Finite Time Analysis of Linear Two-timescale Stochastic Approximation with Markovian Noise
This provides theoretical guarantees for reinforcement learning policy evaluation algorithms, but it is incremental as it extends prior analyses to Markovian noise settings.
The paper tackles the finite-time analysis of linear two-timescale stochastic approximation under Markovian noise, showing that the convergence rate remains the same as with martingale noise, with only constants affected by mixing time, achieving a steady-state error of O(1/k) and a transient term of o(1/k^c).
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 $Ω(1/k)$. A simple numerical experiment is presented to support our theory.