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.