Zeroth-Order Stochastic Block Coordinate Type Methods for Nonconvex Optimization

Abstract: We study (constrained) nonconvex (composite) optimization problems where the decision variables vector can be split into blocks of variables. Random block projection is a popular technique to handle this kind of problem for its remarkable reduction of the computational cost from the projection. However, this powerful method has not been proposed for the situation that first-order information is prohibited and only zeroth-order information is available. In this paper, we propose to develop different classes of zeroth-order stochastic block coordinate type methods. Zeroth-order block coordinate descent (ZS-BCD) is proposed for solving unconstrained nonconvex optimization problem. For composite optimization, we establish the zeroth-order stochastic block mirror descent (ZS-BMD) and its associated two-phase method to achieve the complexity bound for finding $(\epsilon,\Lambda)$-solution. Furthermore, we also establish zeroth-order stochastic block coordinate conditional gradient (ZS-BCCG) method for nonconvex (composite) optimization. By implementing ZS-BCCG method, in each iteration, only (approximate) linear programming subproblem needs to be solved on a random block instead of a rather costly projection subproblem on the whole decision space, in contrast to the existing traditional stochastic approximation methods. In what follows, an approximate ZS-BCCG method and corresponding two-phase ZS-BCCG method are proposed. This is also the first time that a two-phase BCCG method has been developed to achieve the $(\epsilon, \Lambda)$-solution of nonconvex composite optimization problem. To the best of our knowledge, the proposed results in this paper are new in stochastic nonconvex (composite) optimization literature.