Stochastic Adversarial Noise in the "Black Box" Optimization Problem

Abstract: This paper is devoted to the study of the solution of a stochastic convex black box optimization problem. Where the black box problem means that the gradient-free oracle only returns the value of objective function, not its gradient. We consider non-smooth and smooth setting of the solution to the black box problem under adversarial stochastic noise. For two techniques creating gradient-free methods: smoothing schemes via $L_1$ and $L_2$ randomizations, we find the maximum allowable level of adversarial stochastic noise that guarantees convergence. Finally, we analyze the convergence behavior of the algorithms under the condition of a large value of noise level.