Finite-Sample Analysis for Two Time-scale Non-linear TDC with General Smooth Function Approximation

Temporal-difference learning with gradient correction (TDC) is a two time-scale algorithm for policy evaluation in reinforcement learning. This algorithm was initially proposed with linear function approximation, and was later extended to the one with general smooth function approximation. The asymptotic convergence for the on-policy setting with general smooth function approximation was established in [bhatnagar2009convergent], however, the finite-sample analysis remains unsolved due to challenges in the non-linear and two-time-scale update structure, non-convex objective function and the time-varying projection onto a tangent plane. In this paper, we develop novel techniques to explicitly characterize the finite-sample error bound for the general off-policy setting with i.i.d.\ or Markovian samples, and show that it converges as fast as $\mathcal O(1/\sqrt T)$ (up to a factor of $\mathcal O(\log T)$). Our approach can be applied to a wide range of value-based reinforcement learning algorithms with general smooth function approximation.