  We study the problem of off-policy evaluation in the multi-armed bandit model
with bounded rewards, and develop minimax rate-optimal procedures under three
settings. First, when the behavior policy is known, we show that the Switch
estimator, a method that alternates between the plug-in and importance sampling
estimators, is minimax rate-optimal for all sample sizes. Second, when the
behavior policy is unknown, we analyze performance in terms of the competitive
ratio, thereby revealing a fundamental gap between the settings of known and
unknown behavior policies. When the behavior policy is unknown, any estimator
must have mean-squared error larger -- relative to the oracle estimator
equipped with the knowledge of the behavior policy -- by a multiplicative
factor proportional to the support size of the target policy. Moreover, we
demonstrate that the plug-in approach achieves this worst-case competitive
ratio up to a logarithmic factor. Third, we initiate the study of the partial
knowledge setting in which it is assumed that the minimum probability taken by
the behavior policy is known. We show that the plug-in estimator is optimal for
relatively large values of the minimum probability, but is sub-optimal when the
minimum probability is low. In order to remedy this gap, we propose a new
estimator based on approximation by Chebyshev polynomials that provably
achieves the optimal estimation error. Numerical experiments on both simulated
and real data corroborate our theoretical findings.
