PAC-Bayesian Treatment Allocation Under Budget Constraints

Abstract: This paper considers the estimation of treatment assignment rules when the policy maker faces a general budget or resource constraint. Utilizing the PAC-Bayesian framework, we propose new treatment assignment rules that allow for flexible notions of treatment outcome, treatment cost, and a budget constraint. For example, the constraint setting allows for cost-savings, when the costs of non-treatment exceed those of treatment for a subpopulation, to be factored into the budget. It also accommodates simpler settings, such as quantity constraints, and doesn't require outcome responses and costs to have the same unit of measurement. Importantly, the approach accounts for settings where budget or resource limitations may preclude treating all that can benefit, where costs may vary with individual characteristics, and where there may be uncertainty regarding the cost of treatment rules of interest. Despite the nomenclature, our theoretical analysis examines frequentist properties of the proposed rules. For stochastic rules that typically approach budget-penalized empirical welfare maximizing policies in larger samples, we derive non-asymptotic generalization bounds for the target population costs and sharp oracle-type inequalities that compare the rules' welfare regret to that of optimal policies in relevant budget categories. A closely related, non-stochastic, model aggregation treatment assignment rule is shown to inherit desirable attributes.