Finite-Time Analysis for Double Q-learning

Although Q-learning is one of the most successful algorithms for finding the best action-value function (and thus the optimal policy) in reinforcement learning, its implementation often suffers from large overestimation of Q-function values incurred by random sampling. The double Q-learning algorithm proposed in~citethasselt2010double overcomes such an overestimation issue by randomly switching the update between two Q-estimators, and has thus gained significant popularity in practice. However, the theoretical understanding of double Q-learning is rather limited. So far only the asymptotic convergence has been established, which does not characterize how fast the algorithm converges. In this paper, we provide the first non-asymptotic (i.e., finite-time) analysis for double Q-learning. We show that both synchronous and asynchronous double Q-learning are guaranteed to converge to an ε-accurate neighborhood of the global optimum by taking tildeOmegaleft(left( frac1(1-γ)6ε2right)^frac1ω +left(frac11-γright)^frac11-ωright) iterations, where ω∈(0,1) is the decay parameter of the learning rate, and γ is the discount factor. Our analysis develops novel techniques to derive finite-time bounds on the difference between two inter-connected stochastic processes, which is new to the literature of stochastic approximation.