We present for the first time an asymptotic convergence analysis of two-timescale stochastic approximation driven by controlled Markov noise. In particular, both the faster and slower recursions have non-additive 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 invariant probability measures associated with the controlled Markov processes. Finally, we show how to solve the off-policy convergence problem for temporal difference learning with linear function approximation using our results.
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