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

TitleAn Optimal Structured Zeroth-order Algorithm for Non-smooth Optimization
Publication TypeConference Paper
Year of Publication2023
AuthorsRando, M, Molinari, C, Villa, S, Rosasco, L
Conference Name37th Conference on Neural Information Processing Systems (NeurIPS 2023)
Date Published12/2023

Finite-difference methods are a class of algorithms designed to solve black-box optimization problems by approximating a gradient of the target function on a set of directions. In black-box optimization, the non-smooth setting is particularly relevant since, in practice, differentiability and smoothness assumptions cannot be verified. To cope with nonsmoothness, several authors use a smooth approximation of the target function and show that finite difference methods approximate its gradient. Recently, it has been proved that imposing a structure in the directions allows improving performance. However, only the smooth setting was considered. To close this gap, we introduce and analyze O-ZD, the first structured finite- difference algorithm for non-smooth black-box optimization. Our method exploits a smooth approximation of the target function and we prove that it approximates its gradient on a subset of random orthogonal directions. We analyze the convergence of O-ZD under different assumptions. For non-smooth convex functions, we obtain the optimal complexity. In the non-smooth non-convex setting, we characterize the number of iterations needed to bound the expected norm of the smoothed gradient. For smooth functions, our analysis recovers existing results for structured zeroth-order methods for the convex case and extends them to the non-convex setting. We conclude with numerical simulations where assumptions are satisfied, observing that our algorithm has very good practical performances.


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