Gradient Free Methods for Non-Smooth Convex Optimization with Heavy Tails on Convex Compact

Optimization problems, in which only the realization of a function or a zeroth-order oracle is available, have many applications in practice. An effective method for solving such problems is the approximation of the gradient using sampling and finite differences of the function values. However, some noise can be present in the zeroth-order oracle not allowing the exact evaluation of the function value, and this noise can be stochastic or adversarial. In this paper, we propose and study new easy-to-implement algorithms that are optimal in terms of the number of oracle calls for solving non-smooth optimization problems on a convex compact set with heavy-tailed stochastic noise (random noise has $(1+\kappa)$-th bounded moment) and adversarial noise. The first algorithm is based on the heavy-tail-resistant mirror descent and uses special transformation functions that allow controlling the tails of the noise distribution. The second algorithm is based on the gradient clipping technique. The paper provides proof of algorithms' convergence results in terms of high probability and in terms of expectation when a convex function is minimized. For functions satisfying a $r$-growth condition, a faster algorithm is proposed using the restart technique. Particular attention is paid to the question of how large the adversarial noise can be so that the optimality and convergence of the algorithms is guaranteed.