Gradient-Variation Regret Bounds for Unconstrained Online Learning

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 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).

Machine Learning (cs.LG); Machine Learning (stat.ML)

arXiv:2604.11151 [cs.LG]

(or arXiv:2604.11151v2 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2604.11151 Focus to learn more arXiv-issued DOI via DataCite

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