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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Adamuon: Adaptive muon optimizer.arXiv preprint arXiv:2507.11005
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Muon outperforms Adam by reducing curvature penalty via lower Normalized Directional Sharpness, as shown via Taylor approximation on LLM training and proven on stylized quadratic problems with heterogeneous curvature.
AMUSE is a new optimizer integrating Muon orthogonalization with Schedule-Free averaging via adaptive interpolation for schedule-free anytime training that improves Pareto frontiers on vision and LLM tasks.
Pion modifies Muon's Newton-Schulz iterations into a controllable high-pass filter that anchors dominant singular values at 1 while suppressing noisy tails, outperforming Muon and AdamW in VLA and RLVR regimes.
Muon succeeds by guaranteeing local step-size optimality rather than by tracking any ideal global geometry, as random-spectrum and quasi-norm variants match its performance on language models.
LMO-IGT achieves O(ε^{-3.5}) iteration complexity for stochastic LMO optimization via implicit gradient transport with a single gradient per step and introduces the regularized support function as a unified stationarity measure.
A unified stochastic convergence theory is developed for adaptive preconditioned first-order methods including AdaGrad variants, Shampoo, and Muon in nonconvex optimization.
Muon-MVR2 attains the optimal anytime convergence rate of ~O(T^{-1/3}) in stochastic non-convex settings under horizon-free schedules.
One-step gradient delay is optimizer-dependent rather than intrinsically unstable, with Muon and error-feedback correction enabling async pipeline parallelism to match synchronous performance on models up to 10B parameters.
Aurora is a leverage-aware spectral optimizer that enforces uniform row norms in matrix updates while preserving Muon's polar geometry, outperforming Muon and achieving SOTA among spectral methods on modded-nanoGPT.
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.
Double preconditioning (DoPr) improves downstream task performance in test-time feedback settings without consistent gains in validation loss.
Introduces a class of asynchronous adaptive first-order methods and establishes O(1/sqrt t) convergence (up to logs) for non-convex stochastic optimization under reasonable assumptions.
MONA integrates Nesterov acceleration into Muon's orthogonalization framework, reporting better convergence than Muon and AdamW on MoE models up to 68B parameters trained on 1T tokens and SOTA fine-tuning results.
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 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.
OrScale adds a Frobenius-norm trust-ratio layer-wise scaler to Muon’s orthogonalized updates, with per-layer calibration for language models, yielding higher CIFAR-10 accuracy and better language-model pre-training loss than Muon+Moonlight and AdamW.
PolarAdamW disentangles spectral control from gauge-equivariance in matrix optimizers, with experiments demonstrating their distinct roles on standard versus symmetry-aware neural networks.
Parcae stabilizes looped LLMs via spectral norm constraints on injection parameters, enabling power-law scaling for training FLOPs and saturating exponential scaling at test time that improves quality over fixed-depth baselines under fixed parameter budgets.
MuonEq introduces pre-orthogonalization equilibration schemes that improve Muon optimizer performance during large language model pretraining.
Entry-wise clipping achieves spectral control of gradients via localization under heavy-tailed contamination, with O(ε^{-4}) convergence and empirical savings on NanoGPT pretraining.
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 low-rank matrix recovery.
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.
RMNP preconditions matrix updates via row-wise L2 normalization instead of Newton-Schulz iteration, reducing complexity to O(mn) while matching Muon's non-convex convergence rate and empirical performance.
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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.