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.
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Publication details
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- Open access
- Green open access
Cite this article
APA 7
Braverman, M., Liu, J., Mao, J., Schneider, J., & Xue, E. (2026). Optimally Pacing Budget Spending and Learning. https://omanscience.com/en/articles/optimally-pacing-budget-spending-and-learning
MLA 9
Braverman, Mark, et al. "Optimally Pacing Budget Spending and Learning." https://omanscience.com/en/articles/optimally-pacing-budget-spending-and-learning.
Chicago (author–date)
Braverman, Mark, Jingyi Liu, Jieming Mao, Jon Schneider, and Eric Xue. 2026. "Optimally Pacing Budget Spending and Learning." https://omanscience.com/en/articles/optimally-pacing-budget-spending-and-learning.
Harvard
Braverman, M., Liu, J., Mao, J., Schneider, J. and Xue, E. (2026) 'Optimally Pacing Budget Spending and Learning', Available at: https://omanscience.com/en/articles/optimally-pacing-budget-spending-and-learning.
Vancouver
Braverman M, Liu J, Mao J, Schneider J, Xue E. Optimally Pacing Budget Spending and Learning. https://omanscience.com/en/articles/optimally-pacing-budget-spending-and-learning
IEEE
M. Braverman, J. Liu, J. Mao, J. Schneider, and E. Xue, "Optimally Pacing Budget Spending and Learning," https://omanscience.com/en/articles/optimally-pacing-budget-spending-and-learning.