Gradient-Variation Regret Bounds for Unconstrained Online Learning

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 losses, we provide fully-adaptive algorithms achieving regret of \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 for the stochastically-extended adversarial (SEA) model, which significantly improves upon the previous best-known result (Wang et al., 2025).