Stochastic Zeroth order Descent with Structured Directions

We introduce and analyze Structured Stochastic Zeroth order Descent (S-SZD), a finite difference approach which approximates a stochastic gradient on a set of $l\leq d$ orthogonal directions, where $d$ is the dimension of the ambient space. These directions are randomly chosen, and may change at each step. For smooth convex functions we prove almost sure convergence of the iterates and a convergence rate on the function values of the form $O(d/l k^{-c})$ for every $c<1/2$, which is arbitrarily close to the one of Stochastic Gradient Descent (SGD) in terms of number of iterations. Our bound also shows the benefits of using $l$ multiple directions instead of one. For non-convex functions satisfying the Polyak-{\L}ojasiewicz condition, we establish the first convergence rates for stochastic zeroth order algorithms under such an assumption. We corroborate our theoretical findings in numerical simulations where assumptions are satisfied and on the real-world problem of hyper-parameter optimization, observing that S-SZD has very good practical performances.