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from contextlib import nullcontext
from functools import partial
from typing import TYPE_CHECKING, Callable, Iterable, 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.legacy_soap import soap_utils
from emerging_optimizers.scalar_optimizers import update_functions
from emerging_optimizers.utils import FP32MatmulPrecT
__all__ = [
"SOAP",
"project_in",
"project_out",
"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)
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.
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.
stream_list: Optional list of CUDA streams. When provided, each parameter in the inner loop uses a
stream from this list in round-robin fashion.
"""
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,
correct_bias: bool = True,
fp32_matmul_prec: FP32MatmulPrecT = "highest",
use_eigh: bool = False,
qr_fp32_matmul_prec: FP32MatmulPrecT = "high",
power_iter_steps: int = 1,
max_update_rms: float = 0.0,
use_kl_shampoo: bool = False,
correct_shampoo_beta_bias: bool | None = None,
stream_list: list[torch.cuda.Stream] | None = None,
) -> None:
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.power_iter_steps = power_iter_steps
self.max_update_rms = max_update_rms
self.use_kl_shampoo = use_kl_shampoo
self.stream_list = stream_list
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"] = torch.zeros(p.shape[0], p.shape[0], device=p.device)
state["R"] = torch.zeros(p.shape[1], p.shape[1], 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)
state["eigvals_L"] = torch.zeros(p.shape[0], device=p.device)
state["eigvals_R"] = torch.zeros(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: Unsupported; must be ``None``.
"""
if closure is not None:
raise ValueError("closure is not supported")
for group in self.param_groups:
self._init_group(group)
current_stream = torch.cuda.current_stream() if torch.cuda.is_available() else None
if self.stream_list is not None and current_stream is not None:
for stream in self.stream_list:
stream.wait_stream(current_stream)
for group in self.param_groups:
for param_idx, p in enumerate(group["params"]):
if p.grad is None:
continue # pragma: no cover
stream_ctx: torch.cuda.StreamContext | nullcontext[None] = nullcontext()
if self.stream_list is not None and current_stream is not None:
stream = self.stream_list[param_idx % len(self.stream_list)]
stream_ctx = torch.cuda.stream(stream)
with stream_ctx:
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
kronecker_factor_list = [state["L"], state["R"]]
eigenbasis_list = [state["Q_L"], state["Q_R"]]
eigvals_list = [state["eigvals_L"], state["eigvals_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,
eigvals_list=eigvals_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)
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
)
# Always use eigh for the first eigenbasis update
use_eigh = self.use_eigh if state["step"] != 0 else True
with utils.fp32_matmul_precision(self.qr_fp32_matmul_prec):
updated_eigvals_list, 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
state["eigvals_L"], state["eigvals_R"] = updated_eigvals_list
# rebind local ref so project_in() below uses the updated Q
eigenbasis_list = updated_eigenbasis_list
state["exp_avg"] = exp_avg
state["exp_avg_sq"] = exp_avg_sq
self._apply_weight_decay_inplace(
p,
grad,
group["lr"],
group["weight_decay"],
)
# Project gradients to the eigenbases of Shampoo's preconditioner
with utils.fp32_matmul_precision(self.fp32_matmul_prec):
grad_projected = project_in(grad, eigenbasis_list)
# 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"],
betas=group["betas"],
eps=group["eps"],
correct_bias=self.correct_bias,
nesterov=self.nesterov,
step=curr_iter_1_based, # 1-based iteration index is used for bias correction
)
# Projecting back the preconditioned (by ADAM) exponential moving average of gradients
with utils.fp32_matmul_precision(self.fp32_matmul_prec):
precond_update = project_out(adam_update, eigenbasis_list)
_clip_update_rms_in_place(precond_update, self.max_update_rms)
p.add_(precond_update, alpha=-group["lr"])
state["step"] += 1
if self.stream_list is not None and current_stream is not None:
for stream in self.stream_list:
current_stream.wait_stream(stream)
return None
[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: Iterable[torch.Tensor],
grad: torch.Tensor,
shampoo_beta: float,
eigenbasis_list: Iterable[torch.Tensor],
eigvals_list: Iterable[torch.Tensor],
eps: float,
eigval_exp: float = -1.0,
) -> None:
"""Updates the kronecker factor matrices in place using KL-Shampoo correction.
Implements the KullbackāLeibler minimization update from https://arxiv.org/pdf/2509.03378.
For a gradient :math:`G \\in \\mathbb{R}^{m \\times n}`, current kronecker factors
:math:`L_t \\in \\mathbb{R}^{m \\times m}`, :math:`R_t \\in \\mathbb{R}^{n \\times n}`, their
orthonormal eigenbases :math:`Q_L, Q_R`, and the approximate eigenvalues in those eigenbases
.. math::
\\Lambda_L = \\mathrm{diag}(Q_L^{\\top} L_t Q_L), \\quad
\\Lambda_R = \\mathrm{diag}(Q_R^{\\top} R_t Q_R)
(passed as ``eigvals_list``, typically computed and stored at the previous eigenbasis update, see
:func:`~emerging_optimizers.legacy_soap.soap_utils.get_eigenbasis_qr` and
:func:`~emerging_optimizers.legacy_soap.soap_utils.get_eigenbasis_eigh`), the EMA update with momentum
:math:`\\beta` (= ``shampoo_beta``) and exponent :math:`p` (= ``eigval_exp``, default ``-1``) is
.. math::
L_{t+1} = \\beta\\, L_t + \\frac{1-\\beta}{n}\\, G\\, Q_R\\, \\mathrm{diag}(\\Lambda_R^{p})\\, Q_R^{\\top} G^{\\top} \\\\
R_{t+1} = \\beta\\, R_t + \\frac{1-\\beta}{m}\\, G^{\\top}\\, Q_L\\, \\mathrm{diag}(\\Lambda_L^{p})\\, Q_L^{\\top} G
Eigenvalues are clamped to ``eps`` from below before exponentiation for numerical stability.
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
eigvals_list: List of approximate eigenvalues of each kronecker factor in its eigenbasis.
eps: Small offset for numerical stability.
eigval_exp: Exponent applied to the (clamped) 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, (eigenbasis, approx_eigvals) in enumerate(zip(eigenbasis_list, eigvals_list, strict=True)):
scale_factor = 1 / grad.shape[idx] * approx_eigvals.clamp_min(eps) ** eigval_exp
logging.debug("scale_factor[%s]: %s", 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], 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.
Permuted along each kronecker-factor axis on the QR path to track the sorted eigenbasis
columns; returned unchanged on the eigh path.
exp_avg: Inner Adam's first moment tensor, used for tracking gradient momentum.
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:
- List of (approximate) eigenvalues of each kronecker factor in its updated eigenbasis
- 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)
>>> eigvals_list, 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
exp_avg = project_out(exp_avg, eigenbasis_list)
# Step 2: Update eigenbases
if use_eigh:
updated_eigvals_list, updated_eigenbasis_list = soap_utils.get_eigenbasis_eigh(
kronecker_factor_list,
)
else:
updated_eigvals_list, updated_eigenbasis_list = soap_utils.get_eigenbasis_qr(
kronecker_factor_list,
eigenbasis_list,
power_iter_steps,
)
# Step 3: Project exp_avg to the new eigenbasis using the updated eigenbases
exp_avg = project_in(exp_avg, updated_eigenbasis_list)
return updated_eigvals_list, updated_eigenbasis_list, exp_avg, exp_avg_sq
[docs]
@torch.no_grad() # type: ignore[misc]
def project_in(
x: torch.Tensor,
eigenbasis_list: list[torch.Tensor],
) -> torch.Tensor:
"""Projects a tensor into the 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 project into the eigenbasis.
eigenbasis_list: List of eigenbases for preconditioning.
"""
for Q in eigenbasis_list:
x = torch.tensordot(x, Q, dims=[[0], [0]])
return x
[docs]
@torch.no_grad() # type: ignore[misc]
def project_out(
x: torch.Tensor,
eigenbasis_list: list[torch.Tensor],
) -> torch.Tensor:
"""Projects a tensor out of the eigenbases
Note:
Uses ``tensordot`` rather than ``matmul`` for the same numerical-consistency reason described in
:func:`project_in`.
Args:
x: Input tensor to project back to the original space.
eigenbasis_list: List of eigenbases for preconditioning.
"""
for Q in eigenbasis_list:
x = torch.tensordot(x, Q, dims=[[0], [1]])
return x
@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)
