A Single Stepsize Suffices for Unprojected Linear TD(0): Simultaneous Robust and Fast Rates via Polyak--Ruppert Averaging

Abstract

We study linear TD(0) under Markovian sampling, where data are generated along a single trajectory. We provide high-probability guarantees for a plain unprojected TD(0) algorithm with Polyak-Ruppert (PR) averaging, using a single stepsize schedule $\hat{I}{\cdot }_t\propto \frac{1}{\ddot{I}{„}_{\mathrm{mix}}\log (t)\sqrt{t}}$ that depends on the mixing time but requires no prior knowledge of the curvature parameter $\ddot{I}‰$. Our first result shows that such a choice of the stepsize guarantees that the TD(0) iterates are automatically and uniformly bounded with high probability, without projections and without any stability argument based on $\ddot{I}‰$. Building on this result, we establish a simultaneous high-probability convergence guarantee for the PR average: the same stepsize yields both a robust curvature-free $\tilde{\mathcal{O}}\left(\frac{\ddot{I}{„}_{\mathrm{mix}}}{\sqrt{T}}\right)$ rate and a fast curvature-dependent $\tilde{\mathcal{O}}\left(\frac{\ddot{I}{„}_{\mathrm{mix}}^2}{\ddot{I}‰T}\right)$rate, with the bound taking the minimum of the two. The core technical ingredient is a Poisson-equation toolkit for geometrically mixing Markov chains, which decomposes Markov noise into a martingale term plus a controlled remainder and enables a new self-bounding inductive argument for pathwise stability.