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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On the Convergence of Muon and Beyond
16 Pith papers cite this work. Polarity classification is still indexing.
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Pith papers citing it
abstract
The Muon optimizer has demonstrated remarkable empirical success in handling matrix-structured parameters for training neural networks. However, a significant gap remains between its practical performance and theoretical understanding. Existing analyses show that the Muon variants achieve only a suboptimal ergodic convergence rate of $\mathcal{O}(T^{-1/4})$ in stochastic non-convex settings, where $T$ denotes the number of iterations. To study the theoretical limits of Muon, we analyze two momentum-based variance-reduced variants: the one-batch Muon-MVR1 and the two-batch Muon-MVR2. We provide the first rigorous proof that, under \textbf{horizon-free} learning-rate schedules, variance reduction enables Muon-MVR2 to attain the optimal anytime convergence rate $\widetilde{\mathcal{O}}(T^{-1/3})$, matching the lower bound for this problem class. Under the Polyak--\L{}ojasiewicz (PL) condition, we establish anytime guarantees for Muon-MVR1 and Muon-MVR2: they attain best-iterate rates of $\widetilde{\mathcal{O}}(T^{-1/4})$ and $\widetilde{\mathcal{O}}(T^{-1/3})$ for the expected square-root suboptimality, and, given an additional uniform gradient bound along the iterates, achieve last-iterate rates of $\mathcal{O}(T^{-1/4})$ and $\mathcal{O}(T^{-1/3})$ for the objective gap, respectively. Experiments on CIFAR-10 and C4 support the practical effectiveness of the proposed variance-reduced Muon variants. Code is available at \href{https://github.com/MaeChd/MUON-MVR}{Muon-MVR} Codebase.
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UNVERDICTED 16representative citing papers
Spectral clipping of leading singular values in gradient matrices stabilizes SGD for non-convex problems with heavy-tailed noise and achieves the optimal convergence rate O(K^{(2-2α)/(3α-2)}).
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.
Convergence rates are derived for Muon-type methods with inexact LMO in the degenerate case under novel assumptions and layer-wise (L^0, L^1)-smoothness for non-convex and star-convex objectives with weight decay.
OptMuon combines orthogonalized momentum with trajectory-dependent AdaGrad-Norm adaptation to obtain expected-stationarity rates of order T^{-1/2} + sigma^{1/2}T^{-1/4} or T^{-1/2} + sigma^{1/3}T^{-1/3} that reduce to near-optimal deterministic first-order rates in the zero-noise regime.
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.
Muon achieves dimension-free saddle-point escape through non-linear spectral shaping, resolvent calculus, and structural incoherence, yielding an algebraically dimension-free escape bound.
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.
MuonEq introduces pre-orthogonalization equilibration schemes that improve Muon optimizer performance during large language model pretraining.
MGUP augments momentum optimizers with selective larger steps on a fixed proportion of parameters per iteration, claiming convergence guarantees for MGUP-AdamW and superior empirical performance on pretraining and fine-tuning.
Compression of LLMs often decouples accuracy from uncertainty, with larger models absorbing the effect better and inflation occurring in a threshold-like manner.
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.
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.
Compressed Gluon variants using unbiased/contraction compressors and SARAH-style variance reduction achieve convergence guarantees and lower communication costs in federated learning under layer-wise smoothness.
HTMuon modifies Muon to produce heavier-tailed updates and weight spectra via HT-SR theory, yielding up to 0.98 lower perplexity on LLaMA pretraining and serving as a plug-in for other Muon variants.
Derives finite-round upper-tail guarantee on population-empirical gap for client-sampled orthogonalized matrix momentum under heterogeneous data, with Lipschitz condition on the orthogonalizer.
citing papers explorer
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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.
-
Gradient Clipping Beyond Vector Norms: A Spectral Approach for Matrix-Valued Parameters
Spectral clipping of leading singular values in gradient matrices stabilizes SGD for non-convex problems with heavy-tailed noise and achieves the optimal convergence rate O(K^{(2-2α)/(3α-2)}).
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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 Analysis of Muon-type Methods with Inexact LMO in the Degenerate Case
Convergence rates are derived for Muon-type methods with inexact LMO in the degenerate case under novel assumptions and layer-wise (L^0, L^1)-smoothness for non-convex and star-convex objectives with weight decay.
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OptMuon: Closed-Loop Orthogonalized Momentum Methods for Stochastic Optimization with Zero-Noise Optimality
OptMuon combines orthogonalized momentum with trajectory-dependent AdaGrad-Norm adaptation to obtain expected-stationarity rates of order T^{-1/2} + sigma^{1/2}T^{-1/4} or T^{-1/2} + sigma^{1/3}T^{-1/3} that reduce to near-optimal deterministic first-order rates in the zero-noise regime.
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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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Dimension-Free Saddle-Point Escape in Muon
Muon achieves dimension-free saddle-point escape through non-linear spectral shaping, resolvent calculus, and structural incoherence, yielding an algebraically dimension-free escape bound.
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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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MuonEq: Balancing Before Orthogonalization with Lightweight Equilibration
MuonEq introduces pre-orthogonalization equilibration schemes that improve Muon optimizer performance during large language model pretraining.
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MGUP: A Momentum-Gradient Alignment Update Policy for Stochastic Optimization
MGUP augments momentum optimizers with selective larger steps on a fixed proportion of parameters per iteration, claiming convergence guarantees for MGUP-AdamW and superior empirical performance on pretraining and fine-tuning.
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Does Compression Preserve Uncertainty? A Unified Benchmark for Quantized and Sparse LLMs via Conformal Prediction
Compression of LLMs often decouples accuracy from uncertainty, with larger models absorbing the effect better and inflation occurring in a threshold-like manner.
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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.
-
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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Communication-Efficient Gluon in Federated Learning
Compressed Gluon variants using unbiased/contraction compressors and SARAH-style variance reduction achieve convergence guarantees and lower communication costs in federated learning under layer-wise smoothness.
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HTMuon: Improving Muon via Heavy-Tailed Spectral Correction
HTMuon modifies Muon to produce heavier-tailed updates and weight spectra via HT-SR theory, yielding up to 0.98 lower perplexity on LLaMA pretraining and serving as a plug-in for other Muon variants.
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A Note on Stability for Orthogonalized Matrix Momentum with Client Sampling
Derives finite-round upper-tail guarantee on population-empirical gap for client-sampled orthogonalized matrix momentum under heterogeneous data, with Lipschitz condition on the orthogonalizer.