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 $\sum_{t=1}^T \|\nabla \ell(x^\star)\|^2$. 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.

翻译：本文针对光滑损失函数的在线凸优化问题，引入了一种新的问题依赖遗憾度量。该度量称为$G^\star$遗憾，其定义依赖于在事后最优决策处计算的累积平方梯度范数$\sum_{t=1}^T \|\nabla \ell(x^\star)\|^2$。我们证明$G^\star$遗憾严格改进了现有的$L^\star$（小损失）遗憾度量，且当损失函数在事后最优决策附近具有消失曲率时，其度量精度可任意提高。我们建立了$G^\star$遗憾的上界与下界，并将结果推广至动态遗憾和赌博机场景。作为副产品，我们改进了插值条件下随机优化算法的现有收敛性分析。部分实验验证了我们的理论结果。