Gradient-Variation Regret Bounds for Unconstrained Online Learning
About
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].
Yuheng Zhao, Andrew Jacobsen, Nicol\`o Cesa-Bianchi, Peng Zhao• 2026
Related benchmarks
| Task | Dataset | Result | Rank | |
|---|---|---|---|---|
| Static Regret Minimization | Gradient-variation regret static Fully Adaptive | Regret (O-notation)2 | 4 | |
| Static Regret Minimization | Gradient-variation regret static Comparator Adaptive | Regret (O-notation)2 | 3 |
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