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We study a stochastic first order primal-dual method for solvingconvex-concave saddle point problems over real reflexive Banach spaces usingBregman divergences and relative smoothness assumptions, in which we allow forstochastic error in the computation of gradient terms within the algorithm. Weshow ergodic convergence in expectation of the Lagrangian optimality gap with arate of O(1/k) and that every almost sure weak cluster point of the ergodicsequence is a saddle point in expectation under mild assumptions. Underslightly stricter assumptions, we show almost sure weak convergence of thepointwise iterates to a saddle point. Under a relative strong convexityassumption on the objective functions and a total convexity assumption on theentropies of the Bregman divergences, we establish almost sure strongconvergence of the pointwise iterates to a saddle point. Our framework isgeneral and does not need strong convexity of the entropies inducing theBregman divergences in the algorithm. Numerical applications are consideredincluding entropically regularized Wasserstein barycenter problems andregularized inverse problems on the simplex.

A Stochastic Bregman Primal-Dual Splitting Algorithm for Composite Optimization