Dynamics of stochastic approximation with iterate-dependent Markov noise under verifiable conditions in compact state space with the stability of iterates not ensured

TL;DR

This paper extends the analysis of stochastic approximation algorithms with iterate-dependent Markov noise to cases without guaranteed stability, providing convergence conditions, lock-in probability bounds, and applications to adaptive algorithms.

Contribution

It introduces a new framework for analyzing convergence of stochastic approximation with Markov noise without assuming stability, including lock-in probability bounds and multi-timescale extensions.

Findings

Almost sure convergence under weaker conditions

Lower bounds for lock-in probability with bounded Markov noise

Sample complexity estimates for step-size selection

Abstract

This paper compiles several aspects of the dynamics of stochastic approximation algorithms with Markov iterate-dependent noise when the iterates are not known to be stable beforehand. We achieve the same by extending the lock-in probability (i.e. the probability of convergence of the iterates to a specific attractor of the limiting o.d.e. given that the iterates are in its domain of attraction after a sufficiently large number of iterations (say) n 0 ) framework to such recursions. Specifically, with the more restrictive assumption of Markov iterate-dependent noise supported on a bounded subset of the Euclidean space we give a lower bound for the lock-in probability. We use these results to prove almost sure convergence of the iterates to the specified attractor when the iterates satisfy an asymptotic tightness condition. The novelty of our approach is that if the state space of the Markov process is compact we prove almost sure convergence under much weaker assumptions compared to the work by Andrieu et al. which solves the general state space case under much restrictive assumptions. We also extend our single timescale results to the case where there are two separate recursions over two different timescales. This, in turn, is shown to be useful in analyzing the tracking ability of general adaptive algorithms. Additionally, we show that our results can be used to derive a sample complexity estimate of such recursions, which then can be used for step-size selection