A Unifying Framework for Online Optimization with Long-Term Constraints
This work addresses a foundational challenge in online learning for decision-makers needing to balance rewards with constraints, with applications in areas like auction budget management.
The paper tackles the problem of online optimization with long-term constraints by introducing a best-of-both-worlds algorithm that guarantees no-regret performance in both stochastic and adversarial settings, achieving a fraction of the optimal reward and sublinear regret.
We study online learning problems in which a decision maker has to take a sequence of decisions subject to $m$ long-term constraints. The goal of the decision maker is to maximize their total reward, while at the same time achieving small cumulative constraints violation across the $T$ rounds. We present the first best-of-both-world type algorithm for this general class of problems, with no-regret guarantees both in the case in which rewards and constraints are selected according to an unknown stochastic model, and in the case in which they are selected at each round by an adversary. Our algorithm is the first to provide guarantees in the adversarial setting with respect to the optimal fixed strategy that satisfies the long-term constraints. In particular, it guarantees a $ρ/(1+ρ)$ fraction of the optimal reward and sublinear regret, where $ρ$ is a feasibility parameter related to the existence of strictly feasible solutions. Our framework employs traditional regret minimizers as black-box components. Therefore, by instantiating it with an appropriate choice of regret minimizers it can handle the full-feedback as well as the bandit-feedback setting. Moreover, it allows the decision maker to seamlessly handle scenarios with non-convex rewards and constraints. We show how our framework can be applied in the context of budget-management mechanisms for repeated auctions in order to guarantee long-term constraints that are not packing (e.g., ROI constraints).