Sample Complexity of Asynchronous Q-Learning: Sharper Analysis and Variance Reduction
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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 ε-accurate estimate of the Q-function – is at most on the order of 1/μ_𝗆𝗂𝗇(1-γ)^5ε^2+ t_𝗆𝗂𝗑/μ_𝗆𝗂𝗇(1-γ) up to some logarithmic factor, provided that a proper constant learning rate is adopted. Here, t_𝗆𝗂𝗑 and μ_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 complexity in the case with independent samples drawn from the stationary distribution of the trajectory. The second term reflects the expense 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 by a factor of at least |𝒮||𝒜|. Further, the scaling on the discount complexity can be improved by means of variance reduction.
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