q-Learning in Continuous Time

TL;DR

This paper extends Q-learning to continuous-time reinforcement learning using a new q-function approximation, developing theory and algorithms that are independent of time discretization, and demonstrating their effectiveness through simulations.

Contribution

It introduces a continuous-time q-learning framework with martingale-based characterization, unifying actor-critic algorithms and connecting to existing methods like SARSA and policy gradients.

Findings

The proposed algorithms perform competitively with existing PG-based methods.

The continuous-time q-learning approach is robust to time discretization.

Simulation results show improved convergence properties.

Abstract

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.