  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.
