A stochastic Bregman golden ratio algorithm for non-Lipschitz stochastic mixed variational inequalities with application to resource share problems

Abstract

In the study of stochastic mixed variational inequalities(SMVIs), Lipschitz is an indispensable assumption for the convergence analysis. However, practical applications may not satisfy this assumption. In this paper, we propose a stochastic Bregman golden ratio algorithm for solving non-Lipschitz SMVIs. Since our algorithm only requires to calculate one stochastic approximation of the expected mapping per iteration, the computations can be reduced. Under some moderate conditions, we prove the almost surely convergence of the iteration sequence and the $O(1/K)$ convergence rate, where $K$ denotes the maximum iteration. Furthermore, we derive the probabilities of large deviation results, which provide a high probability guarantee for the convergence of the proposed algorithm. Numerical experiments on Logistic regression problems and modified entropy regularized LP boosting problems show that our algorithm is competitive compared with some existing algorithms. Finally, we apply our algorithm to solve a non-Lipschitz resource sharing problem.