import math
from typing import Tuple, Dict
import torch
import torch.distributed as dist
from torch.distributed.tensor import DTensor, Replicate, Shard
def zeropower_via_newtonschulz5(G, steps):
"""
Newton-Schulz iteration to compute the zeroth power / orthogonalization of G. We opt to use a
quintic iteration whose coefficients are selected to maximize the slope at zero. For the purpose
of minimizing steps, it turns out to be empirically effective to keep increasing the slope at
zero even beyond the point where the iteration no longer converges all the way to one everywhere
on the interval. This iteration therefore does not produce UV^T but rather something like US'V^T
where S' is diagonal with S_{ii}' ~ Uniform(0.5, 1.5), which turns out not to hurt model
performance at all relative to UV^T, where USV^T = G is the SVD.
"""
if len(G.shape) != 2:
raise ValueError(
f"zeropower_via_newtonschulz5 expects a 2-D tensor, got shape {G.shape}"
)
a, b, c = (3.4445, -4.7750, 2.0315)
X = G
if G.size(0) > G.size(1):
X = X.T
X = X / (X.norm() + 1e-7)
for _ in range(steps):
A = X @ X.T
B = b * A + c * A @ A
X = a * X + B @ X
if G.size(0) > G.size(1):
X = X.T
return X
def normalize_range(value_range: Tuple[int, int], start: int):
return (value_range[0] - start, value_range[1] - start)
def adjust_lr_wd_for_muon(lr, matched_adamw_rms, param_shape):
A, B = param_shape[:2]
adjusted_ratio = math.sqrt(max(A, B)) * matched_adamw_rms
adjusted_lr = lr * adjusted_ratio
return adjusted_lr
class Muon(torch.optim.Optimizer):
"""
Muon - MomentUm Orthogonalized by Newton-schulz
Muon internally runs standard SGD-momentum, and then performs an orthogonalization post-
processing step, in which each 2D parameter's update is replaced with the nearest orthogonal
matrix. To efficiently orthogonalize each update, we use a Newton-Schulz iteration, which has
the advantage that it can be stably run in bfloat16 on the GPU.
Some warnings:
- We believe this optimizer is unlikely to work well for training with small batch size.
- We believe it may not work well for finetuning pretrained models, but we haven't tested this.
Arguments:
param_groups: The parameters to be optimized.
lr: The learning rate. The updates will have spectral norm of `lr`. (0.02 is a good default)
momentum: The momentum used by the internal SGD. (0.95 is a good default)
matched_adamw_rms: The AdamW Update RMS that Muon is designed to match. (0.2~0.4 recommended)
nesterov: Whether to use Nesterov-style momentum in the internal SGD.
ns_steps: The number of Newton-Schulz iterations to run. (5 is probably always enough).
adamw_betas: The betas for the internal AdamW.
adamw_eps: The epsilon for the internal AdamW.
adamw_wd: The weight decay for the internal AdamW.
"""
def __init__(self,
param_groups,
lr: float = 2e-2,
weight_decay: float = 0.1,
matched_adamw_rms: float = 0.2,
momentum: float = 0.95,
nesterov: bool = True,
ns_steps: int = 5,
adamw_betas: Tuple[float, float] = (0.95, 0.95),
adamw_eps: float = 1e-8):
defaults = dict(
lr=lr,
weight_decay=weight_decay,
matched_adamw_rms=matched_adamw_rms,
momentum=momentum,
nesterov=nesterov,
ns_steps=ns_steps,
adamw_betas=adamw_betas,
adamw_eps=adamw_eps,
)
super().__init__(param_groups, defaults)
@torch.no_grad()
def step(self):
for group in self.param_groups:
if not group.get("use_muon", False):
continue
momentum = group['momentum']
nesterov = group['nesterov']
for p in group["params"]:
if p.grad is None:
continue
state = self.state[p]
if "muon_buffer" not in state:
state["muon_buffer"] = torch.zeros_like(p.grad)
buf = state["muon_buffer"]
buf.mul_(momentum).add_(p.grad)
if nesterov:
g = p.grad.add(buf, alpha=momentum)
else:
g = buf
state["ns_input"] = g.to(torch.bfloat16)
for group in self.param_groups:
if not group.get("use_muon", False):
continue
lr = group["lr"]
ns_steps = group["ns_steps"]
weight_decay = group["weight_decay"]
matched_adamw_rms = group["matched_adamw_rms"]
for p in group["params"]:
if p.grad is None:
continue
state = self.state[p]
ns_input = state.pop("ns_input", None)
if ns_input is None:
continue
if hasattr(p, 'device_mesh'):
device_mesh = ns_input.device_mesh
new_placements = [Replicate() if isinstance(placement, Shard)
else placement for placement in ns_input.placements]
ns_input_full = ns_input.redistribute(device_mesh, new_placements)
ns_input_local = ns_input_full.to_local()
else:
ns_input_local = ns_input
update = zeropower_via_newtonschulz5(ns_input_local, steps=ns_steps)
if hasattr(p, 'device_mesh'):
update_dtensor = DTensor.from_local(update, p.device_mesh, new_placements)
update_sharded = update_dtensor.redistribute(p.device_mesh, p.placements)
update = update_sharded
p.mul_(1 - lr * weight_decay)
adjusted_lr = adjust_lr_wd_for_muon(lr, matched_adamw_rms, ns_input.shape)
p.data.add_(update, alpha=-adjusted_lr)
for group in self.param_groups:
if group.get('use_muon', False):
continue
if 'step' in group:
group['step'] += 1
else:
group['step'] = 1
step = group['step']
lr = group['lr']
weight_decay = group['weight_decay']
beta1, beta2 = group['adamw_betas']
eps = group['adamw_eps']
for p in group["params"]:
if p.grad is None:
continue
g = p.grad
state = self.state[p]
if len(state) == 0:
state['adamw_exp_avg'] = torch.zeros_like(g)
state['adamw_exp_avg_sq'] = torch.zeros_like(g)
buf1 = state['adamw_exp_avg']
buf2 = state['adamw_exp_avg_sq']
buf1.lerp_(g, 1 - beta1)
buf2.lerp_(g.square(), 1 - beta2)
g = buf1 / (eps + buf2.sqrt())
bias_correction1 = 1 - beta1**step
bias_correction2 = 1 - beta2**step
scale = bias_correction1 / bias_correction2**0.5
p.data.mul_(1 - lr * weight_decay)
p.data.add_(g, alpha=- lr / scale)
