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A Note on the Convergence of Muon
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A Note on the Convergence of Muon
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In this note, we inspect the convergence of a new optimizer for pretraining LLMs, namely the Muon optimizer. Such an optimizer is closely related to a specialized steepest descent method where the update direction is the minimizer of the quadratic approximation of the objective function under spectral norm. We provide the convergence analysis on both versions of the optimizer and discuss its implications.
Forward citations
Cited by 21 Pith papers
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When and Why SignSGD Outperforms SGD: A Theoretical Study Based on $\ell_1$-norm Lower Bounds
SignSGD provably beats SGD by a factor of d under sparse noise via matched ℓ1-norm upper and lower bounds, with an equivalent result for Muon on matrices, and this predicts faster GPT-2 pretraining.
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Symmetry-Compatible Principle for Optimizer Design: Embeddings, LM Heads, SwiGLU MLPs, and MoE Routers
Proposes equivariant optimizers matched to the symmetry groups of embeddings, SwiGLU projections and MoE routers, with experiments showing consistent gains over AdamW on language model pre-training.
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Intrinsic Muon: Spectral Optimization on Riemannian Matrix Manifolds
Intrinsic Muon provides closed-form linear maximization oracles on multiple Riemannian matrix manifolds for unitarily invariant norms, with convergence rates depending only on manifold dimension or rank.
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Muon with Nesterov Momentum: Heavy-Tailed Noise and (Randomized) Inexact Polar Decomposition
Muon with Nesterov momentum and inexact polar decomposition achieves optimal convergence rates of O(ε^(-(3α-2)/(α-1))) under heavy-tailed noise for ε-stationary points in non-convex settings.
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Convergence Rate Analysis of SOAP with Arbitrary Orthogonal Projection Matrices
SOAP and its generalizations with arbitrary orthogonal projections converge at a provable rate when the projections are conditionally independent of the current gradient.
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A unified convergence theory for adaptive first-order methods in the nonconvex case, including AdaNorm, full and diagonal AdaGrad, Shampoo and Muo
A unified stochastic convergence theory is developed for adaptive preconditioned first-order methods including AdaGrad variants, Shampoo, and Muon in nonconvex optimization.
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Muse: Representation Geometry of Muon Beyond Normalized Momentum
The matrix shape given to Muon's orthonormalization step is a genuine optimizer axis: square-ish reshapes match native Muon, skinnier reshapes interpolate toward normalized SGD with momentum.
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Reassessing Muon for Matrix Factorization
Muon's advantage over AdamW is problem-dependent: it loses or ties on plain low-rank factorization and completion but wins on nonnegative matrix factorization.
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LionMuon: Alternating Spectral and Sign Descent for Efficient Training
LionMuon alternates Lion and Muon steps with shared dual-EMA buffer to Pareto-dominate existing optimizers in loss and compute on models up to 720M parameters.
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LionMuon: Alternating Spectral and Sign Descent for Efficient Training
LionMuon alternates Lion sign steps and Muon spectral steps with shared dual-EMA momentum to match Lion memory while outperforming both at P=2 on 124M-720M models, backed by heavy-tailed complexity bounds that predict...
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Scale-Invariant Neural Network Optimization: Norm Geometry and Heavy-Tailed Noise
Establishes matching lower and upper oracle complexity bounds for scale-invariant methods with spectral norm under heavy-tailed noise, plus improved rates with higher-order smoothness, and practical tests on neural networks.
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Scale-Invariant Neural Network Optimization: Norm Geometry and Heavy-Tailed Noise
Establishes matching Ω and O(min{m,n} ε^-(3p-2)/(p-1)) bounds for scale-invariant spectral-norm methods under heavy-tailed noise, plus an improved O(min{m,n} ε^-(5p-3)/(2p-2)) rate via transported Scion under Hessian ...
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Symmetry-Compatible Principle for Optimizer Design: Embeddings, LM Heads, SwiGLU MLPs, and MoE Routers
Proposes equivariant optimizer updates matched to layer symmetries for embeddings, SwiGLU MLPs, and MoE routers, with reported gains in validation loss and training stability on several language model architectures.
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Spectral Flattening Is All Muon Needs: How Orthogonalization Controls Learning Rate and Convergence
Muon achieves faster convergence and larger stable learning rates by flattening the singular value spectrum of the momentum buffer through orthogonalization, scaling step size with average rather than maximum singular values.
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Muon Does Not Converge on Convex Lipschitz Functions
Muon does not converge on convex Lipschitz functions regardless of learning rate, while error feedback restores theoretical convergence but degrades performance on CIFAR-10 and nanoGPT tasks.
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Muon in Associative Memory Learning: Training Dynamics and Scaling Laws
In a linear softmax memory model, Muon equalizes learning across frequency tiers and gives exponential (noiseless) or T^{-2} (noisy power-law) convergence, versus polynomial or T^{-(1-1/β)} for gradient descent.
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Convergence of Spectral Descent for Non-smooth Optimization
Proves linear convergence of Spectral Descent (SD) and Truncated SD for non-smooth convex problems under stated conditions, sublinear rates for regularized versions via Frank-Wolfe, and recovery guarantees for robust ...
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Anytime Training with Schedule-Free Spectral Optimization
SF-NorMuon is a new schedule-free spectral optimizer that closes the gap with tuned AdamW on 125M-772M parameter models across 1-8x Chinchilla horizons while providing stationarity guarantees.
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MiMuon: Mixed Muon Optimizer with Improved Generalization for Large Models
MiMuon is a hybrid optimizer that achieves a generalization error bound of O(1/N) independent of the small singular-value gap that limits the original Muon bound, while retaining the same O(1/T^{1/4}) convergence rate.
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A Muon-Accelerated Algorithm for Low Separation Rank Tensor Generalized Linear Models
LSRTR-M integrates Muon updates into the LSRTR algorithm for tensor GLMs, achieving faster convergence, lower estimation errors on synthetic linear/logistic/Poisson models, and competitive performance with better effi...
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On the Convergence Analysis of Muon
Convergence analysis shows Muon outperforms gradient descent by exploiting low-rank structure in neural network Hessians.
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