Sequential stochastic blackbox optimization with zeroth-order gradient estimators

Abstract: This work considers stochastic optimization problems in which the objective function values can only be computed by a blackbox corrupted by some random noise following an unknown distribution. The proposed method is based on sequential stochastic optimization (SSO): the original problem is decomposed into a sequence of subproblems. Each of these subproblems is solved using a zeroth order version of a sign stochastic gradient descent with momentum algorithm (ZO-Signum) and with an increasingly fine precision. This decomposition allows a good exploration of the space while maintaining the efficiency of the algorithm once it gets close to the solution. Under Lipschitz continuity assumption on the blackbox, a convergence rate in expectation is derived for the ZO-signum algorithm. Moreover, if the blackbox is smooth and convex or locally convex around its minima, a convergence rate to an $\epsilon$-optimal point of the problem may be obtained for the SSO algorithm. Numerical experiments are conducted to compare the SSO algorithm with other state-of-the-art algorithms and to demonstrate its competitiveness.