A Stochastic Bregman Primal-Dual Splitting Algorithm for Composite Optimization
This work provides a theoretical framework for stochastic optimization in machine learning and inverse problems, but it is incremental as it extends existing methods with Bregman divergences and stochastic errors.
The authors tackled composite optimization problems in Banach spaces by developing a stochastic primal-dual algorithm using Bregman divergences, achieving an O(1/k) convergence rate for the Lagrangian optimality gap and proving convergence to saddle points under various assumptions.
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.