Optimally Pacing Budget Spending and Learning

2026-10-07

AI Summary

This study addresses the optimization of regret bounds in budget-constrained online learning within adversarial environments. For expert settings under arbitrary budget paces, it proposes a full-information algorithm that matches theoretical lower bounds and extends this approach to online resource allocation tasks. The core innovation lies in achieving, for the first time, regret guarantees superior to O(√T) in fractional allocation settings, thereby breaking through traditional convergence rate limitations. By integrating adversarial online learning frameworks with budget-constrained optimization techniques, this work attains near-optimal regret bounds and establishes a theoretical breakthrough of o(√T) in resource allocation. Ultimately, it provides a novel paradigm for constrained online decision-making.

Abstract

We establish near-optimal regret bounds for budget-constrained online learning against arbitrary classes of budget-pacing experts in the adversarial setting. In particular, given any class of $F$ experts and a candidate budget pacing schedule, we provide a full-information algorithm which obtains regret $O(D \sqrt{\log F}+ \sqrt{T\log F})$ against all experts whose cumulative spending stays within distance $D$ of this schedule, matching lower bounds established by Braverman et al. (2025). We additionally show that our technique extends to various problems in online resource allocation, where the learner gets to see the rewards and costs of the current options available to them, and establish $O(D\sqrt{\log F})$ regret bounds when fractional allocation is allowed. This is the first algorithm we are aware of which can achieve $o(\sqrt{T})$ guarantees for such tasks.

Problem

• budget pacing
• online learning
• regret bounds
• adversarial setting
• online resource allocation

Innovation

• budget pacing
• online learning
• regret bounds
• adversarial setting
• online resource allocation