  Stochastic Approximation (SA) is a widely used algorithmic approach in
various fields, including optimization and reinforcement learning (RL). Among
RL algorithms, Q-learning is particularly popular due to its empirical success.
In this paper, we study asynchronous Q-learning with constant stepsize, which
is commonly used in practice for its fast convergence. By connecting the
constant stepsize Q-learning to a time-homogeneous Markov chain, we show the
distributional convergence of the iterates in Wasserstein distance and
establish its exponential convergence rate. We also establish a Central Limit
Theory for Q-learning iterates, demonstrating the asymptotic normality of the
averaged iterates. Moreover, we provide an explicit expansion of the asymptotic
bias of the averaged iterate in stepsize. Specifically, the bias is
proportional to the stepsize up to higher-order terms and we provide an
explicit expression for the linear coefficient. This precise characterization
of the bias allows the application of Richardson-Romberg (RR) extrapolation
technique to construct a new estimate that is provably closer to the optimal Q
function. Numerical results corroborate our theoretical finding on the
improvement of the RR extrapolation method.
