Entropy Regularization for Mean Field Games with Learning

Entropy regularization has been extensively adopted to improve the efficiency, the stability, and the convergence of algorithms in reinforcement learning. This paper analyzes both quantitatively and qualitatively the impact of entropy regularization for Mean Field Game (MFG) with learning in a finite time horizon. Our study provides a theoretical justification that entropy regularization yields time-dependent policies and, furthermore, helps stabilizing and accelerating convergence to the game equilibrium. In addition, this study leads to a policy-gradient algorithm for exploration in MFG. Under this algorithm, agents are able to learn the optimal exploration scheduling, with stable and fast convergence to the game equilibrium.

Exploration versus exploitation in reinforcement learning: a stochastic control approach

We consider reinforcement learning (RL) in continuous time and study the problem of achieving the best trade-off between exploration of a black box environment and exploitation of current knowledge. We propose an entropy-regularized reward function involving the differential entropy of the distributions of actions, and motivate and devise an exploratory formulation for the feature dynamics that captures repetitive learning under exploration. The resulting optimization problem is a resurrection of the classical relaxed stochastic control. We carry out a complete analysis of the problem in the linear--quadratic (LQ) case and deduce that the optimal control distribution for balancing exploitation and exploration is Gaussian. This in turn interprets and justifies the widely adopted Gaussian exploration in RL, beyond its simplicity for sampling. Moreover, the exploitation and exploration are reflected respectively by the mean and variance of the Gaussian distribution. We also find that a more random environment contains more learning opportunities in the sense that less exploration is needed, other things being equal. As the weight of exploration decays to zero, we prove the convergence of the solution to the entropy-regularized LQ problem to that of the classical LQ problem. Finally, we characterize the cost of exploration, which is shown to be proportional to the entropy regularization weight and inversely proportional to the discount rate in the LQ case.