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We study online learning problems in which a decision maker has to make asequence of costly decisions, with the goal of maximizing their expected rewardwhile adhering to budget and return-on-investment (ROI) constraints. Previouswork requires the decision maker to know beforehand some specific parametersrelated to the degree of strict feasibility of the offline problem. Moreover,when inputs are adversarial, it requires the existence of a strictly feasiblesolution to the offline optimization problem at each round. Both requirementsare unrealistic for practical applications such as bidding in online adauctions. We propose a best-of-both-worlds primal-dual framework whichcircumvents both assumptions by exploiting the notion of interval regret,providing guarantees under both stochastic and adversarial inputs. Our prooftechniques can be applied to both input models with minimal modifications,thereby providing a unified perspective on the two problems. Finally, we showhow to instantiate the framework to optimally bid in various mechanisms ofpractical relevance, such as first- and second-price auctions.

Online Learning under Budget and ROI Constraints and Applications to Bidding in Non-Truthful Auctions