Two Time-Scale Stochastic Approximation with Controlled Markov Noise and Off-Policy Temporal-Difference Learning

We present for the first time an asymptotic convergence analysis of two time-scale stochastic approximation driven by “controlled” Markov noise. In particular, the faster and slower recursions have nonadditive controlled Markov noise components in addition to martingale difference noise. We analyze the asymptotic behavior of our framework by relating it to limiting differential inclusions in both time scales that are defined in terms of the ergodic occupation measures associated with the controlled Markov processes. Finally, we present a solution to the off-policy convergence problem for temporal-difference learning with linear function approximation, using our results.