Optimal and instance-dependent guarantees for Markovian linear stochastic approximation

Abstract

We study stochastic approximation procedures for approximately solving a $d$-dimen­sional linear fixed-point equation based on observing a trajectory of length $n$ from an ergodic Markov chain. We first exhibit a non-asymptotic bound of the order $t_{\mathrm{mix}}\frac{d}{n}$ on the squared error of the last iterate of a standard scheme, where $t_{\mathrm{mix}}$ is a mixing time. We then prove a non-asymptotic instance-dependent bound on a suitably averaged sequence of iterates, with a leading term that matches the local asymptotic minimax limit, including sharp dependence on the parameters $(d,t_{\mathrm{mix}})$ in the higher-order terms. We complement these upper bounds with a non-asymptotic minimax lower bound that establishes the instance-optimality of the averaged SA estimator. We derive corollaries of these results for policy evaluation with Markov noise—covering the $\mathrm{TD}(\lambda )$ family of algorithms for all $\lambda \in [0,1)$—and linear autoregressive models. Our instance-dependent characterizations open the door to the design of fine-grained model selection procedures for hyperparameter tuning (e.g., choosing the value of $\lambda$ when running the $\mathrm{TD}(\lambda )$ algorithm).