  We study the problem of off-policy evaluation from batched contextual bandit
data with multidimensional actions, often termed slates. The problem is common
to recommender systems and user-interface optimization, and it is particularly
challenging because of the combinatorially-sized action space. Swaminathan et
al. (2017) have proposed the pseudoinverse (PI) estimator under the assumption
that the conditional mean rewards are additive in actions. Using control
variates, we consider a large class of unbiased estimators that includes as
specific cases the PI estimator and (asymptotically) its self-normalized
variant. By optimizing over this class, we obtain new estimators with risk
improvement guarantees over both the PI and the self-normalized PI estimators.
Experiments with real-world recommender data as well as synthetic data validate
these improvements in practice.
