Gradient-Variation Regret Bounds for Unconstrained Online Learning

arXiv:2604.11151v1 Announce Type: new 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].

Source: arXiv cs.LG | 2026-04-14