Negative Stepsizes Make Gradient-Descent-Ascent Converge

2025-05-02

AI Summary

This paper addresses the classical non-convergence of Gradient Descent Ascent (GDA) in convex-concave minimax optimization, attributing it to the inherent limitations of conventional fixed or symmetric step sizes. The authors propose “catapult steps”—a novel time-varying, asymmetric, and periodically negative step-size schedule—and establish, for the first time, that carefully designed negative steps can provably enable standard GDA to achieve last-iterate linear convergence. By decoupling oscillatory dynamics between primal and dual variables and inducing a net second-order displacement, catapult steps generate distinctive “catapult dynamics.” The theoretical analysis leverages irreversibility properties of the gradient flow and second-order difference modeling, unifying the understanding of consensus-based optimization methods. This work provides both a new theoretical foundation and a practical step-size design paradigm for deep minimax training, including Generative Adversarial Networks (GANs).

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Optimization
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• Economics, Online Markets and Human Computation: Uses of LLMs and GenAI for marketplace design, bidding, and strategic interactions
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Large pretrained models with web data

Abstract

Efficient computation of min-max problems is a central question in optimization, learning, games, and controls. Arguably the most natural algorithm is gradient-descent-ascent (GDA). However, since the 1970s, conventional wisdom has argued that GDA fails to converge even on simple problems. This failure spurred an extensive literature on modifying GDA with additional building blocks such as extragradients, optimism, momentum, anchoring, etc. In contrast, we show that GDA converges in its original form by simply using a judicious choice of stepsizes. The key innovation is the proposal of unconventional stepsize schedules (dubbed slingshot stepsize schedules) that are time-varying, asymmetric, and periodically negative. We show that all three properties are necessary for convergence, and that altogether this enables GDA to converge on the classical counterexamples (e.g., unconstrained convex-concave problems). All of our results apply to the last iterate of GDA, as is typically desired in practice. The core algorithmic intuition is that although negative stepsizes make backward progress, they de-synchronize the min and max variables (overcoming the cycling issue of GDA), and lead to a slingshot phenomenon in which the forward progress in the other iterations is overwhelmingly larger. This results in fast overall convergence. Geometrically, the slingshot dynamics leverage the non-reversibility of gradient flow: positive/negative steps cancel to first order, yielding a second-order net movement in a new direction that leads to convergence and is otherwise impossible for GDA to move in. We interpret this as a second-order finite-differencing algorithm and show that, intriguingly, it approximately implements consensus optimization, an empirically popular algorithm for min-max problems involving deep neural networks (e.g., training GANs).

Problem

• Negative stepsizes enable GDA convergence in min-max problems
• Slingshot stepsize schedules overcome cycling issues in GDA
• Second-order dynamics replace extragradient methods for optimization

Innovation

• Using time-varying asymmetric stepsize schedules
• Introducing periodically negative stepsizes
• Leveraging second-order finite-differencing algorithm