Work

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

Year: 2021

Type: article

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 _{\mathsf {min}}(1-\gamma)^{5}\varepsilon ^{2}}+ \frac { t_{\mathsf {mix}}}{ \mu _{\mathsf {min}}(1-\gamma)}$up to some logarithmic factor, provided that a proper constant learning rate is adopted. Here,$t_{\mathsf {mix}}$and$\mu _{\mathsf {min}}$denote respectively the mixing time and the minimum state-action occupancy probability of the sample trajectory. The first term of this bound matches the sample complexity in the synchronous case with independent samples drawn from the stationary distribution of the trajectory. The second term reflects the cost taken for the empirical distribution of the Markovian trajectory to reach a steady state, which is incurred at the very beginning and becomes amortized as the algorithm runs. Encouragingly, the above bound improves upon the state-of-the-art result by a factor of at least$|\mathcal {S}||\mathcal {A}|$for all scenarios, and by a factor of at least$t_{\mathsf {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.

Source: IEEE Transactions on Information Theory

DOI: 10.1109/tit.2021.3120096

Authors: Gen Li, Yuting Wei, Yuejie Chi, Yuantao Gu, Yuxin Chen

Institutions: Princeton University, University of Pennsylvania, Carnegie Mellon University, Tsinghua University

Language: English

Topic: Reinforcement Learning in Robotics

Subfield: Artificial Intelligence

Field: Computer Science

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $2,045