  We study the off-policy evaluation problem---estimating the value of a target
policy using data collected by another policy---under the contextual bandit
model. We consider the general (agnostic) setting without access to a
consistent model of rewards and establish a minimax lower bound on the mean
squared error (MSE). The bound is matched up to constants by the inverse
propensity scoring (IPS) and doubly robust (DR) estimators. This highlights the
difficulty of the agnostic contextual setting, in contrast with multi-armed
bandits and contextual bandits with access to a consistent reward model, where
IPS is suboptimal. We then propose the SWITCH estimator, which can use an
existing reward model (not necessarily consistent) to achieve a better
bias-variance tradeoff than IPS and DR. We prove an upper bound on its MSE and
demonstrate its benefits empirically on a diverse collection of data sets,
often outperforming prior work by orders of magnitude.
