  In machine learning we often try to optimise a decision rule that would have
worked well over a historical dataset; this is the so called empirical risk
minimisation principle. In the context of learning from recommender system
logs, applying this principle becomes a problem because we do not have
available the reward of decisions we did not do. In order to handle this
"bandit-feedback" setting, several Counterfactual Risk Minimisation (CRM)
methods have been proposed in recent years, that attempt to estimate the
performance of different policies on historical data. Through importance
sampling and various variance reduction techniques, these methods allow more
robust learning and inference than classical approaches. It is difficult to
accurately estimate the performance of policies that frequently perform actions
that were infrequently done in the past and a number of different types of
estimators have been proposed.
  In this paper, we review several methods, based on different off-policy
estimators, for learning from bandit feedback. We discuss key differences and
commonalities among existing approaches, and compare their empirical
performance on the RecoGym simulation environment. To the best of our
knowledge, this work is the first comparison study for bandit algorithms in a
recommender system setting.
