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].

翻译：我们针对无约束在线学习问题开发了无参数算法，其遗憾保证随梯度变化量 $V_T(u) = \sum_{t=2}^T \|\nabla f_t(u)-\nabla f_{t-1}(u)\|^2$ 缩放。对于 $L$-光滑凸损失函数，我们提供了完全自适应的算法，可在无需预先知晓比较器范数 $\|u\|$、Lipschitz 常数 $G$ 或光滑度 $L$ 的情况下实现 $\widetilde{O}(\|u\|\sqrt{V_T(u)} + L\|u\|^2+G^4)$ 量级的遗憾。每轮更新可通过闭式表达式高效计算。我们的结果可推广至动态遗憾，并对随机扩展对抗（SEA）模型产生直接影响，显著改进了先前最优结果 [Wang et al., 2025]。