error.
      preconditioners_init = new_stacked_padded_statistics
      n = new_stacked_padded_statistics.shape[0]
      errors_init = jnp.ones([n], jnp.float32) * inverse_failure_threshold
      init_state = [preconditioners_init, errors_init]
      perform_step = state.count % preconditioning_compute_steps == 0
      new_preconditioners, errors = efficient_cond(
          perform_step, _internal_inverse_pth_root_all, init_state)

    errors = errors.reshape((-1, 1, 1))
    predicate = jnp.logical_or(
        jnp.isnan(errors),
        errors >= inverse_failure_threshold).astype(new_preconditioners.dtype)
    # TODO(rohananil): Check for numerical instabilities.
    new_conditional_preconditioners = (
        predicate * global_stats.preconditioners +
        (1.0 - predicate) * new_preconditioners)
    new_global_stats = GlobalShardedParameterStats(
        new_stacked_padded_statistics, new_conditional_preconditioners,
        global_stats.exponents)
    new_shampoo_state = ShampooState(
        count=state.count + 1,
        stats=ShardedShampooStats(new_global_stats, new_local_stats))
    return updates, new_shampoo_state

  def init_fn(params):
    """Initialise the optimiser's state."""

    def _init(param):
      preconditioner = Preconditioner(param, block_size,
                                      best_effort_shape_interpretation)
      statistics = []
      preconditioners = []
      if not _skip_preconditioning(param):
        shapes = preconditioner.shapes_for_preconditioners()
        statistics = [matrix_epsilon * jnp.eye(s[0]) for s in shapes]
        preconditioners = [jnp.eye(s[0]) for s in shapes]

      diagonal_statistics = []
      if graft_type != GraftingType.SGD:
        diagonal_statistics = jnp.zeros_like(param)
      return ParameterStats(
          _quantize_diagonal_statistics(diagonal_statistics),
          _maybe_quantize_statistics(statistics),
          _maybe_quantize_preconditioners(preconditioners),
          _quantize_momentum(jnp.zeros_like(param)),
          _quantize_momentum(jnp.zeros_like(param)))
    return ShampooState(
        count=jnp.zeros([], jnp.int32), stats=jax.tree_map(_init, params))

  def _skip_preconditioning(param):
    return len(param.shape) < 1 or any(
        [s > skip_preconditioning_dim_size_gt for s in param.shape])

  def _compute_stats(grad, state, param, step):
    """Compute per-parameter statistics."""
    preconditioner = Preconditioner(param, block_size,
                                    best_effort_shape_interpretation)
    new_statistics = [[]] * len(state.statistics)
    w1 = beta2
    w2 = beta2 if beta2 == 1.0 else (1.0 - beta2)
    if not _skip_preconditioning(param):

      def compute_updated_statistics():
        new_stats = preconditioner.statistics_from_grad(grad)
        new_stats_accumulators = []
        for stat, stat_accumulator in zip(new_stats, state.statistics):
          new_stats_accumulators.append(w1 * _to_float(stat_accumulator) +
                                        w2 * stat)
        return _maybe_quantize_statistics(new_stats_accumulators)

      if statistics_compute_steps > 1:
        perform_step = step % statistics_compute_steps == 0
        init_state = state.statistics
        new_statistics = list(
            efficient_cond(perform_step, compute_updated_statistics,
                           init_state))
      else:
        new_statistics = compute_updated_statistics()
    return ParameterStats(state.diagonal_statistics, new_statistics,
                          state.preconditioners, state.diagonal_momentum,
                          state.momentum)

  def _matrix_inverse_pth_root_vmap(xs, ps):
    mi_pth_root = functools.partial(
        matrix_inverse_pth_root,
        ridge_epsilon=matrix_epsilon,
        precision=precision)
    return jax.vmap(mi_pth_root)(xs, ps)

  def _quantized_matrix_inverse_pth_root_vmap(qxs, qds, qbs, ps):

    def _quantized_to_float(qx, qd, qb):
      qv = QuantizedValue(qx, qd, qb, qx.dtype, True, list(qx.shape))
      return qv.to_float()

    def matrix_inverse_pth_root_wrapper(qx, qd, qb, p):
      v = _quantized_to_float(qx, qd, qb)
      preconditioner, error = matrix_inverse_pth_root(
          v, p, ridge_epsilon=matrix_epsilon, precision=precision)
      qp = QuantizedValue.from_float_value(preconditioner, qx.dtype, True)
      return qp.quantized, qp.diagonal, qp.bucket_size, error

    return jax.vmap(matrix_inverse_pth_root_wrapper)(qxs, qds, qbs, ps)

  def _matrix_inverse_pth_root_pjit(xs, ps, statistics_partition_spec=None):
    # Partition the concatenated statistics matrix across all cores.
    pspec_for_partition = preconditioner_partition_spec
    partitioned_xs = pjit.with_sharding_constraint(xs, pspec_for_partition)
    partitioned_ps = pjit.with_sharding_constraint(
        ps, pjit.PartitionSpec(preconditioner_partition_spec[0]))
    # Run matrix inverse pth root on each shard.
    partitioned_preconditioners, partitioned_errors = (
        _matrix_inverse_pth_root_vmap(partitioned_xs, partitioned_ps))
    # Reshard output to have the same PSpec as input. This is required to avoid
    # vmap seeing the full set of statistics.
    partitioned_preconditioners = pjit.with_sharding_constraint(
        partitioned_preconditioners, pspec_for_partition)
    # Recombine the outputs at each core.
    preconditioners = pjit.with_sharding_constraint(partitioned_preconditioners,
                                                    statistics_partition_spec)
    errors = pjit.with_sharding_constraint(partitioned_errors,
                                           pjit.PartitionSpec())
    return preconditioners, errors

  def _pmap_compute_preconditioners(states, step, statistics,
                                    num_statistics_per_state, original_shapes,
                                    exponents, max_size, prev_preconditioners):
    """Computes preconditioners for given statistics in states in PMAP mode.

    Args:
      states: A list of optimizer states.
      step: Current step number
      statistics: A list of statistics for all variables (for every dim)
      num_statistics_per_state: Number of statistis per state to reconstruct
        output states.
      original_shapes: A list of shapes of the statistics.
      exponents: Exponent power to use for inverse-pth roots.
      max_size: Maximum dim of the statistics to pad.
      prev_preconditioners: Previously available preconditioner.

    Returns:
      New optimizer states after computing the preconditioner.
    """
    num_devices = lax.psum(1, batch_axis_name)
    num_statistics = len(statistics)
    # Pad statistics and exponents to next multiple of num_devices.
    packed_statistics = [pad_matrix(stat, max_size) for stat in statistics]
    to_pad = -num_statistics % num_devices
    packed_statistics.extend([
        jnp.eye(max_size, dtype=packed_statistics[0].dtype)
        for _ in range(to_pad)
    ])
    exponents.extend([1 for _ in range(to_pad)])

    if not packed_statistics:
      return states

    all_statistics = batch(packed_statistics, num_devices)
    all_exponents = batch(exponents, num_devices)

    def _internal_inverse_pth_root_all():
      current_replica = lax.axis_index(batch_axis_name)
      preconditioners, errors = _matrix_inverse_pth_root_vmap(
          all_statistics[current_replica], all_exponents[current_replica])
      preconditioners = jax.lax.all_gather(preconditioners, batch_axis_name)
      errors = jax.lax.all_gather(errors, batch_axis_name)
      preconditioners_flat = unbatch(preconditioners)
      errors_flat = unbatch(errors)
      return preconditioners_flat, errors_flat

    if preconditioning_compute_steps == 1:
      preconditioners_flat, errors_flat = _internal_inverse_pth_root_all()
    else:
      # Passing statistics instead of preconditioners as they are similarly
      # shaped tensors. Note statistics will be ignored as we are passing in
      # a large init value for error.
      preconditioners_init = packed_statistics
      errors_init = ([inverse_failure_threshold] * len(packed_statistics))
      init_state = [preconditioners_init, errors_init]
      perform_step = step % preconditioning_compute_steps == 0
      preconditioners_flat, errors_flat = efficient_cond(
          perform_step, _internal_inverse_pth_root_all, init_state)

    def _skip(error):
      condition = jnp.logical_or(
          jnp.isnan(error), error >= inverse_failure_threshold)
      return condition.astype(error.dtype)

    def _select_preconditioner(error, new_p, old_p):
      return lax.cond(
          _skip(error), lambda _: old_p, lambda _: new_p, operand=None)

    new_preconditioners_flat = []
    for p, shape, prev_p, error in zip(preconditioners_flat, original_shapes,
                                       prev_preconditioners, errors_flat):
      new_preconditioners_flat.append(
          _select_preconditioner(error, p[:shape[0], :shape[1]], prev_p))

    assert len(states) == len(num_statistics_per_state)
    assert len(new_preconditioners_flat) == num_statistics

    # Add back empty preconditioners so we that we can set the optimizer state.
    preconditioners_for_states = []
    idx = 0
    for num_statistics, state in zip(num_statistics_per_state, states):
      if num_statistics == 0:
        preconditioners_for_states.append([])
      else:
        preconditioners_for_state = new_preconditioners_flat[idx:idx +
                                                             num_statistics]
        assert len(state.statistics) == len(preconditioners_for_state)
        preconditioners_for_states.append(preconditioners_for_state)
        idx += num_statistics
    new_states = []
    for state, new_preconditioners in zip(states, preconditioners_for_states):
      new_states.append(
          ParameterStats(state.diagonal_statistics, state.statistics,
                         new_preconditioners, state.diagonal_momentum,
                         state.momentum))

    return new_states

  def _pmap_quantized_compute_preconditioners(states, step, statistics,
                                              num_statistics_per_state,
                                              original_shapes, exponents,
                                              max_size, prev_preconditioners):
    """Computes preconditioners for given statistics in states in PMAP mode.

    For quantization, each statistic is represented by three values:
      quantized matrix, diagonal, and bucket sizes, we run inverse pth-roots
      without ever recreating the original matrix in f32.

    Args:
      states: A list of optimizer states.
      step: Current step number
      statistics: A list of statistics for all variables (for every dim)
      num_statistics_per_state: Number of statistis per state to reconstruct
        output states.
      original_shapes: A list of shapes of the statistics.
      exponents: Exponent power to use for inverse-pth roots.
      max_size: Maximum dim of the statistics to pad.
      prev_preconditioners: Previously available preconditioner.

    Returns:
      New optimizer states after computing the preconditioner.
    """
    num_devices = lax.psum(1, batch_axis_name)
    num_statistics = len(statistics)
    quantized_dtype = quantized_dtype_for_second_moment_statistics_buffers()
    # Complexity here is around: shapes needing be statically shaped,
    # our custom quantization type requires a different type of packing.

    # Parallel tensors:
    # quantized [dxd]
    # diagonals [d] f32
    # bucket_sizes [d] f32
    packed_quantized_statistics = [
        pad_matrix(stat.quantized, max_size) for stat in statistics
    ]
    packed_quantized_diagonals = [
        pad_vector(stat.diagonal, max_size) for stat in statistics
    ]
    packed_quantized_bucket_sizes = [
        pad_vector(stat.bucket_size, max_size) for stat in statistics
    ]

    to_pad = -num_statistics % num_devices
    padded_eye = jnp.eye(max_size, dtype=jnp.float32)
    quantized_eye = QuantizedValue.from_float_value(padded_eye, quantized_dtype,
                                                    True)
    packed_quantized_statistics.extend(
        [quantized_eye.quantized for _ in range(to_pad)])
    packed_quantized_diagonals.extend(
        [quantized_eye.diagonal for _ in range(to_pad)])
    packed_quantized_bucket_sizes.extend(
        [quantized_eye.bucket_size for _ in range(to_pad)])
    exponents.extend([1 for _ in range(to_pad)])

    if not packed_quantized_statistics:
      return states

    all_quantized_statistics = batch(packed_quantized_statistics, num_devices)
    all_quantized_diagonals = batch(packed_quantized_diagonals, num_devices)
    all_quantized_bucket_sizes = batch(packed_quantized_bucket_sizes,
                                       num_devices)
    all_exponents = batch(exponents, num_devices)

    def _internal_inverse_pth_root_all():
      current_replica = lax.axis_index(batch_axis_name)
      (quantized_preconditioners, quantized_diagonals, quantized_bucket_sizes,
       errors) = (
           _quantized_matrix_inverse_pth_root_vmap(
               all_quantized_statistics[current_replica],
               all_quantized_diagonals[current_replica],
               all_quantized_bucket_sizes[current_replica],
               all_exponents[current_replica]))
      quantized_preconditioners = jax.lax.all_gather(quantized_preconditioners,
                                                     batch_axis_name)
      quantized_diagonals = jax.lax.all_gather(quantized_diagonals,
                                               batch_axis_name)
      quantized_bucket_sizes = jax.lax.all_gather(quantized_bucket_sizes,
                                                  batch_axis_name)
      errors = jax.lax.all_gather(errors, batch_axis_name)
      quantized_preconditioners_flat = unbatch(quantized_preconditioners)
      quantized_diagonals_flat = unbatch(quantized_diagonals)
      quantized_bucket_sizes_flat = unbatch(quantized_bucket_sizes)
      errors_flat = unbatch(errors)
      return (quantized_preconditioners_flat, quantized_diagonals_flat,
              quantized_bucket_sizes_flat, errors_flat)

    if preconditioning_compute_steps == 1:
      (quantized_preconditioners_flat, quantized_diagonals_flat,
       quantized_bucket_sizes_flat, errors_flat) = (
           _internal_inverse_pth_root_all())
    else:
      # Passing statistics instead of preconditioners as they are similarly
      # shaped tensors. Note statistics will be ignored as we are passing in
      # a large init value for error.
      quantized_preconditioners_init = packed_quantized_statistics
      quantized_diagonals_init = packed_quantized_diagonals
      quantized_bucket_sizes_init = packed_quantized_bucket_sizes
      errors_init = ([inverse_failure_threshold] *
                     len(quantized_preconditioners_init))
      init_state = [
          quantized_preconditioners_init, quantized_diagonals_init,
          quantized_bucket_sizes_init, errors_init
      ]
      perform_step = step % preconditioning_compute_steps == 0
      (quantized_preconditioners_flat, quantized_diagonals_flat,
       quantized_bucket_sizes_flat, errors_flat) = (
           efficient_cond(perform_step, _internal_inverse_pth_root_all,
                          init_state))

    def _skip(error):
      condition = jnp.logical_or(
          jnp.isnan(error), error >= inverse_failure_threshold)
      return condition.astype(error.dtype)

    def _select_preconditioner(error, new_p, old_p):
      return lax.cond(
          _skip(error), lambda _: old_p, lambda _: new_p, operand=None)

    new_quantized_preconditioners_flat = []
    new_quantized_diagonals_flat = []
    new_quantized_bucket_sizes_flat = []
    for p, d, b, shape, prev_p, error in zip(quantized_preconditioners_flat,
                                             quantized_diagonals_flat,
                                             quantized_bucket_sizes_flat,
                                             original_shapes,
                                             prev_preconditioners, errors_flat):
      new_quantized_preconditioners_flat.append(
          _select_preconditioner(error, p[:shape[0], :shape[1]],
                                 prev_p.quantized))
      new_quantized_diagonals_flat.append(
          _select_preconditioner(error, d[:shape[0]], prev_p.diagonal))
      new_quantized_bucket_sizes_flat.append(
          _select_preconditioner(error, b[:shape[0]], prev_p.bucket_size))

    assert len(states) == len(num_statistics_per_state)
    assert len(new_quantized_preconditioners_flat) == num_statistics
    assert len(new_quantized_diagonals_flat) == num_statistics
    assert len(new_quantized_bucket_sizes_flat) == num_statistics

    # Add back empty preconditioners so we that we can set the optimizer state.
    preconditioners_for_states = []
    idx = 0
    for num_statistics, state in zip(num_statistics_per_state, states):
      if num_statistics == 0:
        preconditioners_for_states.append([])
      else:
        quantized_preconditioners_for_state = new_quantized_preconditioners_flat[
            idx:idx + num_statistics]
        quantized_diagonals_for_state = new_quantized_diagonals_flat[
            idx:idx + num_statistics]
        quantized_bucket_sizes_for_state = new_quantized_bucket_sizes_flat[
            idx:idx + num_statistics]

        assert len(state.statistics) == len(quantized_preconditioners_for_state)
        assert len(state.statistics) == len(quantized_diagonals_for_state)
        assert len(state.statistics) == len(quantized_bucket_sizes_for_state)

        quantized_preconditioners = []
        for qv, qd, qb in zip(quantized_preconditioners_for_state,
                              quantized_diagonals_for_state,
                              quantized_bucket_sizes_for_state):
          quantized_preconditioners.append(
              QuantizedValue(qv, qd, qb, qv.dtype, True, list(qv.shape)))
        preconditioners_for_states.append(quantized_preconditioners)
        idx += num_statistics
    new_states = []
    for state, new_preconditioners in zip(states, preconditioners_for_states):
      new_states.append(
          ParameterStats(state.diagonal_statistics, state.statistics,
                         new_preconditioners, state.diagonal_momentum,
                         state.momentum))

    return new_states

  def _pjit_compute_preconditioners(states, step, statistics,
                                    num_statistics_per_state, original_shapes,
                                    exponents, max_size, prev_preconditioners):
    """Computes preconditioners for given statistics in states in PJIT mode.

    Args:
      states: A list of optimizer states.
      step: Current step number
      statistics: A list of statistics for all variables (for every dim)
      num_statistics_per_state: Number of statistis per state to reconstruct
        output states.
      original_shapes: A list of shapes of the statistics.
      exponents: Exponent power to use for inverse-pth roots.
      max_size: Maximum dim of the statistics to pad.
      prev_preconditioners: Previously available preconditioner.

    Returns:
      New optimizer states after computing the preconditioner.
    """
    num_statistics = len(statistics)
    to_pad = -num_statistics % num_devices_for_pjit
    padded_statistics = [pad_matrix(stat, max_size) for stat in statistics]
    padded_statistics.extend([
        jnp.eye(max_size, dtype=padded_statistics[0].dtype)
        for _ in range(to_pad)
    ])
    exponents.extend([1 for _ in range(to_pad)])
    all_statistics = jnp.stack(padded_statistics)
    all_exponents = jnp.stack(exponents)

    def _internal_inverse_pth_root_all():
      preconditioners, errors = _matrix_inverse_pth_root_pjit(
          all_statistics, all_exponents)
      b1 = preconditioners.shape[0]

      def split(batched_values):
        return [
            jnp.squeeze(v)
            for v in jnp.split(batched_values, indices_or_sections=b1, axis=0)
        ]

      return split(preconditioners), split(errors)

    if preconditioning_compute_steps == 1:
      preconditioners_flat, errors_flat = _internal_inverse_pth_root_all()
    else:
      # Passing statistics instead of preconditioners as they are similarly
      # shaped tensors. Note statistics will be ignored as we are passing in
      # a large init value for error.
      preconditioners_init = padded_statistics
      errors_init = [inverse_failure_threshold] * len(padded_statistics)
      init_state = [preconditioners_init, errors_init]
      perform_step = step % preconditioning_compute_steps == 0
      preconditioners_flat, errors_flat = efficient_cond(
          perform_step, _internal_inverse_pth_root_all, init_state)

    def _skip(error):
      condition = jnp.logical_or(
          jnp.isnan(error), error >= inverse_failure_threshold)
      return condition.astype(error.dtype)

    def _select_preconditioner(error, new_p, old_p):
      return lax.cond(
          _skip(error), lambda _: old_p, lambda _: new_p, operand=None)

    new_preconditioners_flat = []
    for p, shape, prev_p, error in zip(preconditioners_flat, original_shapes,
                                       prev_preconditioners, errors_flat):
      new_preconditioners_flat.append(
          _select_preconditioner(error, p[:shape[0], :shape[1]], prev_p))

    assert len(states) == len(num_statistics_per_state)
    assert len(new_preconditioners_flat) == num_statistics

    # Add back empty preconditioners so we that we can set the optimizer state.
    preconditioners_for_states = []
    idx = 0
    for num_statistics, state in zip(num_statistics_per_state, states):
      if num_statistics == 0:
        preconditioners_for_states.append([])
      else:
        preconditioners_for_state = new_preconditioners_flat[idx:idx +
                                                             num_statistics]
        assert len(state.statistics) == len(preconditioners_for_state)
        preconditioners_for_states.append(preconditioners_for_state)
        idx += num_statistics
    new_states = []
    for state, new_preconditioners in zip(states, preconditioners_for_states):
      new_states.append(
          ParameterStats(state.diagonal_statistics, state.statistics,
                         new_preconditioners, state.diagonal_momentum,
                         state.momentum))

    return new_states

  def _compute_preconditioners(states, params, step):
    """Computes preconditioners for given statistics in states.

    Args:
      states: A list of optimizer states.
      params: A list of params.
      step: Current step number

    Returns:
      New optimizer states after computing the preconditioner.
    """
    statistics = []
    num_statistics_per_state = []
    original_shapes = []
    exponents = []
    max_size = 0
    prev_preconditioners = []

    for state, param in zip(states, params):
      num_statistics = len(state.statistics)
      num_statistics_per_state.append(num_statistics)
      original_shapes_for_state = []
      if num_statistics > 0:
        preconditioner = Preconditioner(param, block_size,
                                        best_effort_shape_interpretation)
        for statistic in state.statistics:
          exponents.append(preconditioner.exponent_for_preconditioner(
          ) if exponent_override == 0 else exponent_override)
          original_shapes_for_state.append(statistic.shape)
          max_size = max(max_size, statistic.shape[0])

        statistics.extend(state.statistics)
        prev_preconditioners.extend(state.preconditioners)
        original_shapes.extend(original_shapes_for_state)

    if batch_axis_name:
      # Quantization is only enabled if batch_axis_name is not set.
      quantized_dtype = quantized_dtype_for_second_moment_statistics_buffers()

      if quantized_dtype == jnp.float32:
        return _pmap_compute_preconditioners(states, step, statistics,
                                             num_statistics_per_state,
                                             original_shapes, exponents,
                                             max_size, prev_preconditioners)
      else:
        return _pmap_quantized_compute_preconditioners(
            states, step, statistics, num_statistics_per_state, original_shapes,
            exponents, max_size, prev_preconditioners)

    else:
      return _pjit_compute_preconditioners(states, step, statistics,
                                           num_statistics_per_state,
                                           original_shapes, exponents, max_size,
                                           prev_preconditioners)

  def _transform_grad(grad, state, param, step):
    """Transform per-parameter gradients."""
    preconditioner = Preconditioner(param, block_size,
                                    best_effort_shape_interpretation)
    sgd_update = grad
    new_diagonal_statistics = state.diagonal_statistics.to_float()
    if graft_type == GraftingType.ADAGRAD:
      new_diagonal_statistics = state.diagonal_statistics.to_float(
      ) + jnp.square(grad)
      adagrad_update = grad / (
          jnp.sqrt(new_diagonal_statistics) + diagonal_epsilon)
      grafting_update = adagrad_update
    elif (graft_type == GraftingType.RMSPROP or
          graft_type == GraftingType.RMSPROP_NORMALIZED):

      scaled_grad = grad
      if graft_type == GraftingType.RMSPROP_NORMALIZED:
        scaled_grad = grad / jnp.linalg.norm(grad)

      w1 = beta2
      w2 = beta2 if beta2 == 1.0 else (1.0 - beta2)

      new_diagonal_statistics = (
          w1 * state.diagonal_statistics.to_float() +
          w2 * jnp.square(scaled_grad))
      rmsprop_update = scaled_grad / (
          jnp.sqrt(new_diagonal_statistics) + diagonal_epsilon)

      if clip_by_scaled_gradient_norm:
        scaled_grad_norm = jnp.linalg.norm(rmsprop_update) / (
            jnp.sqrt(float(rmsprop_update.size)))
        clipping_denom = jnp.maximum(
            1., scaled_grad_norm / clip_by_scaled_gradient_norm)
        rmsprop_update /= clipping_denom

      grafting_update = rmsprop_update
    else:
      grafting_update = sgd_update

    precond_grad = grad
    if not _skip_preconditioning(param):
      precond_grad = preconditioner.preconditioned_grad(
          precond_grad,
          _maybe_dequantize_preconditioners(state.preconditioners))
    else:
      precond_grad = grafting_update

    grafting_update_norm = jnp.linalg.norm(grafting_update)
    precond_grad_norm = jnp.linalg.norm(precond_grad)

    multiplier = (grafting_update_norm / (precond_grad_norm + 1e-16))
    shampoo_update = precond_grad * multiplier

    shampoo_update_with_wd = shampoo_update
    grafting_update_with_wd = grafting_update
    if weight_decay != 0:
      shampoo_update_with_wd = shampoo_update + weight_decay * param
      grafting_update_with_wd = grafting_update + weight_decay * param

    w = (1.0 - beta1) if moving_average_for_momentum else 1.0
    shampoo_update_with_wd_momentum = (
        state.momentum.to_float() * beta1 + w * shampoo_update_with_wd)
    grafting_update_with_wd_momentum = (
        state.diagonal_momentum.to_float() * beta1 +
        w * grafting_update_with_wd)

    run_shampoo = (step >= start_preconditioning_step).astype(
        grafting_update_with_wd_momentum.dtype)

    momentum_update = (
        run_shampoo * shampoo_update_with_wd_momentum +
        (1.0 - run_shampoo) * grafting_update_with_wd_momentum)

    wd_update = (
        run_shampoo * shampoo_update_with_wd +
        (1.0 - run_shampoo) * grafting_update_with_wd)

    if nesterov:
      momentum_update = w * wd_update + beta1 * momentum_update

    lr = learning_rate
    if callable(learning_rate):
      lr = learning_rate(step)
    transformed_update = -1.0 * lr * momentum_update

    param_stats = ParameterStats(
        _quantize_diagonal_statistics(new_diagonal_statistics),
        state.statistics, state.preconditioners,
        _quantize_momentum(grafting_update_with_wd_momentum),
        _quantize_momentum(shampoo_update_with_wd_momentum))
    return transformed_update, param_stats

  def update_fn(grads, state, params):
    """Transform the input gradient and update all statistics.

    Args:
      grads: the gradient tensors for the parameters.
      state: a named tuple containing the state of the optimizer
      params: the parameters that should be updated.

    Returns:
      A tuple containing the new parameters and the new optimizer state.
    """
    params_flat, treedef = jax.tree_flatten(params)
    stats_flat = treedef.flatten_up_to(state.stats)
    grads_flat = treedef.flatten_up_to(grads)

    new_stats_flat = jax.tree_multimap(
        lambda g, s, p: _compute_stats(g, s, p, state.count), grads_flat,
        stats_flat, params_flat)
    new_stats_flat = _compute_preconditioners(new_stats_flat, params_flat,
                                              state.count)

    outputs = jax.tree_multimap(
        lambda g, s, p: _transform_grad(g, s, p, state.count), grads_flat,
        new_stats_flat, params_flat)
    updates_flat, new_stats_flat = list(zip(*outputs)) if outputs else ((), ())

    updates = jax.tree_unflatten(treedef, updates_flat)
    new_stats = jax.tree_unflatten(treedef, new_stats_flat)

    new_state = ShampooState(
        count=state.count+1, stats=new_stats)
    return updates, new_state

  if shard_optimizer_states:
    # Hijacks the init_fn signature so we can return an OptState with
    # appropriate init_fns.
    def _init_fns(unused_params):
      return InitFnState(
          init_fn=sharded_init_fn,
          pspec_fn=sharded_init_partition_spec_fn,
          shape_and_dtype_fn=sharded_init_shape_and_dtype_fn)

    return optax.GradientTransformation(_init_fns, sharded_update_fn)
  else:
    return optax.GradientTransformation(init_fn, update_fn)