  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.
