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 \citet{hasselt2010double} 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 \epsilon -accurate neighborhood of the global optimum by taking \tilde{\Omega}\left(\left( \frac{1}{(1-\gamma) 6\epsilon 2}\right) {\frac{1}{\omega}} \left(\frac{1}{1-\gamma}\right) {\frac{1}{1-\omega}}\right) iterations, where \omega\in(0,1) is the decay parameter of the learning rate, and \gamma is the discount factor.