Stochastic recursive inclusion in two timescales with an application to the Lagrangian dual problem

In this paper we present a framework to analyze the asymptotic behavior of two timescale stochastic approximation algorithms including those with set-valued mean fields. This paper builds on the works of Borkar and Perkins & Leslie. The framework presented herein is more general as compared to the synchronous two timescale framework of Perkins \& Leslie, however the assumptions involved are easily verifiable. As an application, we use this framework to analyze the two timescale stochastic approximation algorithm corresponding to the Lagrangian dual problem in optimization theory

Two Timescale 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, both the faster and slower recursions have non-additive 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.Comment: 23 pages (relaxed some important assumptions from the previous version), accepted in Mathematics of Operations Research in Feb, 201