A New Policy Iteration Algorithm For Reinforcement Learning in Zero-Sum Markov Games

TL;DR

This paper introduces a simple, efficient policy iteration algorithm for zero-sum Markov games that converges exponentially fast, with practical implementation in function approximation settings and implications for reinforcement learning.

Contribution

It proposes a novel variant of naive policy iteration using lookahead policies that guarantees exponential convergence in zero-sum Markov games.

Findings

The new algorithm converges exponentially fast.

Lookahead policies can be efficiently implemented in linear function approximation.

Provides bounds for policy-based reinforcement learning algorithms.

Abstract

Optimal policies in standard MDPs can be obtained using either value iteration or policy iteration. However, in the case of zero-sum Markov games, there is no efficient policy iteration algorithm; e.g., it has been shown that one has to solve Omega(1/(1-alpha)) MDPs, where alpha is the discount factor, to implement the only known convergent version of policy iteration. Another algorithm, called naive policy iteration, is easy to implement but is only provably convergent under very restrictive assumptions. Prior attempts to fix naive policy iteration algorithm have several limitations. Here, we show that a simple variant of naive policy iteration for games converges exponentially fast. The only addition we propose to naive policy iteration is the use of lookahead policies, which are anyway used in practical algorithms. We further show that lookahead can be implemented efficiently in the function approximation setting of linear Markov games, which are the counterpart of the much-studied linear MDPs. We illustrate the application of our algorithm by providing bounds for policy-based RL (reinforcement learning) algorithms. We extend the results to the function approximation setting.