Small Gradient Norm Regret for Online Convex Optimization

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.