RNC-LM extends geodesic-accelerated Levenberg-Marquardt to arbitrary-order Riemann normal coordinate corrections, reusing the LM matrix factorization for all orders and achieving large speedups on PINN and potential-fitting benchmarks.
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Old Optimizer, New Norm: An Anthology
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abstract
Deep learning optimizers are often motivated through a mix of convex and approximate second-order theory. We select three such methods -- Adam, Shampoo and Prodigy -- and argue that each method can instead be understood as a squarely first-order method without convexity assumptions. In fact, after switching off exponential moving averages, each method is equivalent to steepest descent under a particular norm. By generalizing this observation, we chart a new design space for training algorithms. Different operator norms should be assigned to different tensors based on the role that the tensor plays within the network. For example, while linear and embedding layers may have the same weight space of $\mathbb{R}^{m\times n}$, these layers play different roles and should be assigned different norms. We hope that this idea of carefully metrizing the neural architecture might lead to more stable, scalable and indeed faster training.
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representative citing papers
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.
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.
On power-law covariance least squares problems, SignSVD (Muon) and SignSGD (Adam proxy) show three phases of relative performance depending on data exponent α and target exponent β.
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.
Steepest descent under divergence-induced quadratic models equals an LQR problem, enabling learning of diagonal or Kronecker-factored inverse preconditioners via a global layerwise objective for scalable geometry-aware training.
A unified stochastic convergence theory is developed for adaptive preconditioned first-order methods including AdaGrad variants, Shampoo, and Muon in nonconvex optimization.
Muon achieves higher storage capacity than SGD and matches Newton's method in one-step recovery rates for associative memory under power-law distributions, while saturating at larger critical batch sizes and showing faster initial multi-step dynamics.
Muon-MVR2 attains the optimal anytime convergence rate of ~O(T^{-1/3}) in stochastic non-convex settings under horizon-free schedules.
Scion is a new stochastic LMO-based optimizer family that unifies existing methods, supports unconstrained problems, and delivers hyperparameter transferability plus speedups on nanoGPT training.
Optimizer choice during LLM fine-tuning produces up to 7x variation in emergent misalignment rates, with spectral regularization on LoRA adapters substantially mitigating misalignment for prone optimizers.
Muon in matrix factorization avoids saddle-to-saddle dynamics, learns top modes simultaneously, conserves sqrt(P^TP) - sqrt(Q^TQ), and reaches balanced solutions from small initialization with a two-step alignment schedule.
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.
LoRA-Muon applies Muon's spectral steepest descent to low-rank factors with split weight decay, acting as a transferable proxy for full-rank Muon and Shampoo optimizers.
Feedback alignment in deep networks is limited by low-rank error signals; orthogonal weight updates and activity normalization raise effective rank and boost performance.
FOGO introduces spectral orthogonalization of momentum updates plus a random-projection codebook memory to detect and correct gradient interference, improving convergence and retention over Adam and Muon on imbalanced, continual, and large-model tasks.
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.
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.
SGD is reformulated via a master equation from discrete updates, producing a discrete Fokker-Planck equation that predicts non-stationary variance growth proportional to learning rate in flat Hessian directions.
The same Transformer architecture follows different spectral scaling laws under different optimizers, with Muon achieving linear hard-rank scaling on tail representations while AdamW shows weak scaling, even when perplexity is matched.
Introduces Distance-Adaptive Muon, Scale-Calibrated Muon, and Distance-Free Muon with stationarity and O(1/T) objective-gap guarantees, shown to match or improve fixed-scale Muon on GPT-124M and ViT-Tiny models.
Rescaled ASGD recovers convergence to the true global objective by rescaling worker stepsizes proportional to computation times, matching the known time lower bound in the leading term under non-convex smoothness and bounded heterogeneity.
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.
citing papers explorer
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Higher-Order Geometric Updates for Levenberg-Marquardt Method via Riemann Normal Coordinates
RNC-LM extends geodesic-accelerated Levenberg-Marquardt to arbitrary-order Riemann normal coordinate corrections, reusing the LM matrix factorization for all orders and achieving large speedups on PINN and potential-fitting benchmarks.
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Why Muon Outperforms Adam: A Curvature Perspective
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.
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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.
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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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Phases of Muon: When Muon Eclipses SignSGD
On power-law covariance least squares problems, SignSVD (Muon) and SignSGD (Adam proxy) show three phases of relative performance depending on data exponent α and target exponent β.
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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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Layerwise LQR for Geometry-Aware Optimization of Deep Networks
Steepest descent under divergence-induced quadratic models equals an LQR problem, enabling learning of diagonal or Kronecker-factored inverse preconditioners via a global layerwise objective for scalable geometry-aware training.
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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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Sharp Capacity Scaling of Spectral Optimizers in Learning Associative Memory
Muon achieves higher storage capacity than SGD and matches Newton's method in one-step recovery rates for associative memory under power-law distributions, while saturating at larger critical batch sizes and showing faster initial multi-step dynamics.
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On the Convergence of Muon and Beyond
Muon-MVR2 attains the optimal anytime convergence rate of ~O(T^{-1/3}) in stochastic non-convex settings under horizon-free schedules.
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Training Deep Learning Models with Norm-Constrained LMOs
Scion is a new stochastic LMO-based optimizer family that unifies existing methods, supports unconstrained problems, and delivers hyperparameter transferability plus speedups on nanoGPT training.
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Evil Spectra: How Optimisers can Amplify or Suppress Emergent Misalignment
Optimizer choice during LLM fine-tuning produces up to 7x variation in emergent misalignment rates, with spectral regularization on LoRA adapters substantially mitigating misalignment for prone optimizers.
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Muon learns balanced solutions in matrix factorization without slow saddle-to-saddle dynamics
Muon in matrix factorization avoids saddle-to-saddle dynamics, learns top modes simultaneously, conserves sqrt(P^TP) - sqrt(Q^TQ), and reaches balanced solutions from small initialization with a two-step alignment schedule.
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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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LoRA-Muon: Spectral Steepest Descent on the Low-Rank Manifold
LoRA-Muon applies Muon's spectral steepest descent to low-rank factors with split weight decay, acting as a transferable proxy for full-rank Muon and Shampoo optimizers.
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Overcoming Rank Collapse in Feedback Alignment
Feedback alignment in deep networks is limited by low-rank error signals; orthogonal weight updates and activity normalization raise effective rank and boost performance.
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FOGO: Forgetting-aware Orthogonalization Optimizer
FOGO introduces spectral orthogonalization of momentum updates plus a random-projection codebook memory to detect and correct gradient interference, improving convergence and retention over Adam and Muon on imbalanced, continual, and large-model tasks.
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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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Stochastic convergence of parallel asynchronous adaptive first-order methods
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.
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Why SGD is not Brownian Motion: A New Perspective on Stochastic Dynamics
SGD is reformulated via a master equation from discrete updates, producing a discrete Fokker-Planck equation that predicts non-stationary variance growth proportional to learning rate in flat Hessian directions.
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Same Architecture, Different Capacity: Optimizer-Induced Spectral Scaling Laws
The same Transformer architecture follows different spectral scaling laws under different optimizers, with Muon achieving linear hard-rank scaling on tail representations while AdamW shows weak scaling, even when perplexity is matched.
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Distance-Aware Muon: Adaptive Step Scaling for Normalized Optimization
Introduces Distance-Adaptive Muon, Scale-Calibrated Muon, and Distance-Free Muon with stationarity and O(1/T) objective-gap guarantees, shown to match or improve fixed-scale Muon on GPT-124M and ViT-Tiny models.
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Rescaled Asynchronous SGD: Optimal Distributed Optimization under Data and System Heterogeneity
Rescaled ASGD recovers convergence to the true global objective by rescaling worker stepsizes proportional to computation times, matching the known time lower bound in the leading term under non-convex smoothness and bounded heterogeneity.
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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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MuonQ: Enhancing Low-Bit Muon Quantization via Directional Fidelity Optimization
A 4-bit quantization scheme for the Muon optimizer — combining pre-quantization normalization, singular-structure-aligned decomposition, and µ-law companding — recovers most full-precision training quality with up to 7.3x optimizer-state memory savings.
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Optimistic Dual Averaging Unifies Modern Optimizers
SODA unifies several modern optimizers under optimistic dual averaging and supplies a 1/k decay wrapper that improves performance without weight decay tuning.
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PolarAdamW: Disentangling Spectral Control and Schur Gauge-Equivariance in Matrix Optimisation
PolarAdamW disentangles spectral control from gauge-equivariance in matrix optimizers, with experiments demonstrating their distinct roles on standard versus symmetry-aware neural networks.
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Orth-Dion: Eliminating Geometric Mismatch in Distributed Low-Rank Spectral Optimization
Orth-Dion uses QR factorization on the right factor instead of column normalization to eliminate the geometric mismatch in low-rank approximations of spectral optimizers like Muon, achieving O(sqrt(L_r/T)) rate under non-Euclidean smoothness.
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Optimizer-Model Consistency: Full Finetuning with the Same Optimizer as Pretraining Forgets Less
Full finetuning with the pretraining optimizer reduces forgetting compared to other optimizers or LoRA while achieving comparable new-task performance.
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Demystifying Manifold Constraints in LLM Pre-training
Manifold constraints via the new MACRO optimizer independently bound activation scales and enforce rotational equilibrium in LLM pre-training, subsuming RMS normalization and decoupled weight decay while delivering competitive performance with convergence guarantees.
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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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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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Memory-Efficient LLM Pretraining via Minimalist Optimizer Design
SCALE matches Adam performance in LLM pretraining from 60M to 7B parameters by combining column-wise gradient normalization with last-layer-only momentum, using 35-45% of Adam's memory.
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On the Convergence Analysis of Muon
Muon's convergence rate depends on an average Hessian curvature along its update directions, which can be much smaller than the worst-case Lipschitz constant when Hessians are low-rank.
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Improving Neural Network Training by Decoupling the Magnitude and Direction of Weight Vectors
Splitting weight matrices into a fixed-norm direction and learnable per-row/column magnitudes improves LLM training over AdamW/Muon, removes weight decay and warmup, and transfers the optimal LR across width.
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Zero-order Parameter-free Optimization for LMO-based Methods: Novel Approach for Efficient Fine-tuning
AdaNAGED combines zeroth-order gradient-free training, automatic parameter adaptation, and LMO-based non-Euclidean geometry with claimed convergence guarantees, demonstrated on OPT-1.3B fine-tuning.
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Zeta: Dual Whitening for Matrix Optimization via Coordinate-Adaptive Preconditioning
Zeta applies coordinate whitening followed by spectral whitening in a fixed order to reduce orthogonalization error in matrix optimization for neural networks.
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Muon Learns More Robust and Transferable Features than Adam
Muon learns more robust and transferable features than Adam and SGD, shown via corruption robustness tests, transfer experiments, layer-wise probes, effective rank measurements, and a theoretical proof on margins in a multi-component classification problem.
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Softsign: Smooth Sign in Your Optimizer For Better Parameter Heterogeneity Handling
SoftSignum replaces hard sign with soft-sign in optimizers via temperature control and quantile scheduling, extends to SoftMuon, provides a convergence proof for stochastic non-convex settings, and reports better performance than sign-based methods and AdamW on deep learning tasks.
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Can Entry-Wise Clipping Give Spectral Control of Stochastic Gradients?
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.
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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 low-rank matrix recovery.
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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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Pion: A Spectrum-Preserving Optimizer via Orthogonal Equivalence Transformation
Pion is an optimizer that preserves the singular values of weight matrices in LLM training by applying orthogonal equivalence transformations.
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Flatness and Gradient Alignment Are Both Necessary: Spectral-Aware Gradient-Aligned Exploration for Multi-Distribution Learning
SAGE, an optimizer combining spectral polar-factor perturbation with gradient-agreement-scaled noise, reports 78.9% average on DomainBed.
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Representation Before Training: A Fixed-Budget Benchmark for Generative Medical Event Models
Fused code-value tokenization improves mortality AUROC from 0.891 to 0.915 and other clinical outcome predictions, while certain temporal encodings like event order match or exceed time tokens with shorter sequences.
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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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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.
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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 as a Residual Connection
Muon is interpreted as an implicit residual connection that sacrifices local gradient fidelity to improve downstream layer usability in neural network training.