A Stochastic Variance-reduced Accelerated Primal-dual Method for Finite-sum Saddle-point Problems

TL;DR

This paper introduces a new variance-reduced primal-dual algorithm with Bregman distance for convex-concave saddle-point problems, achieving improved oracle complexity and demonstrating effectiveness in distributionally robust optimization.

Contribution

The paper presents a novel variance-reduced primal-dual method with Bregman distance that improves sample complexity for finite-sum saddle-point problems.

Findings

Achieves oracle complexity of O(√n/ε) and O(n/√ε + 1/ε^{1.5}) with different parameter settings.

Significantly reduces the number of primal-dual gradient samples needed for ε-accuracy.

Demonstrates effectiveness through experiments on distributionally robust optimization.

Abstract

In this paper, we propose a variance-reduced primal-dual algorithm with Bregman distance for solving convex-concave saddle-point problems with finite-sum structure and nonbilinear coupling function. This type of problems typically arises in machine learning and game theory. Based on some standard assumptions, the algorithm is proved to converge with oracle complexity of $O\left(\frac{\sqrt n}{\epsilon}\right)$ and $O\left(\frac{n}{\sqrt \epsilon}+\frac{1}{\epsilon^{1.5}}\right)$ using constant and non-constant parameters, respectively where $n$ is the number of function components. Compared with existing methods, our framework yields a significant improvement over the number of required primal-dual gradient samples to achieve $\epsilon$-accuracy of the primal-dual gap. We tested our method for solving a distributionally robust optimization problem to show the effectiveness of the algorithm.