A Tale of Two-Timescale Reinforcement Learning with the Tightest Finite-Time Bound

TL;DR

This paper provides the first finite-time convergence rate bounds for two-timescale reinforcement learning algorithms, showing they converge at specific rates and that these bounds are tight, applicable to a broad range of stepsizes.

Contribution

It establishes the first finite-time bounds for two-timescale RL algorithms, demonstrating tight convergence rates and broader stepsize applicability.

Findings

Convergence rates are O(n^{-eta/2}) and O(n^{-eta/2}) for the two iterates.

Bounds are shown to be tight via lower bounds.

Finite-time analysis confirms the decoupling of two timescale components from some finite time.

Abstract

Policy evaluation in reinforcement learning is often conducted using two-timescale stochastic approximation, which results in various gradient temporal difference methods such as GTD(0), GTD2, and TDC. Here, we provide convergence rate bounds for this suite of algorithms. Algorithms such as these have two iterates, $\theta_n$ and $w_n,$ which are updated using two distinct stepsize sequences, $\alpha_n$ and $\beta_n,$ respectively. Assuming $\alpha_n = n^{-\alpha}$ and $\beta_n = n^{-\beta}$ with $1 > \alpha > \beta > 0,$ we show that, with high probability, the two iterates converge to their respective solutions $\theta^*$ and $w^*$ at rates given by $\|\theta_n - \theta^*\| = \tilde{O}( n^{-\alpha/2})$ and $\|w_n - w^*\| = \tilde{O}(n^{-\beta/2});$ here, $\tilde{O}$ hides logarithmic terms. Via comparable lower bounds, we show that these bounds are, in fact, tight. To the best of our knowledge, ours is the first finite-time analysis which achieves these rates. While it was known that the two timescale components decouple asymptotically, our results depict this phenomenon more explicitly by showing that it in fact happens from some finite time onwards. Lastly, compared to existing works, our result applies to a broader family of stepsizes, including non-square summable ones.