Sample Complexity of Asynchronous Q-Learning: Sharper Analysis and Variance Reduction

TL;DR

This paper provides a sharper analysis of asynchronous Q-learning's sample complexity, showing improved bounds and variance reduction techniques for learning optimal Q-functions in Markov decision processes from Markovian samples.

Contribution

It offers a novel, tighter sample complexity bound for asynchronous Q-learning and introduces variance reduction to improve the effective horizon scaling.

Findings

Sample complexity bound is improved by a factor of at least |S||A| over previous results.

The bound accounts for the mixing time and minimum state-action occupancy probability.

Variance reduction techniques improve the scaling with respect to the effective horizon.

Abstract

Asynchronous Q-learning aims to learn the optimal action-value function (or Q-function) of a Markov decision process (MDP), based on a single trajectory of Markovian samples induced by a behavior policy. Focusing on a $\gamma$-discounted MDP with state space $\mathcal{S}$ and action space $\mathcal{A}$, we demonstrate that the $\ell_{\infty}$-based sample complexity of classical asynchronous Q-learning --- namely, the number of samples needed to yield an entrywise $\varepsilon$-accurate estimate of the Q-function --- is at most on the order of $\frac{1}{\mu_{\min}(1-\gamma)^5\varepsilon^2}+ \frac{t_{mix}}{\mu_{\min}(1-\gamma)}$ up to some logarithmic factor, provided that a proper constant learning rate is adopted. Here, $t_{mix}$ and $\mu_{\min}$ denote respectively the mixing time and the minimum state-action occupancy probability of the sample trajectory. The first term of this bound matches the sample complexity in the synchronous case with independent samples drawn from the stationary distribution of the trajectory. The second term reflects the cost taken for the empirical distribution of the Markovian trajectory to reach a steady state, which is incurred at the very beginning and becomes amortized as the algorithm runs. Encouragingly, the above bound improves upon the state-of-the-art result \cite{qu2020finite} by a factor of at least $|\mathcal{S}||\mathcal{A}|$ for all scenarios, and by a factor of at least $t_{mix}|\mathcal{S}||\mathcal{A}|$ for any sufficiently small accuracy level $\varepsilon$. Further, we demonstrate that the scaling on the effective horizon $\frac{1}{1-\gamma}$ can be improved by means of variance reduction.