Q-learning with Uniformly Bounded Variance: Large Discounting is Not a Barrier to Fast Learning

Sample complexity bounds are a common performance metric in the Reinforcement Learning literature. In the discounted cost, infinite horizon setting, all of the known bounds have a factor that is a polynomial in $1/(1-\gamma)$, where $\gamma < 1$ is the discount factor. For a large discount factor, these bounds seem to imply that a very large number of samples is required to achieve an $\varepsilon$-optimal policy. The objective of the present work is to introduce a new class of algorithms that have sample complexity uniformly bounded for all $\gamma < 1$. One may argue that this is impossible, due to a recent min-max lower bound. The explanation is that this previous lower bound is for a specific problem, which we modify, without compromising the ultimate objective of obtaining an $\varepsilon$-optimal policy. Specifically, we show that the asymptotic covariance of the Q-learning algorithm with an optimized step-size sequence is a quadratic function of $1/(1-\gamma)$; an expected, and essentially known result. The new relative Q-learning algorithm proposed here is shown to have asymptotic covariance that is a quadratic in $1/(1- \rho^* \gamma)$, where $1 - \rho^* > 0$ is an upper bound on the spectral gap of an optimal transition matrix.