Regularized Online Allocation Problems: Fairness and Beyond

Online allocation problems with resource constraints have a rich history in computer science and operations research. In this paper, we introduce the \emph{regularized online allocation problem}, a variant that includes a non-linear regularizer acting on the total resource consumption. In this problem, requests repeatedly arrive over time and, for each request, a decision maker needs to take an action that generates a reward and consumes resources. The objective is to simultaneously maximize total rewards and the value of the regularizer subject to the resource constraints. Our primary motivation is the online allocation of internet advertisements wherein firms seek to maximize additive objectives such as the revenue or efficiency of the allocation. By introducing a regularizer, firms can account for the fairness of the allocation or, alternatively, punish under-delivery of advertisements---two common desiderata in internet advertising markets. We design an algorithm when arrivals are drawn independently from a distribution that is unknown to the decision maker. Our algorithm is simple, fast, and attains the optimal order of sub-linear regret compared to the optimal allocation with the benefit of hindsight. Numerical experiments confirm the effectiveness of the proposed algorithm and of the regularizers in an internet advertising application.