# Constrained Stochastic Spectral Preconditioning Converges for Nonconvex Objectives

arXiv: https://arxiv.org/abs/2605.11850 · HTML: https://mlsanity.com/paper/2605.11850/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 50 of its 68 references that are in the map.
- First posted on arXiv 12 May 2026.

## Builds on (most cited first, 30 of 50)

- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets](https://mlsanity.com/paper/2201.02177/) (2022)
- [signSGD: Compressed Optimisation for Non-Convex Problems](https://mlsanity.com/paper/1802.04434/) (2018)
- [Symbolic Discovery of Optimization Algorithms](https://mlsanity.com/paper/2302.06675/) (2023)
- [Shampoo: Preconditioned Stochastic Tensor Optimization](https://mlsanity.com/paper/1802.09568/) (2018)
- [Analyzing and Improving the Training Dynamics of Diffusion Models](https://mlsanity.com/paper/2312.02696/) (2023)
- [Why gradient clipping accelerates training: A theoretical justification for adaptivity](https://mlsanity.com/paper/1905.11881/) (2019)
- [Momentum-Based Variance Reduction in Non-Convex SGD](https://mlsanity.com/paper/1905.10018/) (2019)
- [Training Deep Learning Models with Norm-Constrained LMOs](https://mlsanity.com/paper/2502.07529/) (2025)
- [Momentum Improves Normalized SGD](https://mlsanity.com/paper/2002.03305/) (2020)
- [The Polar Express: Optimal Matrix Sign Methods and Their Application to the Muon Algorithm](https://mlsanity.com/paper/2505.16932/) (2025)
- [Fantastic Pretraining Optimizers and Where to Find Them](https://mlsanity.com/paper/2509.02046/) (2025)
- [Understanding Gradient Orthogonalization for Deep Learning via Non-Euclidean Trust-Region Optimization](https://mlsanity.com/paper/2503.12645/) (2025)
- [Improved Analysis of Clipping Algorithms for Non-convex Optimization](https://mlsanity.com/paper/2010.02519/) (2020)
- [Muon Optimizes Under Spectral Norm Constraints](https://mlsanity.com/paper/2506.15054/) (2025)
- [Revisiting Gradient Clipping: Stochastic bias and tight convergence guarantees](https://mlsanity.com/paper/2305.01588/) (2023)
- [High-probability Bounds for Non-Convex Stochastic Optimization with Heavy Tails](https://mlsanity.com/paper/2106.14343/) (2021)
- [Gluon: Making Muon & Scion Great Again! (Bridging Theory and Practice of LMO-based Optimizers for LLMs)](https://mlsanity.com/paper/2505.13416/) (2025)
- [nGPT: Normalized Transformer with Representation Learning on the Hypersphere](https://mlsanity.com/paper/2410.01131/) (2024)
- [Dion: Distributed Orthonormalized Updates](https://mlsanity.com/paper/2504.05295/) (2025)
- [MARS: Unleashing the Power of Variance Reduction for Training Large Models](https://mlsanity.com/paper/2411.10438/) (2024)
- [Convergence of AdaGrad for Non-convex Objectives: Simple Proofs and Relaxed Assumptions](https://mlsanity.com/paper/2305.18471/) (2023)
- [PolarGrad: A Class of Matrix-Gradient Optimizers from a Unifying Preconditioning Perspective](https://mlsanity.com/paper/2505.21799/) (2025)
- [Lion Secretly Solves Constrained Optimization: As Lyapunov Predicts](https://mlsanity.com/paper/2310.05898/) (2023)
- [Preconditioned Stochastic Gradient Descent](https://mlsanity.com/paper/1512.04202/) (2015)
- [From Gradient Clipping to Normalization for Heavy Tailed SGD](https://mlsanity.com/paper/2410.13849/) (2024)
- [Convex and Non-convex Optimization Under Generalized Smoothness](https://mlsanity.com/paper/2306.01264/) (2023)
- [Nonconvex Stochastic Optimization under Heavy-Tailed Noises: Optimal Convergence without Gradient Clipping](https://mlsanity.com/paper/2412.19529/) (2024)
- [Lions and Muons: Optimization via Stochastic Frank-Wolfe under Heavy-Tailed Noise](https://mlsanity.com/paper/2506.04192/) (2025)
- [Generalized-Smooth Nonconvex Optimization is As Efficient As Smooth Nonconvex Optimization](https://mlsanity.com/paper/2303.02854/) (2023)

## Abstract (arXiv)

In this work, we develop proximal preconditioned gradient methods with a focus on spectral gradient methods providing a proximal extension to the Muon and Scion optimizers. We introduce a family of stochastic algorithms that can handle a wide variety of convex and nonconvex constraints and study its convergence under heavy-tailed noise, through a novel analysis tailored to the geometry of the proposed methods. We further propose a variance-reduced version, which achieves faster convergence under standard noise assumptions. Finally, we show that the polynomial iterations used in Muon are more accurately captured by a nonlinear preconditioner than by the ideal matrix sign, leading to a convergence analysis that more faithfully reflects practical implementations.