Small Gradient Norm Regret for Online Convex Optimization

Abstract:This paper introduces a new problem-dependent regret measure for online convex optimization with smooth losses. The notion, which we call the $G^\star$ regret, depends on the cumulative squared gradient norm evaluated at the decision in hindsight. We show that the $G^\star$ regret strictly refines the existing $L^\star$ (small loss) regret, and that it can be arbitrarily sharper when the losses have vanishing curvature around the hindsight decision. We establish upper and lower bounds on the $G^\star$ regret and extend our results to dynamic regret and bandit settings. As a byproduct, we refine the existing convergence analysis of stochastic optimization algorithms in the interpolation regime. Some experiments validate our theoretical findings.

Machine Learning (stat.ML); Machine Learning (cs.LG); Optimization and Control (math.OC)

arXiv:2601.13519 [stat.ML]

(or arXiv:2601.13519v4 [stat.ML] for this version)

https://doi.org/10.48550/arXiv.2601.13519 Focus to learn more arXiv-issued DOI via DataCite

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