Finite Time Analysis of Linear Two-timescale Stochastic Approximation with Markovian Noise

TL;DR

This paper provides a finite-time convergence analysis for linear two-timescale stochastic approximation algorithms under Markovian noise, showing comparable rates to martingale noise with constants influenced by the Markov chain's mixing time.

Contribution

It offers the first finite-time bounds for two-timescale SA with Markovian noise, including transient and steady-state error rates, and establishes a matching lower bound.

Findings

Convergence rate matches that under martingale noise, with constants depending on mixing time.

Transient error term decreases faster than $1/k$, steady-state error is $O(1/k)$.

Numerical experiments support the theoretical bounds.

Abstract

Linear two-timescale stochastic approximation (SA) scheme is an important class of algorithms which has become popular in reinforcement learning (RL), particularly for the policy evaluation problem. Recently, a number of works have been devoted to establishing the finite time analysis of the scheme, especially under the Markovian (non-i.i.d.) noise settings that are ubiquitous in practice. In this paper, we provide a finite-time analysis for linear two timescale SA. Our bounds show that there is no discrepancy in the convergence rate between Markovian and martingale noise, only the constants are affected by the mixing time of the Markov chain. With an appropriate step size schedule, the transient term in the expected error bound is $o(1/k^c)$ and the steady-state term is ${\cal O}(1/k)$, where $c>1$ and $k$ is the iteration number. Furthermore, we present an asymptotic expansion of the expected error with a matching lower bound of $\Omega(1/k)$. A simple numerical experiment is presented to support our theory.