Small Gradient Norm Regret for Online Convex Optimization

TL;DR

This paper proposes a new $G^*$ regret measure for online convex optimization that depends on the cumulative squared gradient norm, providing sharper bounds and insights especially when losses have vanishing curvature.

Contribution

It introduces the $G^*$ regret, a problem-dependent measure that refines existing regret notions and extends analysis to dynamic and bandit settings, with improved theoretical bounds.

Findings

$G^*$ regret offers tighter bounds than $L^*$ regret.

The measure adapts to losses with vanishing curvature.

Experimental results support theoretical claims.

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.