Gradient-Variation Regret Bounds for Unconstrained Online Learning

TL;DR

This paper introduces parameter-free algorithms for unconstrained online learning that adapt to gradient variation, achieving regret bounds without prior knowledge of problem parameters.

Contribution

The authors develop fully adaptive, efficient algorithms with regret bounds based on gradient variation, extending to dynamic regret and improving SEA model results.

Findings

Achieve regret of order  ext{O}( ext{ extbar}u ext{ extbar}\u00a0\u221aV_T(u)+L ext{ extbar}u ext{ extbar}^2+G^4)

No prior knowledge of comparator norm, Lipschitz constant, or smoothness needed

Efficient closed-form update in each round

Abstract

We develop parameter-free algorithms for unconstrained online learning with regret guarantees that scale with the gradient variation $V_T(u) = \sum_{t=2}^T \|\nabla f_t(u)-\nabla f_{t-1}(u)\|^2$. For $L$-smooth convex loss, we provide fully-adaptive algorithms achieving regret of order $\widetilde{O}(\|u\|\sqrt{V_T(u)} + L\|u\|^2+G^4)$ without requiring prior knowledge of comparator norm $\|u\|$, Lipschitz constant $G$, or smoothness $L$. The update in each round can be computed efficiently via a closed-form expression. Our results extend to dynamic regret and find immediate implications to the stochastically-extended adversarial (SEA) model, which significantly improves upon the previous best-known result [Wang et al., 2025].