q-Learning in Continuous Time

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 …

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