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

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 γ-discounted MDP with state space S and action space A, we demonstrate that the ℓ∞-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 frac1μmin(1-γ)5varepsilon2+ fractmixμmin(1-γ) up to some logarithmic factor, provided that a proper constant learning rate is adopted. Here, tmix and μmin 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 citequ2020finite by a factor of at least |S||A| for all scenarios, and by a factor of at least tmix|S||A| for any sufficiently small accuracy level varepsilon. Further, we demonstrate that the scaling on the effective horizon frac11-γ can be improved by means of variance reduction.