  We propose a new framework for designing estimators for off-policy evaluation
in contextual bandits. Our approach is based on the asymptotically optimal
doubly robust estimator, but we shrink the importance weights to minimize a
bound on the mean squared error, which results in a better bias-variance
tradeoff in finite samples. We use this optimization-based framework to obtain
three estimators: (a) a weight-clipping estimator, (b) a new weight-shrinkage
estimator, and (c) the first shrinkage-based estimator for combinatorial action
sets. Extensive experiments in both standard and combinatorial bandit benchmark
problems show that our estimators are highly adaptive and typically outperform
state-of-the-art methods.
