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import torch.distributed as dist
def zeropower_via_newtonschulz5(G, steps: int):
"""
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.
"""
assert G.ndim >= 2 # batched Muon implementation by @scottjmaddox, and put into practice in the record by @YouJiacheng
a, b, c = (3.4445, -4.7750, 2.0315)
X = G.bfloat16()
if G.size(-2) > G.size(-1):
X = X.mT
# Ensure spectral norm is at most 1
X = X / (X.norm(dim=(-2, -1), keepdim=True) + 1e-7)
# Perform the NS iterations
for _ in range(steps):
A = X @ X.mT
B = b * A + c * A @ A # quintic computation strategy adapted from suggestion by @jxbz, @leloykun, and @YouJiacheng
X = a * X + B @ X
if G.size(-2) > G.size(-1):
X = X.mT
return X
def muon_update(grad, momentum, beta=0.95, ns_steps=5, nesterov=True):
momentum.lerp_(grad, 1 - beta)
update = grad.lerp_(momentum, beta) if nesterov else momentum
if update.ndim == 4: # for the case of conv filters
update = update.view(len(update), -1)
update = zeropower_via_newtonschulz5(update, steps=ns_steps)
update *= max(1, grad.size(-2) / grad.size(-1))**0.5
return update
class Muon(torch.optim.Optimizer):
"""
Muon - MomentUm Orthogonalized by Newton-schulz
https://kellerjordan.github.io/posts/muon/
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. For efficient orthogonalization we use a Newton-Schulz iteration, which has the
advantage that it can be stably run in bfloat16 on the GPU.
Muon should only be used for hidden weight layers. The input embedding, final output layer,
and any internal gains or biases should be optimized using a standard method such as AdamW.
Hidden convolutional weights can be trained using Muon by viewing them as 2D and then
collapsing their last 3 dimensions.
Arguments:
lr: The learning rate, in units of spectral norm per update.
weight_decay: The AdamW-style weight decay.
momentum: The momentum. A value of 0.95 here is usually fine.
"""
def __init__(self, params, lr=0.02, weight_decay=0, momentum=0.95):
defaults = dict(lr=lr, weight_decay=weight_decay, momentum=momentum)
assert isinstance(params, list) and len(params) >= 1 and isinstance(params[0], torch.nn.Parameter)
params = sorted(params, key=lambda x: x.size(), reverse=True)
super().__init__(params, defaults)
@torch.no_grad()
def step(self, closure=None):
loss = None
if closure is not None:
with torch.enable_grad():
loss = closure()
for group in self.param_groups:
params = group["params"]
params_pad = params + [torch.empty_like(params[-1])] * (dist.get_world_size() - len(params) % dist.get_world_size())
for base_i in range(len(params))[::dist.get_world_size()]:
if base_i + dist.get_rank() < len(params):
p = params[base_i + dist.get_rank()]
if p.grad is None:
# continue
p.grad = torch.zeros_like(p) # Force synchronization
state = self.state[p]
if len(state) == 0:
state["momentum_buffer"] = torch.zeros_like(p)
update = muon_update(p.grad, state["momentum_buffer"], beta=group["momentum"])
p.mul_(1 - group["lr"] * group["weight_decay"])
p.add_(update.reshape(p.shape), alpha=-group["lr"])
dist.all_gather(params_pad[base_i:base_i + dist.get_world_size()], params_pad[base_i + dist.get_rank()])
return loss
class SingleDeviceMuon(torch.optim.Optimizer):
"""
Muon variant for usage in non-distributed settings.
"""
def __init__(self, params, lr=0.02, weight_decay=0, momentum=0.95):
defaults = dict(lr=lr, weight_decay=weight_decay, momentum=momentum)
super().__init__(params, defaults)
@torch.no_grad()
def step(self, closure=None):
loss = None
if closure is not None:
with torch.enable_grad():
loss = closure()
for group in self.param_groups:
for p in group["params"]:
if p.grad is None:
# continue
p.grad = torch.zeros_like(p) # Force synchronization
state = self.state[p]
if len(state) == 0:
state["momentum_buffer"] = torch.zeros_like(p)
update = muon_update(p.grad, state["momentum_buffer"], beta=group["momentum"])
p.mul_(1 - group["lr"] * group["weight_decay"])
p.add_(update.reshape(p.shape), alpha=-group["lr"])
return loss
def adam_update(grad, buf1, buf2, step, betas, eps):
buf1.lerp_(grad, 1 - betas[0])
buf2.lerp_(grad.square(), 1 - betas[1])
buf1c = buf1 / (1 - betas[0]**step)
buf2c = buf2 / (1 - betas[1]**step)
return buf1c / (buf2c.sqrt() + eps)
class MuonWithAuxAdam(torch.optim.Optimizer):
"""
Distributed Muon variant that can be used for all parameters in the network, since it runs an
internal AdamW for the parameters that are not compatible with Muon. The user must manually
specify which parameters shall be optimized with Muon and which with Adam by passing in a
list of param_groups with the `use_muon` flag set.
The point of this class is to allow the user to have a single optimizer in their code, rather
than having both a Muon and an Adam which each need to be stepped.
You can see an example usage below:
https://github.com/KellerJordan/modded-nanogpt/blob/master/records/052525_MuonWithAuxAdamExample/b01550f9-03d8-4a9c-86fe-4ab434f1c5e0.txt#L470
```
hidden_matrix_params = [p for n, p in model.blocks.named_parameters() if p.ndim >= 2 and "embed" not in n]
embed_params = [p for n, p in model.named_parameters() if "embed" in n]
scalar_params = [p for p in model.parameters() if p.ndim < 2]
head_params = [model.lm_head.weight]
from muon import MuonWithAuxAdam
adam_groups = [dict(params=head_params, lr=0.22), dict(params=embed_params, lr=0.6), dict(params=scalar_params, lr=0.04)]
adam_groups = [dict(**g, betas=(0.8, 0.95), eps=1e-10, use_muon=False) for g in adam_groups]
muon_group = dict(params=hidden_matrix_params, lr=0.05, momentum=0.95, use_muon=True)
param_groups = [*adam_groups, muon_group]
optimizer = MuonWithAuxAdam(param_groups)
```
"""
def __init__(self, param_groups):
for group in param_groups:
assert "use_muon" in group
if group["use_muon"]:
group["params"] = sorted(group["params"], key=lambda x: x.size(), reverse=True)
# defaults
group["lr"] = group.get("lr", 0.02)
group["momentum"] = group.get("momentum", 0.95)
group["weight_decay"] = group.get("weight_decay", 0)
assert set(group.keys()) == set(["params", "lr", "momentum", "weight_decay", "use_muon"])
else:
# defaults
group["lr"] = group.get("lr", 3e-4)
group["betas"] = group.get("betas", (0.9, 0.95))
group["eps"] = group.get("eps", 1e-10)
group["weight_decay"] = group.get("weight_decay", 0)
assert set(group.keys()) == set(["params", "lr", "betas", "eps", "weight_decay", "use_muon"])
super().__init__(param_groups, dict())
@torch.no_grad()
def step(self, closure=None):
loss = None
if closure is not None:
with torch.enable_grad():
loss = closure()
for group in self.param_groups:
if group["use_muon"]:
params = group["params"]
params_pad = params + [torch.empty_like(params[-1])] * (dist.get_world_size() - len(params) % dist.get_world_size())
for base_i in range(len(params))[::dist.get_world_size()]:
if base_i + dist.get_rank() < len(params):
p = params[base_i + dist.get_rank()]
if p.grad is None:
# continue
p.grad = torch.zeros_like(p) # Force synchronization
state = self.state[p]
if len(state) == 0:
state["momentum_buffer"] = torch.zeros_like(p)
update = muon_update(p.grad, state["momentum_buffer"], beta=group["momentum"])
p.mul_(1 - group["lr"] * group["weight_decay"])
p.add_(update.reshape(p.shape), alpha=-group["lr"])
dist.all_gather(params_pad[base_i:base_i + dist.get_world_size()], params_pad[base_i + dist.get_rank()])
else:
for p in group["params"]:
if p.grad is None:
# continue
p.grad = torch.zeros_like(p) # Force synchronization
state = self.state[p]
if len(state) == 0:
state["exp_avg"] = torch.zeros_like(p)
state["exp_avg_sq"] = torch.zeros_like(p)
state["step"] = 0
state["step"] += 1
update = adam_update(p.grad, state["exp_avg"], state["exp_avg_sq"],
state["step"], group["betas"], group["eps"])
p.mul_(1 - group["lr"] * group["weight_decay"])
p.add_(update, alpha=-group["lr"])
return loss
class SingleDeviceMuonWithAuxAdam(torch.optim.Optimizer):
"""
Non-distributed variant of MuonWithAuxAdam.
"""
def __init__(self, param_groups):
for group in param_groups:
assert "use_muon" in group
if group["use_muon"]:
# defaults
group["lr"] = group.get("lr", 0.02)
group["momentum"] = group.get("momentum", 0.95)
group["weight_decay"] = group.get("weight_decay", 0)
assert set(group.keys()) == set(["params", "lr", "momentum", "weight_decay", "use_muon"])
else:
# defaults
group["lr"] = group.get("lr", 3e-4)
group["betas"] = group.get("betas", (0.9, 0.95))
group["eps"] = group.get("eps", 1e-10)
group["weight_decay"] = group.get("weight_decay", 0)
assert set(group.keys()) == set(["params", "lr", "betas", "eps", "weight_decay", "use_muon"])
super().__init__(param_groups, dict())
@torch.no_grad()
def step(self, closure=None):
loss = None
if closure is not None:
with torch.enable_grad():
loss = closure()
for group in self.param_groups:
if group["use_muon"]:
for p in group["params"]:
if p.grad is None:
# continue
p.grad = torch.zeros_like(p) # Force synchronization
state = self.state[p]
if len(state) == 0:
state["momentum_buffer"] = torch.zeros_like(p)
update = muon_update(p.grad, state["momentum_buffer"], beta=group["momentum"])
p.mul_(1 - group["lr"] * group["weight_decay"])
p.add_(update.reshape(p.shape), alpha=-group["lr"])
else:
for p in group["params"]:
if p.grad is None:
# continue
p.grad = torch.zeros_like(p) # Force synchronization
state = self.state[p]
if len(state) == 0:
state["exp_avg"] = torch.zeros_like(p)
state["exp_avg_sq"] = torch.zeros_like(p)
state["step"] = 0
state["step"] += 1
update = adam_update(p.grad, state["exp_avg"], state["exp_avg_sq"],
state["step"], group["betas"], group["eps"])
p.mul_(1 - group["lr"] * group["weight_decay"])
p.add_(update, alpha=-group["lr"])
return loss
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