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Zeroth-order (ZO) optimization is a subset of gradient-free optimization that emerges in many signal processing and machine learning (ML) applications. It is used for solving optimization problems similarly to gradient-based methods. However, it does not require the gradient, using only function evaluations. Specifically, ZO optimization iteratively performs three major steps: gradient estimation, descent direction computation, and the solution update. In this article, we provide a comprehensive review of ZO optimization, with an emphasis on showing the underlying intuition, optimization principles, and recent advances in convergence analysis.

Many contemporary applications in signal processing and machine learning give rise to structured nonconvex nonsmooth optimization problems that can often be tackled by simple iterative methods quite effectively. One of the keys to understanding such a phenomenon-and, in fact, a very difficult conundrum even for experts-lies in the study of "stationary points" of the problem in question. Unlike smooth optimization, for which the definition of a stationary point is rather standard, there are myriad definitions of stationarity in nonsmooth optimization.