import math import torch # This code snippet is a modified version adapted from the following GitHub repository: # https://github.com/KellerJordan/Muon/blob/master/muon.py @torch.compile 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. """ assert len(G.shape) == 2 a, b, c = (3.4445, -4.7750, 2.0315) X = G.bfloat16() if G.size(0) > G.size(1): X = X.T # Ensure spectral norm is at most 1 X = X / (X.norm() + 1e-7) # Perform the NS iterations for _ in range(steps): A = X @ X.T B = ( b * A + c * A @ A ) # adapted from suggestion by @jxbz, @leloykun, and @YouJiacheng X = a * X + B @ X if G.size(0) > G.size(1): X = X.T return X class MuonKIMI(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: muon_params: The parameters to be optimized by Muon. 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) nesterov: Whether to use Nesterov-style momentum in the internal SGD. (recommended) ns_steps: The number of Newton-Schulz iterations to run. (6 is probably always enough) {0, 1}-D or are detected as being the embed or lm_head will be optimized by AdamW as well. adamw_lr: The learning rate for the internal AdamW. 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, params, lr=1e-3, weight_decay=0.01, momentum=0.95, nesterov=True, ns_steps=5, adamw_betas=(0.9, 0.95), adamw_eps=1e-8, ): wd = weight_decay defaults = dict( lr=lr, wd=wd, momentum=momentum, nesterov=nesterov, ns_steps=ns_steps, adamw_betas=adamw_betas, adamw_eps=adamw_eps, ) super().__init__(params, defaults) # # Sort parameters into those for which we will use Muon, and those for which we will not for group in self.param_groups: force_adamw = group.get("force_adamw", False) for p in group["params"]: if p.requires_grad: if force_adamw or p.ndim < 2: self.state[p]["use_muon"] = False else: self.state[p]["use_muon"] = True else: self.state[p]["use_muon"] = False def adjust_lr_for_muon(self, lr, param_shape): A, B = param_shape[:2] # We adjust the learning rate and weight decay based on the size of the parameter matrix # as described in the paper adjusted_ratio = 0.2 * math.sqrt(max(A, B)) adjusted_lr = lr * adjusted_ratio return adjusted_lr def step(self, closure=None): """Perform a single optimization step. Args: closure (Callable, optional): A closure that reevaluates the model and returns the loss. """ loss = None if closure is not None: with torch.enable_grad(): loss = closure() for group in self.param_groups: ############################ # Muon # ############################ params = [p for p in group["params"] if self.state[p]["use_muon"]] lr = group["lr"] wd = group["wd"] momentum = group["momentum"] # Distribute parameters for p in params: g = p.grad if g is None: continue if g.ndim > 2: g = g.view(g.size(0), -1) assert g is not None # calc update state = self.state[p] if "momentum_buffer" not in state: state["momentum_buffer"] = torch.zeros_like(g) buf = state["momentum_buffer"] buf.mul_(momentum).add_(g) if group["nesterov"]: g = g.add(buf, alpha=momentum) else: g = buf u = zeropower_via_newtonschulz5(g, steps=group["ns_steps"]) # scale update adjusted_lr = self.adjust_lr_for_muon(lr, g.shape) # apply weight decay p.data.mul_(1 - lr * wd) # apply update p.data.add_(u.reshape(p.shape), alpha=-adjusted_lr) ############################ # AdamW backup # ############################ params = [p for p in group["params"] if not self.state[p]["use_muon"]] lr = group["lr"] beta1, beta2 = group["adamw_betas"] eps = group["adamw_eps"] weight_decay = group["wd"] for p in params: g = p.grad if g is None: continue state = self.state[p] if "step" not in state: state["step"] = 0 state["moment1"] = torch.zeros_like(g) state["moment2"] = torch.zeros_like(g) state["step"] += 1 step = state["step"] buf1 = state["moment1"] buf2 = state["moment2"] 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) return loss