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# SPDX-License-Identifier: Apache-2.0
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
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#
# http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
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from functools import partial
from typing import TYPE_CHECKING, Callable, override
if TYPE_CHECKING:
from typing import overload
import torch
from absl import logging
from torch import optim
from torch.optim.optimizer import ParamsT
from emerging_optimizers import mixin as opt_mixin
from emerging_optimizers import registry, utils
from emerging_optimizers.scalar_optimizers import update_functions
from emerging_optimizers.soap import soap_utils
from emerging_optimizers.utils import FP32MatmulPrecT
__all__ = [
"SOAP",
"precondition",
"init_kronecker_factors",
"update_kronecker_factors",
"update_eigenbasis_and_exp_avgs",
]
[docs]
@registry.register_optimizer("soap")
class SOAP(opt_mixin.WeightDecayMixin, optim.Optimizer):
"""Implements a variant of SOAP (ShampoO with Adam in the Preconditioner eigenbasis) algorithm.
SOAP (https://arxiv.org/abs/2409.11321) is a preconditioned optimizer that combines the benefits of Shampoo's
non-diagonal preconditioning with Adam's adaptive learning rates. It uses
gradient correlation matrix eigenbasis-based preconditioning to adapt to the local geometry of the
optimization landscape.
Args:
params: Iterable of parameters to optimize or dicts defining parameter groups
lr: The learning rate to use
betas: Inner Adam's betas parameters (b1, b2)
shampoo_beta: Beta for the kronecker factor matrices (L and R in paper) moving average
instead of betas[1] if >= 0
eps: Inner Adam's epsilon for numerical stability
weight_decay: Weight decay coefficient
weight_decay_method: Method to apply weight decay, see :class:`~emerging_optimizers.mixin.WeightDecayMixin`
for more details.
nesterov: uses Nesterov momentum in Adam (https://cs229.stanford.edu/proj2015/054_report.pdf,
https://openreview.net/forum?id=OM0jvwB8jIp57ZJjtNEZ)
precondition_frequency: How often to update the preconditioner. Can be an integer for fixed frequency
or a callable function that takes the current step as input and returns the frequency.
adam_warmup_steps: How many steps to skip preconditioning in the beginning (i.e. use standard AdamW updates)
correct_bias: Whether to use bias correction in Inner Adam and Kronecker factor matrices EMA
fp32_matmul_prec: Precision of the matmul operations in optimizer states GEMM operations
use_eigh: Whether to use full symmetric eigendecomposition (eigh) to compute the eigenbasis.
If False, use orthogonal iteration to compute the eigenbasis.
qr_fp32_matmul_prec: Precision of the matmul operations in QR decomposition.
use_adaptive_criteria: Whether to use criteria to determine if eigenbasis update is needed
adaptive_update_tolerance: Tolerance threshold for the update criteria.
Only used if use_adaptive_criteria is True.
power_iter_steps: Number of power iteration steps to perform before QR decomposition.
More steps can lead to better convergence but increased computation time.
max_update_rms: Clip the update RMS to this value (0 means no clipping).
use_kl_shampoo: Whether to use KL-Shampoo correction.
correct_shampoo_beta_bias: Whether to correct shampoo beta bias. Decoupled it from correct_bias for
testability because reference implementation of Soap doesn't bias correct shampoo beta.
"""
def __init__(
self,
params: ParamsT,
lr: float,
betas: tuple[float, float] = (0.9, 0.95),
shampoo_beta: float = 0.95,
eps: float = 1e-8,
weight_decay: float = 0.01,
*,
weight_decay_method: opt_mixin.WeightDecayT = "decoupled",
nesterov: bool = False,
precondition_frequency: int | Callable[[int], int] = 1,
adam_warmup_steps: int = 0,
correct_bias: bool = True,
fp32_matmul_prec: FP32MatmulPrecT = "high",
use_eigh: bool = False,
qr_fp32_matmul_prec: FP32MatmulPrecT = "high",
use_adaptive_criteria: bool = False,
adaptive_update_tolerance: float = 1e-7,
power_iter_steps: int = 1,
max_update_rms: float = 0.0,
use_kl_shampoo: bool = False,
correct_shampoo_beta_bias: bool | None = None,
) -> None:
self.precondition_frequency = precondition_frequency
self.adam_warmup_steps = adam_warmup_steps
self.nesterov = nesterov
self.correct_bias = correct_bias
self.weight_decay_method = weight_decay_method
self.fp32_matmul_prec = fp32_matmul_prec
self.use_eigh = use_eigh
self.qr_fp32_matmul_prec = qr_fp32_matmul_prec
self.use_adaptive_criteria = use_adaptive_criteria
self.adaptive_update_tolerance = adaptive_update_tolerance
self.power_iter_steps = power_iter_steps
self.max_update_rms = max_update_rms
self.use_kl_shampoo = use_kl_shampoo
if correct_shampoo_beta_bias is not None:
self.correct_shampoo_beta_bias = correct_shampoo_beta_bias
else:
self.correct_shampoo_beta_bias = correct_bias
defaults = {
"lr": lr,
"betas": betas,
"shampoo_beta": shampoo_beta,
"eps": eps,
"weight_decay": weight_decay,
}
super().__init__(params, defaults)
@torch.no_grad() # type: ignore[misc]
def _init_group(
self,
group: dict,
skip_non_grad_params: bool = True,
) -> None:
"""Performs lazy state initialization for parameters with gradients.
Args:
group: Parameter group dictionary.
skip_non_grad_params: Whether to skip parameters with no gradients.
Raises:
TypeError: If the parameter is not a 2D tensor.
"""
for p in group["params"]:
if skip_non_grad_params and p.grad is None:
continue
if p.dim() != 2:
raise TypeError("SOAP is only supported for 2D tensors")
state = self.state[p]
if len(state) == 0:
state["step"] = 0
state["exp_avg"] = torch.zeros_like(p.data, dtype=torch.float32)
state["exp_avg_sq"] = torch.zeros_like(p.data, dtype=torch.float32)
# Use shape of p instead of grad for initialization because of the introduction of skip_non_grad_params
# for megatron-lm distributed checkpointing use. _init_group can be called without grad.
state["L"], state["R"] = init_kronecker_factors(p.shape, device=p.device)
state["Q_L"] = torch.eye(p.shape[0], device=p.device)
state["Q_R"] = torch.eye(p.shape[1], device=p.device)
if TYPE_CHECKING:
@overload
def step(self, closure: None = ...) -> None: ...
@overload
def step(self, closure: Callable[[], float]) -> float: ...
[docs]
@torch.no_grad() # type: ignore[misc]
@override
def step(self, closure: Callable[[], float] | None = None) -> float | None:
"""Performs a single optimization step.
Args:
closure: A closure that reevaluates the model and returns the loss.
"""
if closure is None:
loss = None
else:
loss = closure()
for group in self.param_groups:
self._init_group(group)
for p in group["params"]:
if p.grad is None:
continue
grad = p.grad.to(torch.float32)
state = self.state[p]
# NOTE: The upstream PyTorch implementations increment the step counter in the middle of the loop
# to be used in bias correction. But this is confusing and error prone if anything else needs to use
# the step counter.
# We decided to follow Python and C convention to increment the step counter at the end of the loop.
# An explicitly named 1-based iteration/step counter is created for bias correction and other terms
# in the math equation that needs 1-based iteration count.
curr_iter_1_based = state["step"] + 1
# Define kronecker_factor_update_fn based on whether to use KL-Shampoo here
# because it needs access to eigenbasis_list and group
kronecker_factor_list = [state["L"], state["R"]]
eigenbasis_list = [state["Q_L"], state["Q_R"]]
if not self.use_kl_shampoo:
kronecker_factor_update_fn = update_kronecker_factors
else:
kronecker_factor_update_fn = partial(
update_kronecker_factors_kl_shampoo,
eigenbasis_list=eigenbasis_list,
eps=group["eps"],
)
shampoo_beta = group["shampoo_beta"]
if self.correct_shampoo_beta_bias:
shampoo_beta = 1 - (1 - shampoo_beta) / (1 - shampoo_beta**curr_iter_1_based)
torch.cuda.nvtx.range_push("update_kronecker_factors")
with utils.fp32_matmul_precision(self.fp32_matmul_prec):
kronecker_factor_update_fn(
kronecker_factor_list=kronecker_factor_list, grad=grad, shampoo_beta=shampoo_beta
)
torch.cuda.nvtx.range_pop()
# After the adam_warmup_steps are completed , update eigenbases at precondition_frequency steps
torch.cuda.nvtx.range_push("Update eigen basis")
if _is_eigenbasis_update_step(
state["step"],
self.adam_warmup_steps,
self.precondition_frequency,
):
# Always use eigh for the first eigenbasis update
use_eigh = self.use_eigh if state["step"] != self.adam_warmup_steps else True
# Skip eigenbasis update if use_adaptive_criteria is True and all eigenbases meet the criteria.
# Never skip the first eigenbasis update (step == adam_warmup_steps) since QL/QR are still identity.
skip_update = (
self.use_adaptive_criteria
and state["step"] > self.adam_warmup_steps
and soap_utils.all_eigenbases_met_criteria(
kronecker_factor_list, eigenbasis_list, self.adaptive_update_tolerance
)
)
if not skip_update:
with utils.fp32_matmul_precision(self.qr_fp32_matmul_prec):
updated_eigenbasis_list, exp_avg, exp_avg_sq = update_eigenbasis_and_exp_avgs(
kronecker_factor_list=kronecker_factor_list,
eigenbasis_list=eigenbasis_list,
exp_avg_sq=state["exp_avg_sq"],
exp_avg=state["exp_avg"],
use_eigh=use_eigh,
power_iter_steps=self.power_iter_steps,
)
state["Q_L"], state["Q_R"] = updated_eigenbasis_list
# rebind local ref so precondition() below uses the updated Q
eigenbasis_list = updated_eigenbasis_list
state["exp_avg"] = exp_avg
state["exp_avg_sq"] = exp_avg_sq
torch.cuda.nvtx.range_pop()
self._apply_weight_decay_inplace(
p,
grad,
group["lr"],
group["weight_decay"],
)
grad_projected = grad
# Project gradients to the eigenbases of Shampoo's preconditioner
torch.cuda.nvtx.range_push("precondition")
if state["step"] >= self.adam_warmup_steps:
with utils.fp32_matmul_precision(self.fp32_matmul_prec):
grad_projected = precondition(
grad,
eigenbasis_list=eigenbasis_list,
dims=[[0], [0]],
)
torch.cuda.nvtx.range_pop()
# Calculate the Adam update for the projected gradient tensor
adam_update = update_functions.calculate_adam_update(
grad_projected,
state["exp_avg"],
state["exp_avg_sq"],
group["betas"],
self.correct_bias,
self.nesterov,
curr_iter_1_based, # 1-based iteration index is used for bias correction
group["eps"],
)
# Projecting back the preconditioned (by ADAM) exponential moving average of gradients
torch.cuda.nvtx.range_push("precondition")
if state["step"] >= self.adam_warmup_steps:
with utils.fp32_matmul_precision(self.fp32_matmul_prec):
precond_update = precondition(
adam_update,
eigenbasis_list=eigenbasis_list,
dims=[[0], [1]],
)
else:
precond_update = adam_update
torch.cuda.nvtx.range_pop()
_clip_update_rms_in_place(precond_update, self.max_update_rms)
p.add_(precond_update, alpha=-group["lr"])
state["step"] += 1
return loss
[docs]
@torch.no_grad() # type: ignore[misc]
def init_kronecker_factors(
grad_shape: torch.Size,
device: torch.device | None = None,
) -> tuple[torch.Tensor, torch.Tensor]:
"""Initializes the kronecker factor matrices for the SOAP optimizer.
This function creates the initial Kronecker factor matrices (L and R) used for
preconditioning. It creates a square kronecker factor matrix for each dimension
of the 2D gradient shape.
Note:
The Kronecker factors are always initialized to float32 (unless default precision is set otherwise) as its
accumulation and decomposition are not safe in lower precisions.
Args:
grad_shape: Shape of the gradient tensor. Must be 2D.
Determines the size of the kronecker factor matrices.
device: Device on which to create the kronecker factor matrices.
Returns:
Tuple of kronecker factor matrices (L and R in paper).
Example:
>>> # For a 2D tensor (weight matrix)
>>> grad_shape = torch.Size([10, 20])
>>> precond_2d = init_kronecker_factors(grad_shape)
>>> print(len(precond_2d)) # 2
>>> print(precond_2d[0].shape) # (10, 10)
>>> print(precond_2d[1].shape) # (20, 20)
"""
if len(grad_shape) != 2:
raise TypeError("init_kronecker_factors is only supported for 2D tensors")
# Create a square kronecker factor matrix for each dimension
L = torch.zeros(grad_shape[0], grad_shape[0], device=device)
R = torch.zeros(grad_shape[1], grad_shape[1], device=device)
return L, R
[docs]
@torch.no_grad() # type: ignore[misc]
def update_kronecker_factors(
kronecker_factor_list: list[torch.Tensor],
grad: torch.Tensor,
shampoo_beta: float,
) -> None:
"""Updates the preconditioner matrices using gradient outer products.
This function updates the Kronecker factor matrices (L and R) used for preconditioning
by computing and accumulating gradient outer products. kronecker_factor_list is updated in place.
Args:
kronecker_factor_list: List of preconditioner matrices (L and R) to update.
Each matrix should be square and match the corresponding dimension of grad.
grad: Gradient tensor of the parameter being optimized
shampoo_beta: Momentum coefficient for updating preconditioners.
Controls how much weight to give to new vs old gradient statistics.
Example:
>>> grad = torch.randn(10, 20)
>>> L = torch.zeros(10, 10)
>>> R = torch.zeros(20, 20)
>>> update_kronecker_factors([L, R], grad, shampoo_beta=0.95)
"""
# L = G @ G.T, R = G.T @ G
kronecker_factor_list[0].lerp_(grad @ grad.T, 1 - shampoo_beta)
kronecker_factor_list[1].lerp_(grad.T @ grad, 1 - shampoo_beta)
[docs]
@torch.no_grad() # type: ignore[misc]
def update_kronecker_factors_kl_shampoo(
kronecker_factor_list: list[torch.Tensor],
grad: torch.Tensor,
shampoo_beta: float,
eigenbasis_list: list[torch.Tensor],
eps: float,
eigval_exp: float = -1.0,
) -> None:
"""Updates the kronecker factor matrices in place using KL-Shampoo correction.
Implement KullbackāLeibler Minimization from https://arxiv.org/pdf/2509.03378
Args:
kronecker_factor_list: List of preconditioner matrices (L and R) to update.
grad: Gradient tensor of the parameter being optimized
shampoo_beta: Momentum coefficient for updating preconditioners.
eigenbasis_list: List of orthonormal eigenbases of the kronecker factor matrices
eps: Small offset for numerical stability.
eigenval_exp: Exponent of the eigenvalues.
"""
if grad.dim() != 2:
raise TypeError("KL-Shampoo mathematical correction is only supported for 2D tensors")
# Scale the gradient matrix by the approximate eigenvalues and the eigenbasis
# G@Q_R@Ī»_R^(ā1)@Q_R.T@G.T/dim(GG.T) and G.T@Q_L@Ī»_L^(ā1)@Q_L.T@G/dim(G.TG)
updates = []
for idx, (kronecker_factor, eigenbasis) in enumerate(zip(kronecker_factor_list, eigenbasis_list, strict=True)):
approx_eigvals = utils.eig.conjugate(kronecker_factor, eigenbasis, diag=True)
scale_factor = 1 / grad.shape[idx] * approx_eigvals.clamp_min(eps) ** eigval_exp
logging.debug(f"scale_factor[{idx}]: {scale_factor}")
correction = (eigenbasis * scale_factor[None, :]) @ eigenbasis.T
maybe_transpose_grad = grad.T if idx == 1 else grad
updates.append(utils.eig.conjugate(correction, maybe_transpose_grad))
# Note that updates caculated in previous loop are in reverse order of the kronecker factor list they apply to
for kronecker_factor, update in zip(kronecker_factor_list, updates[::-1], strict=True):
kronecker_factor.lerp_(update, 1 - shampoo_beta)
[docs]
@torch.no_grad() # type: ignore[misc]
def update_eigenbasis_and_exp_avgs(
kronecker_factor_list: list[torch.Tensor],
eigenbasis_list: list[torch.Tensor],
exp_avg_sq: torch.Tensor,
exp_avg: torch.Tensor,
use_eigh: bool = False,
power_iter_steps: int = 1,
) -> tuple[list[torch.Tensor], torch.Tensor, torch.Tensor]:
"""Updates the eigenbases and moving averages.
This function performs an update of the eigenbases (QL and QR)
used for preconditioning. It follows these steps:
1. Projects exp_avg back to the original basis
2. Updates the eigenbases using QR decomposition and power iteration (orthogonal iteration)
3. Projects exp_avg back to the new eigenbasis
Args:
kronecker_factor_list: List of preconditioner matrices (L and R) that define
the optimization landscape. These are updated with gradient statistics.
eigenbasis_list: List of current eigenbases (QL and QR)
used for preconditioning. These will be updated by this function.
exp_avg_sq: Inner Adam's second moment tensor, used for scaling the preconditioner updates.
This tensor is modified in-place.
exp_avg: Inner Adam's first moment tensor, used for tracking gradient momentum.
This tensor is modified in-place.
use_eigh: Whether to use full symmetric eigendecomposition (eigh) to compute the eigenbasis.
If False, use orthogonal iteration to compute the eigenbasis.
power_iter_steps: Number of power iteration steps to perform before QR decomposition.
More steps can lead to better convergence but increased computation time.
Returns:
A tuple containing:
- Updated list of eigenbases (QL and QR)
- Updated exp_avg tensor projected to the new eigenbasis
- Updated exp_avg_sq tensor
Example:
>>> L = torch.randn(10, 10)
>>> R = torch.randn(20, 20)
>>> QL = torch.randn(10, 10)
>>> QR = torch.randn(20, 20)
>>> exp_avg_sq = torch.randn(10, 20)
>>> exp_avg = torch.randn(10, 20)
>>> updated_eigenbasis_list, updated_exp_avg, updated_exp_avg_sq = update_eigenbasis_and_exp_avgs(
... [L, R], [QL, QR], exp_avg_sq, exp_avg)
"""
# Step 1: Project exp_avg back to the original basis
torch.cuda.nvtx.range_push("eigenbasis update step 1: precondition")
exp_avg = precondition(
exp_avg,
eigenbasis_list,
dims=[[0], [1]],
)
torch.cuda.nvtx.range_pop()
# Step 2: Update eigenbases
torch.cuda.nvtx.range_push("eigenbasis update step 2: update Q")
if use_eigh:
updated_eigenbasis_list = soap_utils.get_eigenbasis_eigh(
kronecker_factor_list,
)
else:
# Use QR decomposition and power iteration (orthogonal iteration)
updated_eigenbasis_list, exp_avg_sq = soap_utils.get_eigenbasis_qr(
kronecker_factor_list,
eigenbasis_list,
exp_avg_sq,
power_iter_steps,
)
torch.cuda.nvtx.range_pop()
# Step 3: Project exp_avg to the new eigenbasis using the updated eigenbases
torch.cuda.nvtx.range_push("eigenbasis update step 3: project exp_avg")
exp_avg = precondition(
exp_avg,
updated_eigenbasis_list,
dims=[[0], [0]],
)
torch.cuda.nvtx.range_pop()
return updated_eigenbasis_list, exp_avg, exp_avg_sq
[docs]
@torch.no_grad() # type: ignore[misc]
def precondition(
x: torch.Tensor,
eigenbasis_list: list[torch.Tensor] | None = None,
dims: list[list[int]] | None = None,
) -> torch.Tensor:
"""Projects the gradient to and from the eigenbases of the kronecker factor matrices.
This function performs tensor contractions between the input gradient
and kronecker factor eigenbases.
Note:
For 2D tensors, we can use matmul instead of tensordot for code legibility. However, the code has
been using tensordot historically, so does the reference implementation. It is difficult to match
matmul and tensordot outputs exactly because of underlying floating point arithmetic differences.
Therefore, we decided to keep using tensordot for consistency.
Args:
x: Input tensor to be preconditioned
eigenbasis_list: List of eigenbases for preconditioning.
Each matrix should be a square matrix of eigenvectors.
dims: Dimensions for tensor contraction. Default is [[0], [0]] which contracts
the first dimension of grad with the first dimension of each eigenbasis matrix,
for projecting into the eigenbasis. Use [[0], [1]] for projecting back to original space.
Example:
>>> x = torch.randn(10, 20)
>>> Q = torch.randn(10, 10)
>>> precondition(x, [Q], dims=[[0], [0]])
"""
if dims is None:
# Pick contraction dims to project to the eigenbasis
dims = [[0], [0]]
if eigenbasis_list is None:
# If eigenbases are not provided, return the gradient without any preconditioning
return x
for Q in eigenbasis_list:
x = torch.tensordot(x, Q, dims=dims)
return x
def _is_eigenbasis_update_step(
step: int,
adam_warmup_steps: int,
precondition_frequency: int | Callable[[int], int],
) -> bool:
"""Checks if amortized computation of the eigenbasis should be recomputed.
Args:
step: Current step of the optimizer
adam_warmup_steps: Number of steps to skip preconditioning in the beginning (i.e. use standard AdamW updates)
precondition_frequency: How often to update the preconditioner. Can be an integer for fixed frequency
or a callable function that takes the current step as input and returns the frequency.
"""
if step < adam_warmup_steps:
return False
current_frequency = (
precondition_frequency if not callable(precondition_frequency) else precondition_frequency(step)
)
return step % current_frequency == 0
@torch.compile # type: ignore[misc]
def _clip_update_rms_in_place(u: torch.Tensor, max_rms: float, eps: float = 1e-7) -> None:
"""Clip the update root mean square (RMS) to a maximum value, in place.
Do not clip if max_rms is 0.
Inspired by Adafactor (https://arxiv.org/abs/1804.04235) and RMS_t (https://arxiv.org/abs/2304.13013)
Args:
u: The update tensor.
max_rms: The maximum RMS value.
eps: The epsilon value to prevent division by zero.
"""
if max_rms == 0:
return
# compute current update RMS
rms = u.square().mean().sqrt()
# compute scale factor = min(1.0, max_rms/(rms + eps))
scale = (max_rms / (rms + eps)).clamp(max=1.0)
# ināplace scale
u.mul_(scale)
