  We study the problem of estimating the fixed point of a contractive operator
defined on a separable Banach space. Focusing on a stochastic query model that
provides noisy evaluations of the operator, we analyze a variance-reduced
stochastic approximation scheme, and establish non-asymptotic bounds for both
the operator defect and the estimation error, measured in an arbitrary
semi-norm. In contrast to worst-case guarantees, our bounds are
instance-dependent, and achieve the local asymptotic minimax risk
non-asymptotically. For linear operators, contractivity can be relaxed to
multi-step contractivity, so that the theory can be applied to problems like
average reward policy evaluation problem in reinforcement learning. We
illustrate the theory via applications to stochastic shortest path problems,
two-player zero-sum Markov games, as well as policy evaluation and $Q$-learning
for tabular Markov decision processes.
