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.

研究探讨了在连续时间内通过采用熵正则化奖励函数促进探索和利用现有知识之间达到的最佳折衷方案，提出使用行为分布的微分熵来规范化奖励函数的熵正则化，并通过高斯分布表征推导出最佳反馈控制分布来平衡利用和探索性搜索，最后通过熵正则化 LQ 问题的解法证明当探索比重衰减至零时，解法能够收敛于经典 LQ 问题的解。

强化学习中的探索与利用：一种随机控制方法