Improved Dimension Dependence for Bandit Convex Optimization with Gradient Variations

Abstract:Gradient-variation online learning has drawn increasing attention due to its deep connections to game theory and optimization. It has been studied extensively in the full-information setting, but is underexplored with bandit feedback. In this work, we focus on gradient variation in Bandit Convex Optimization (BCO) with two-point feedback. By proposing a refined analysis of the non-consecutive gradient variation, a fundamental quantity in gradient variation with bandit feedback, we improve the dimension dependence for both convex and strongly convex functions compared with the best known results (Chiang et al., 2013). Our improved analysis of the non-consecutive gradient variation also implies other favorable problem-dependent guarantees, such as gradient-variance and small-loss regret bounds. Beyond the two-point setup, we demonstrate the versatility of our technique by achieving the first gradient-variation bound for one-point bandit linear optimization over hyper-rectangular domains. Finally, we validate the effectiveness of our results in more challenging tasks such as dynamic and universal regret minimization, establishing the first gradient-variation dynamic and universal regret bounds for two-point BCO.

Machine Learning (cs.LG); Machine Learning (stat.ML)

arXiv:2602.04761 [cs.LG]

(or arXiv:2602.04761v2 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2602.04761 Focus to learn more arXiv-issued DOI via DataCite

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.