  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.
