We study the continuous-time counterpart of Q-learning for reinforcement learning (RL) under the entropy-regularized, exploratory diffusion process formulation introduced by Wang et al. (2020). As the conventional (big) Q-function collapses in continuous time, we consider its first-order approximation and coin the term ``(little) q-function". This function is related to the instantaneous advantage rate function as well as the Hamiltonian. We develop a ``q-learning" theory around the q-function that is independent of time discretization. Given a stochastic policy, we jointly characterize the associated q-function and value function by martingale conditions of certain stochastic processes, in both on-policy and off-policy settings. We then apply the theory to devise different actor-critic algorithms for solving underlying RL problems, depending on whether or not the density function of the Gibbs measure generated from the q-function can be computed explicitly. One of our algorithms interprets the well-known Q-learning algorithm SARSA, and another recovers a policy gradient (PG) based continuous-time algorithm proposed in Jia and Zhou (2022b). Finally, we conduct simulation experiments to compare the performance of our algorithms with those of PG-based algorithms in Jia and Zhou (2022b) and time-discretized conventional Q-learning algorithms.

翻译：我们研究了在Wang等人(2020)提出的熵正则化探索性扩散过程框架下，强化学习中Q-学习的连续时间对应方法。由于传统（大）Q-函数在连续时间中失效，我们考虑其一阶近似并引入术语“（小）q-函数”。该函数与瞬时优势率函数及哈密顿量相关联。我们围绕q-函数建立了独立于时间离散化的“q-学习”理论。在给定随机策略的条件下，我们通过某些随机过程的鞅条件联合刻画了同策略和异策略设定中相关的q-函数与值函数。随后，我们应用该理论设计了不同的演员-评论家算法来解决基础强化学习问题，具体取决于由q-函数生成的吉布斯测度密度函数能否显式计算。其中一种算法诠释了著名的Q-学习算法SARSA，另一种则复现了Jia与Zhou(2022b)提出的基于策略梯度的连续时间算法。最后，我们通过仿真实验比较了所提算法与Jia和Zhou(2022b)中基于策略梯度的算法及时间离散化传统Q-学习算法的性能。