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arXiv Machine Learning By Pengcheng Xie

Derivative-Free Structured Updates for Muon

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The paper introduces a derivative‑free framework for Muon‑style updates, replacing gradient‑based momentum with structured finite differences. Four variants—full entrywise recovery, random low‑rank surrogates, basis‑aligned rank‑one probing, and direct structured search—are explored, with basis‑aligned probing shown to be equivalent to coordinate finite differences up to scaling. Experiments on matrix regression, noisy‑gradient regression, a neural network, and a CartPole task demonstrate that random rank‑one probing can significantly reduce function evaluations, though at the expense of update accuracy, and that accurate function values can sometimes offset unreliable gradient oracles.

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