In many settings, a decision maker wishes to learn a rule, or policy, that maps from observable characteristics of an individual to an action. Examples include selecting offers, prices, advertisements, or emails to send to consumers, choosing a bid to submit in a contextual first-price auctions, and determining which medication to prescribe to a patient. In this paper, we study the offline multi-action policy learning problem with observational data and where the policy may need to respect budget constraints or belong to a restricted policy class such as decision trees. By using the standard augmented inverse propensity weight estimator, we design and implement a policy learning algorithm that achieves asymptotically minimax-optimal regret. To the best of our knowledge, this is the first result of this type in the multi-action setup, and it provides a substantial performance improvement over the existing learning algorithms. We then consider additional computational challenges that arise in implementing our method for the case where the policy is restricted to take the form of a decision tree. We propose two different approaches: one using a mixed integer program formulation and the other using a tree-search based algorithm.
