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A Single-Loop Smoothed Gradient Descent-Ascent Algorithm for Nonconvex-Concave Min-Max Problems

Jiawei Zhang, Peijun Xiao, Ruoyu Sun, Zhi-Quan Luo

TL;DR

This work tackles nonconvex-concave min–max problems, where standard gradient descent-ascent can oscillate and fail to converge. It proposes a single-loop Smoothed-GDA method that injects proximal smoothing via an auxiliary sequence $\{z^t\}$ and a quadratic term in the primal update, ensuring convergence to stationary points. The authors prove that the Smoothed-GDA framework achieves $O\left(\varepsilon^{-2}\right)$ iteration complexity for the finite-max special case and $O\left(\varepsilon^{-4}\right)$ for general nonconvex-concave problems, with an extension to multi-block settings. They validate the approach experimentally on robust neural-network training, where the method exhibits faster convergence and competitive robustness compared to existing algorithms, indicating practical impact for scalable, stable minimax optimization in machine learning.

Abstract

Nonconvex-concave min-max problem arises in many machine learning applications including minimizing a pointwise maximum of a set of nonconvex functions and robust adversarial training of neural networks. A popular approach to solve this problem is the gradient descent-ascent (GDA) algorithm which unfortunately can exhibit oscillation in case of nonconvexity. In this paper, we introduce a "smoothing" scheme which can be combined with GDA to stabilize the oscillation and ensure convergence to a stationary solution. We prove that the stabilized GDA algorithm can achieve an $O(1/ε^2)$ iteration complexity for minimizing the pointwise maximum of a finite collection of nonconvex functions. Moreover, the smoothed GDA algorithm achieves an $O(1/ε^4)$ iteration complexity for general nonconvex-concave problems. Extensions of this stabilized GDA algorithm to multi-block cases are presented. To the best of our knowledge, this is the first algorithm to achieve $O(1/ε^2)$ for a class of nonconvex-concave problem. We illustrate the practical efficiency of the stabilized GDA algorithm on robust training.

A Single-Loop Smoothed Gradient Descent-Ascent Algorithm for Nonconvex-Concave Min-Max Problems

TL;DR

This work tackles nonconvex-concave min–max problems, where standard gradient descent-ascent can oscillate and fail to converge. It proposes a single-loop Smoothed-GDA method that injects proximal smoothing via an auxiliary sequence and a quadratic term in the primal update, ensuring convergence to stationary points. The authors prove that the Smoothed-GDA framework achieves iteration complexity for the finite-max special case and for general nonconvex-concave problems, with an extension to multi-block settings. They validate the approach experimentally on robust neural-network training, where the method exhibits faster convergence and competitive robustness compared to existing algorithms, indicating practical impact for scalable, stable minimax optimization in machine learning.

Abstract

Nonconvex-concave min-max problem arises in many machine learning applications including minimizing a pointwise maximum of a set of nonconvex functions and robust adversarial training of neural networks. A popular approach to solve this problem is the gradient descent-ascent (GDA) algorithm which unfortunately can exhibit oscillation in case of nonconvexity. In this paper, we introduce a "smoothing" scheme which can be combined with GDA to stabilize the oscillation and ensure convergence to a stationary solution. We prove that the stabilized GDA algorithm can achieve an iteration complexity for minimizing the pointwise maximum of a finite collection of nonconvex functions. Moreover, the smoothed GDA algorithm achieves an iteration complexity for general nonconvex-concave problems. Extensions of this stabilized GDA algorithm to multi-block cases are presented. To the best of our knowledge, this is the first algorithm to achieve for a class of nonconvex-concave problem. We illustrate the practical efficiency of the stabilized GDA algorithm on robust training.

Paper Structure

This paper contains 27 sections, 37 theorems, 211 equations, 2 figures, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Representative Applications
  3. Smoothed GDA Algorithm and Its Convergence
  4. Smoothed Gradient Descent-Ascent (Smoothed-GDA)
  5. Iteration Complexity for Nonconvex-Concave Problems
  6. The Iteration Complexity of Achieving $\epsilon$-Optimization-Stationary Point
  7. Convergence Results for Minimizing the Point-Wise Maximum of Finite Functions
  8. Proof Sketch
  9. The Potential Function and Basic Estimates
  10. Numerical Results on Robust Neural Network Training
  11. Conclusion
  12. Notations
  13. Proof of the two main theorems: one-block case
  14. The potential function and basic estimate
  15. General nonconvex-concave case
  16. ...and 12 more sections

Key Result

Theorem 3.4

Consider solving problem minimax1 by Algorithm Alg2 (or Algorithm Alg3). Suppose Assumption basic-ass holds, and we choose the algorithm parameters to satisfy $p>3L,\ c<1/(p+L)$ and Then, the following holds:

Figures (2)

  • Figure 1: Convergence speed of Smoothed-GDA and the algorithm in nouiehed2019solving.
  • Figure 2: Convergence speed of Smoothed-GDA and the algorithm in nouiehed2019solving on CIFAR10.

Theorems & Definitions (42)

  • Definition 3.1
  • Definition 3.2
  • Theorem 3.4
  • Definition 3.5
  • Theorem 3.6
  • Theorem 3.10
  • Proposition 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Lemma B.1
  • ...and 32 more