# Accelerated Primal-dual Scheme for a Class of Stochastic Nonconvex-concave Saddle Point Problems

## TL;DR

This paper introduces a novel single-loop accelerated primal-dual algorithm for stochastic nonconvex-concave saddle point problems, achieving improved convergence rates and addressing slow convergence issues of existing methods.

## Contribution

It proposes the first single-loop accelerated primal-dual method with new convergence rate results for a class of nonconvex saddle point problems satisfying the Polyak-{ extL}ojasiewicz condition.

## Key findings

- Achieves a stochastic convergence rate of O(ε^{-4}) for ε-gap solutions.
- Improves to an O(ε^{-2}) rate in deterministic settings.
- Addresses slow convergence and multi-loop issues of prior algorithms.

## Abstract

Stochastic nonconvex-concave min-max saddle point problems appear in many machine learning and control problems including distributionally robust optimization, generative adversarial networks, and adversarial learning. In this paper, we consider a class of nonconvex saddle point problems where the objective function satisfies the Polyak-{\L}ojasiewicz condition with respect to the minimization variable and it is concave with respect to the maximization variable. The existing methods for solving nonconvex-concave saddle point problems often suffer from slow convergence and/or contain multiple loops. Our main contribution lies in proposing a novel single-loop accelerated primal-dual algorithm with new convergence rate results appearing for the first time in the literature, to the best of our knowledge. In particular, in the stochastic regime, we demonstrate a convergence rate of $\mathcal O(\epsilon^{-4})$ to find an $\epsilon$-gap solution which can be improved to $\mathcal O(\epsilon^{-2})$ in deterministic setting.