On the Effect of Log-Barrier Regularization in Decentralized Softmax Gradient Play in Multiagent Systems

Softmax policy gradient is a popular algorithm for policy optimization in single-agent reinforcement learning, particularly since projection is not needed for each gradient update. However, in multi-agent systems, the lack of central coordination introduces significant additional difficulties in the convergence analysis. Even for a stochastic game with identical interest, there can be multiple Nash Equilibria (NEs), which disables proof techniques that rely on the existence of a unique global optimum. Moreover, the softmax parameterization introduces non-NE policies with zero gradient, making NE-seeking difficult for gradient-based algorithms. In this paper, we study the finite time convergence of decentralized softmax gradient play in a special form of game, Markov Potential Games (MPGs), which includes the identical interest game as a special case. We investigate both gradient play and natural gradient play, with and without $\log$-barrier regularization. Establishing convergence for the unregularized cases relies on an assumption that the stationary policies are isolated, and yields convergence bounds that contain a trajectory dependent constant that can be arbitrarily large. We introduce the $\log$-barrier regularization to overcome these drawbacks, with the cost of slightly worse dependence on other factors such as the action set size. An empirical study on an identical interest matrix game confirms the theoretical findings.