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This research proposes a derivative-free framework to construct Muon-style updates using structured finite differences, addressing limitations where gradient information is unavailable or unreliable. Experiments on matrix regression and noisy-gradient scenarios suggest that specific probing strategies can reduce function evaluations for black-box problems, though they do not offer general convergence guarantees over standard gradient methods. The work advances optimization techniques for neural networks by exploring trade-offs between update accuracy and computational cost in derivative-free settings.
Derivative-Free Structured Updates for Muon
arXiv:2609.17759v1 Announce Type: cross Abstract: Muon updates matrix-valued neural-network parameters by orthogonalizing a gradient-based momentum matrix. Its reliance on derivatives limits its use when gradients are unavailable or unreliable. We develop a derivative-free framework that constructs Muon-style updates from structured finite differences. Four variants are considered: full entrywise recovery, random low-rank surrogates, basis-aligned rank-one probing, and direct structured search. Exhaustive basis-aligned probing is equivalent, up to positive scaling before ideal polar orthogonalization, to coordinate finite differences. Matrix-regression experiments show that random rank-one probing can reduce the number of function evaluations substantially, at the cost of less accurate updates. Controlled noisy-gradient experiments on regression and a neural network illustrate when accurate function values can compensate for an unreliable gradient oracle. A small CartPole study further examines orthogonal rank-one probes under a fixed episode budget. These results support structured probing as a practical option for selected black-box problems; they do not establish a general convergence guarantee or an advantage over accurate, inexpensive gradients.
Abstract:Muon updates matrix-valued neural-network parameters by orthogonalizing a gradient-based momentum matrix. Its reliance on derivatives limits its use when gradients are unavailable or unreliable. We develop a derivative-free framework that constructs Muon-style updates from structured finite differences. Four variants are considered: full entrywise recovery, random low-rank surrogates, basis-aligned rank-one probing, and direct structured search. Exhaustive basis-aligned probing is equivalent, up to positive scaling before ideal polar orthogonalization, to coordinate finite differences. Matrix-regression experiments show that random rank-one probing can reduce the number of function evaluations substantially, at the cost of less accurate updates. Controlled noisy-gradient experiments on regression and a neural network illustrate when accurate function values can compensate for an unreliable gradient oracle. A small CartPole study further examines orthogonal rank-one probes under a fixed episode budget. These results support structured probing as a practical option for selected black-box problems; they do not establish a general convergence guarantee or an advantage over accurate, inexpensive gradients.
Optimization and Control (math.OC); Machine Learning (cs.LG)
arXiv:2609.17759 [math.OC]
(or arXiv:2609.17759v1 [math.OC] for this version)
https://doi.org/10.48550/arXiv.2609.17759 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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