Zeroth-order optimization with orthogonal random directions

We propose and analyze a randomized zeroth-order optimization method based on approximating the exact gradient by finite differences computed in a set of orthogonal random directions that changes with each iteration. A number of previously proposed methods are recovered as special cases including spherical smoothing, coordinate descent, as well as discretized gradient descent. Our main contribution is proving convergence guarantees as well as convergence rates under different parameter choices and assumptions. In particular, we consider convex objectives, but also possibly non-convex objectives satisfying the Polyak-Lojasiewicz (PL) condition. Theoretical results are complemented and illustrated by numerical experiments.