# Zeroth-Order Algorithms for Smooth Saddle-Point Problems

arXiv: https://arxiv.org/abs/2009.09908 · HTML: https://mlsanity.com/paper/2009.09908/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Sep 2020.

## Built on by (most cited first, 7 of 7)

- [Stochastic Gradient Descent-Ascent: Unified Theory and New Efficient Methods](https://mlsanity.com/paper/2202.07262/) (2022)
- [Zeroth-Order Methods for Convex-Concave Minmax Problems: Applications to Decision-Dependent Risk Minimization](https://mlsanity.com/paper/2106.09082/) (2021)
- [Derivative-free Alternating Projection Algorithms for General Nonconvex-Concave Minimax Problems](https://mlsanity.com/paper/2108.00473/) (2021)
- [A Fully Parameter-Free Second-Order Algorithm for Convex-Concave Minimax Problems](https://mlsanity.com/paper/2407.03571/) (2024)
- [Two trust region type algorithms for solving nonconvex-strongly concave minimax problems](https://mlsanity.com/paper/2402.09807/) (2024)
- [Zeroth-Order Stochastic Mirror Descent Algorithms for Minimax Excess Risk Optimization](https://mlsanity.com/paper/2408.12209/) (2024)
- [Gradient Norm Regularization Second-Order Algorithms for Solving Nonconvex-Strongly Concave Minimax Problems](https://mlsanity.com/paper/2411.15769/) (2024)

## Abstract (arXiv)

Saddle-point problems have recently gained increased attention from the machine learning community, mainly due to applications in training Generative Adversarial Networks using stochastic gradients. At the same time, in some applications only a zeroth-order oracle is available. In this paper, we propose several algorithms to solve stochastic smooth (strongly) convex-concave saddle-point problems using zeroth-order oracles and estimate their convergence rate and its dependence on the dimension n of the variable. In particular, our analysis shows that in the case when the feasible set is a direct product of two simplices, our convergence rate for the stochastic term is only by a log n factor worse than for the first-order methods. We also consider a mixed setup and develop 1/2th-order methods that use zeroth-order oracle for the minimization part and first-order oracle for the maximization part. Finally, we demonstrate the practical performance of our zeroth-order and 1/2th-order methods on practical problems.