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 the optimal period.
Sign-based optimizers are effective under heavy-tailed noise.arXiv preprint arXiv:2602.07425
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abstract
While adaptive gradient methods are the workhorse of modern machine learning, sign-based optimization algorithms such as Lion and Muon have recently demonstrated superior empirical performance over AdamW in training large language models (LLM). However, a theoretical understanding of why sign-based updates outperform variance-adapted methods remains elusive. In this paper, we aim to bridge the gap between theory and practice through the lens of heavy-tailed gradient noise, a phenomenon frequently observed in language modeling tasks. Theoretically, we introduce a novel generalized heavy-tailed noise condition that captures the behavior of LLMs more accurately than standard finite variance assumptions. Under this noise model, we establish sharp convergence rates of SignSGD and Lion for generalized smooth function classes, matching or surpassing previous best-known bounds. Furthermore, we extend our analysis to Muon and Muonlight, providing what is, to our knowledge, the first rigorous analysis of matrix optimization under heavy-tailed stochasticity. These results offer a strong theoretical justification for the empirical superiority of sign-based optimizers, showcasing that they are naturally suited to handle the noisy gradients associated with heavy tails. Empirically, LLM pretraining experiments validate our theoretical insights and confirm that our proposed noise models are well-aligned with practice.
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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.
StoSignSGD resolves SignSGD divergence on non-smooth objectives via structural stochasticity, matching optimal convex rates and improving non-convex bounds while delivering 1.44-2.14x speedups in FP8 LLM pretraining.
CLion achieves O(1/N) generalization error and O(√d / T^{1/4}) convergence for nonconvex stochastic optimization, improving on Lion's O(1/(N τ^T)) bound.
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.
citing papers explorer
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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 the optimal period.
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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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StoSignSGD: Unbiased Structural Stochasticity Fixes SignSGD for Training Large Language Models
StoSignSGD resolves SignSGD divergence on non-smooth objectives via structural stochasticity, matching optimal convex rates and improving non-convex bounds while delivering 1.44-2.14x speedups in FP8 LLM pretraining.
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CLion: Efficient Cautious Lion Optimizer with Enhanced Generalization
CLion achieves O(1/N) generalization error and O(√d / T^{1/4}) convergence for nonconvex stochastic optimization, improving on Lion's O(1/(N τ^T)) bound.
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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.