  Motivated by the study of $Q$-learning algorithms in reinforcement learning,
we study a class of stochastic approximation procedures based on operators that
satisfy monotonicity and quasi-contractivity conditions with respect to an
underlying cone. We prove a general sandwich relation on the iterate error at
each time, and use it to derive non-asymptotic bounds on the error in terms of
a cone-induced gauge norm. These results are derived within a deterministic
framework, requiring no assumptions on the noise. We illustrate these general
bounds in application to synchronous $Q$-learning for discounted Markov
decision processes with discrete state-action spaces, in particular by deriving
non-asymptotic bounds on the $\ell_\infty$-norm for a range of stepsizes. These
results are the sharpest known to date, and we show via simulation that the
dependence of our bounds cannot be improved in a worst-case sense. These
results show that relative to a model-based $Q$-iteration, the
$\ell_\infty$-based sample complexity of $Q$-learning is suboptimal in terms of
the discount factor $\gamma$.
