# A Fully Stochastic Primal-Dual Algorithm

## TL;DR

This paper introduces a novel stochastic primal-dual algorithm designed for composite optimization problems where functions are given as unknown statistical expectations, with proven convergence to a saddle point.

## Contribution

It presents a fully stochastic primal-dual method with convergence guarantees, extending the stochastic Forward Backward algorithm to new composite optimization settings.

## Key findings

- Proven convergence to saddle points under stochastic conditions
- Applicable to convex optimization with stochastic linear constraints
- Utilizes recent advances in stochastic monotone operator theory

## Abstract

A new stochastic primal--dual algorithm for solving a composite optimization problem is proposed. It is assumed that all the functions/operators that enter the optimization problem are given as statistical expectations. These expectations are unknown but revealed across time through i.i.d. realizations. The proposed algorithm is proven to converge to a saddle point of the Lagrangian function. In the framework of the monotone operator theory, the convergence proof relies on recent results on the stochastic Forward Backward algorithm involving random monotone operators. An example of convex optimization under stochastic linear constraints is considered.