  We study a stochastic first order primal-dual method for solving
convex-concave saddle point problems over real reflexive Banach spaces using
Bregman divergences and relative smoothness assumptions, in which we allow for
stochastic error in the computation of gradient terms within the algorithm. We
show ergodic convergence in expectation of the Lagrangian optimality gap with a
rate of O(1/k) and that every almost sure weak cluster point of the ergodic
sequence is a saddle point in expectation under mild assumptions. Under
slightly stricter assumptions, we show almost sure weak convergence of the
pointwise iterates to a saddle point. Under a relative strong convexity
assumption on the objective functions and a total convexity assumption on the
entropies of the Bregman divergences, we establish almost sure strong
convergence of the pointwise iterates to a saddle point. Our framework is
general and does not need strong convexity of the entropies inducing the
Bregman divergences in the algorithm. Numerical applications are considered
including entropically regularized Wasserstein barycenter problems and
regularized inverse problems on the simplex.
