Regularized Muon induces a damped Hamiltonian flow on probability measures over matrix parameters, yielding exponential convergence under gradient dominance assumptions.
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LOSCAR-SGD combines local updates, sparse model averaging, and communication-computation overlap with a delay-corrected merge rule, providing convergence rates for smooth non-convex objectives under worker heterogeneity.
Ringmaster LMO extends delay-thresholding from ASGD to LMO-based momentum updates, providing convergence guarantees under (L0, L1)-smoothness and time-complexity bounds that recover optimal rates in the Euclidean case.
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.
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.
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.
Muon achieves dimension-free saddle-point escape through non-linear spectral shaping, resolvent calculus, and structural incoherence, yielding an algebraically dimension-free escape bound.
Proving stability of Leon's preconditioner enables the first tuning-free Nesterov-accelerated projection-free adaptive SGD variant with improved non-smooth non-convex rates.
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.
Proximal stochastic spectral preconditioning converges for nonconvex constrained objectives under heavy-tailed noise, with a variance-reduced version achieving faster rates and a refined analysis of Muon iterations.
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.
Proposes low-rank orthogonalization and derives low-rank Muon and MSGD variants that outperform standard Muon on GPT-2 and LLaMA pretraining while providing iteration complexity bounds.
Convergence analysis shows Muon outperforms gradient descent by exploiting low-rank structure in neural network Hessians.
Constraining fine-tuning updates with LoRA mitigates performance degradation when switching from Adam to Muon on pretrained models.
Muon+ adds one normalization step after polar orthogonalization in the Muon optimizer, yielding lower training and validation perplexity and faster pre-training across 60M-7B models.
citing papers explorer
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Move on Muon : A Hamiltonian probability gradient flow perspective of Muon optimizer
Regularized Muon induces a damped Hamiltonian flow on probability measures over matrix parameters, yielding exponential convergence under gradient dominance assumptions.
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LOSCAR-SGD: Local SGD with Communication-Computation Overlap and Delay-Corrected Sparse Model Averaging
LOSCAR-SGD combines local updates, sparse model averaging, and communication-computation overlap with a delay-corrected merge rule, providing convergence rates for smooth non-convex objectives under worker heterogeneity.
-
Ringmaster LMO: Asynchronous Linear Minimization Oracle Momentum Method
Ringmaster LMO extends delay-thresholding from ASGD to LMO-based momentum updates, providing convergence guarantees under (L0, L1)-smoothness and time-complexity bounds that recover optimal rates in the Euclidean case.
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Muon is Not That Special: Random or Inverted Spectra Work Just as Well
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.
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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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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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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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Optimal Projection-Free Adaptive SGD for Matrix Optimization
Proving stability of Leon's preconditioner enables the first tuning-free Nesterov-accelerated projection-free adaptive SGD variant with improved non-smooth non-convex rates.
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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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Constrained Stochastic Spectral Preconditioning Converges for Nonconvex Objectives
Proximal stochastic spectral preconditioning converges for nonconvex constrained objectives under heavy-tailed noise, with a variance-reduced version achieving faster rates and a refined analysis of Muon iterations.
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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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Low-rank Orthogonalization for Large-scale Matrix Optimization with Applications to Foundation Model Training
Proposes low-rank orthogonalization and derives low-rank Muon and MSGD variants that outperform standard Muon on GPT-2 and LLaMA pretraining while providing iteration complexity bounds.
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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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Can Muon Fine-tune Adam-Pretrained Models?
Constraining fine-tuning updates with LoRA mitigates performance degradation when switching from Adam to Muon on pretrained models.
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MUON+: Towards More Effective Muon via One Additional Normalization Step for LLM Pre-training
Muon+ adds one normalization step after polar orthogonalization in the Muon optimizer, yielding lower training and validation perplexity and faster pre-training across 60M-7B models.
- Symmetry-Compatible Principle for Optimizer Design: Embeddings, LM Heads, SwiGLU MLPs, and MoE Routers