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Finite-difference methods are a class of algorithms designed to solveblack-box optimization problems by approximating a gradient of the targetfunction on a set of directions. In black-box optimization, the non-smoothsetting is particularly relevant since, in practice, differentiability andsmoothness assumptions cannot be verified. To cope with nonsmoothness, severalauthors use a smooth approximation of the target function and show that finitedifference methods approximate its gradient. Recently, it has been proved thatimposing a structure in the directions allows improving performance. However,only the smooth setting was considered. To close this gap, we introduce andanalyze O-ZD, the first structured finite-difference algorithm for non-smoothblack-box optimization. Our method exploits a smooth approximation of thetarget function and we prove that it approximates its gradient on a subset ofrandom {\em orthogonal} directions. We analyze the convergence of O-ZD underdifferent assumptions. For non-smooth convex functions, we obtain the optimalcomplexity. In the non-smooth non-convex setting, we characterize the number ofiterations needed to bound the expected norm of the smoothed gradient. Forsmooth functions, our analysis recovers existing results for structuredzeroth-order methods for the convex case and extends them to the non-convexsetting. We conclude with numerical simulations where assumptions aresatisfied, observing that our algorithm has very good practical performances.

An Optimal Structured Zeroth-order Algorithm for Non-smooth Optimization