On the Convergence of Continuous Single-timescale Actor-critic

Actor-critic algorithms have been instrumental in boosting the performance of numerous challenging applications involving continuous control, such as highly robust and agile robot motion control. However, their theoretical understanding remains largely underdeveloped. Existing analyses mostly focus on finite state-action spaces and on simplified variants of actor-critic, such as doubleloop updates with i.i.d. sampling, which are often impractical for real-world applications. We consider the canonical and widely adopted singletimescale updates with Markovian sampling in continuous state-action space. Specifically, we establish finite-time convergence by introducing a novel Lyapunov analysis framework, which provides a unified convergence characterization of both the actor and the critic. Our approach is less conservative than previous methods and offers new insights into the coupled dynamics of actor-critic updates.