Polymatrix Competitive Gradient Descent

TL;DR

This paper introduces Polymatrix Competitive Gradient Descent (PCGD), a new method for solving multi-agent competitive optimization problems that converges locally and outperforms existing algorithms in multi-agent reinforcement learning tasks.

Contribution

The paper proposes PCGD, a novel algorithm for general-sum multi-agent games, with proven local convergence and efficient computation, applicable to multi-agent reinforcement learning.

Findings

PCGD converges locally to stable fixed points in n-player general-sum games.

PCGD outperforms existing methods like simultaneous gradient descent and extragradient in speed and effectiveness.

Agents trained with PCGD achieve better performance in multi-agent RL environments.

Abstract

Many economic games and machine learning approaches can be cast as competitive optimization problems where multiple agents are minimizing their respective objective function, which depends on all agents' actions. While gradient descent is a reliable basic workhorse for single-agent optimization, it often leads to oscillation in competitive optimization. In this work we propose polymatrix competitive gradient descent (PCGD) as a method for solving general sum competitive optimization involving arbitrary numbers of agents. The updates of our method are obtained as the Nash equilibria of a local polymatrix approximation with a quadratic regularization, and can be computed efficiently by solving a linear system of equations. We prove local convergence of PCGD to stable fixed points for $n$-player general-sum games, and show that it does not require adapting the step size to the strength of the player-interactions. We use PCGD to optimize policies in multi-agent reinforcement learning and demonstrate its advantages in Snake, Markov soccer and an electricity market game. Agents trained by PCGD outperform agents trained with simultaneous gradient descent, symplectic gradient adjustment, and extragradient in Snake and Markov soccer games and on the electricity market game, PCGD trains faster than both simultaneous gradient descent and the extragradient method.