  What is the most statistically efficient way to do off-policy evaluation and
optimization with batch data from bandit feedback? For log data generated by
contextual bandit algorithms, we consider offline estimators for the expected
reward from a counterfactual policy. Our estimators are shown to have lowest
variance in a wide class of estimators, achieving variance reduction relative
to standard estimators. We then apply our estimators to improve advertisement
design by a major advertisement company. Consistent with the theoretical
result, our estimators allow us to improve on the existing bandit algorithm
with more statistical confidence compared to a state-of-the-art benchmark.
