Second-Order Algorithms for Finding Local Nash Equilibria in Zero-Sum Games
Kushagra Gupta, Xinjie Liu, Ross Allen, Ufuk Topcu, David Fridovich-Keil
TL;DR
This paper introduces a second-order algorithm for finding local Nash equilibria in zero-sum games, ensuring convergence with linear or superlinear rates even in complex nonconvex-nonconcave settings.
Contribution
The paper develops a novel second-order method with proven convergence guarantees for local Nash equilibria in nonconvex, nonconcave zero-sum games, extending to constrained scenarios.
Findings
Guarantees convergence to local Nash equilibria with linear rate.
Interprets the method as a modified Gauss-Newton algorithm with superlinear convergence.
Extends naturally to constrained, coupled settings.
Abstract
Zero-sum games arise in a wide variety of problems, including robust optimization and adversarial learning. However, algorithms deployed for finding a local Nash equilibrium in these games often converge to non-Nash stationary points. This highlights a key challenge: for any algorithm, the stability properties of its underlying dynamical system can cause non-Nash points to be potential attractors. To overcome this challenge, algorithms must account for subtleties involving the curvatures of players' costs. To this end, we leverage dynamical system theory and develop a second-order algorithm for finding a local Nash equilibrium in the smooth, possibly nonconvex-nonconcave, zero-sum game setting. First, we prove that this novel method guarantees convergence to only local Nash equilibria with an asymptotic local \textit{linear} convergence rate. We then interpret a version of this method…
Peer Reviews
Decision·Submitted to ICLR 2025
It's noteworthy that the authors developed an entire family of algorithms for, both, unconstrained and constrained spaces.
1. An obvious weakness is that the suggested algorithms are applicable only to two-player zero-sum games. Could they be modified to support multi-player games? 2. Assumptions 1, 2, and 4 seem reasonable. I am not quite sure how common Assumption 3 is. I understand that the authors consider the relaxation of Assumption 3 as future work. Could they provide some possible intuition? For example, is their natural subclass of two-player zero-sum games that this assumption is satisfied?
I think that the problem that the authors study is significant and this paper gives new directions to understand it. Their results are good since they give new dynamics and prove local rate of convergence using techniques such as bounding by strictly less than 1 the eigenvalues of the Jacobian matrix. Furthermore, the idea to find a dynamical system s.t. any LASE is a local Nash equilibrium of the initial game is nice.
The results hold under specific assumptions and they only guarantee local convergence. I think the paper needs more elaboration how strict are these assumptions and for the latter how far the initial point should be at most from the LASE point in order to have convergence.
* The algorithm leverages second-order dynamics to achieve higher convergence rates and avoid common pitfalls like oscillations around equilibrium points. * The authors validate their approach on a GAN training task, demonstrating the algorithms' robustness and practical utility in adversarial machine learning settings.
* The explanation of assumptions in Section 3.1 is weak, and the example in the numerical experiment does not satisfy Assumption 3. * The author does not claim the technique contribution.
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Taxonomy
TopicsGame Theory and Applications · Game Theory and Voting Systems · Economic theories and models