Learning rates for Q-learning

In this paper we derive convergence rates for Q-learning. We show an interesting relationship between the convergence rate and the learning rate used in Q-learning. For a polynomial learning rate, one which is 1/t(omega) at time t where omega epsilon ( 1/2, 1), we show that the convergence rate is polynomial in 1/(1-gamma), where g is the discount factor. In contrast we show that for a linear learning rate, one which is 1/t at time t, the convergence rate has an exponential dependence on 1/(1-gamma). In addition we show a simple example that proves this exponential behavior is inherent for linear learning rates.