We study unconstrained Online Linear Optimization with Lipschitz losses. The goal is to simultaneously achieve ($i$) second order gradient adaptivity; and ($ii$) comparator norm adaptivity also known as "parameter freeness" in the literature. Existing regret bounds (Cutkosky and Orabona, 2018; Mhammedi and Koolen, 2020; Jacobsen and Cutkosky, 2022) have the suboptimal $O(\sqrt{V_T\log V_T})$ dependence on the gradient variance $V_T$, while the present work improves it to the optimal rate $O(\sqrt{V_T})$ using a novel continuous-time-inspired algorithm, without any impractical doubling trick. This result can be extended to the setting with unknown Lipschitz constant, eliminating the range ratio problem from prior works (Mhammedi and Koolen, 2020). Concretely, we first show that the aimed simultaneous adaptivity can be achieved fairly easily in a continuous time analogue of the problem, where the environment is modeled by an arbitrary continuous semimartingale. Then, our key innovation is a new discretization argument that preserves such adaptivity in the discrete time adversarial setting. This refines a non-gradient-adaptive discretization argument from (Harvey et al., 2023), both algorithmically and analytically, which could be of independent interest.

翻译：我们研究具有Lipschitz损失的无约束在线线性优化问题。目标是同时实现：(i) 二阶梯度自适应性；(ii) 比较器范数自适应性（文献中亦称“参数无关性”）。现有遗憾界（Cutkosky和Orabona，2018；Mhammedi和Koolen，2020；Jacobsen和Cutkosky，2022）对梯度方差$V_T$呈现次优的$O(\sqrt{V_T\log V_T})$依赖，而本文通过一种新颖的连续时间启发式算法将其改进为最优速率$O(\sqrt{V_T})$，且无需任何不实用的加倍技巧。该结果可推广至Lipschitz常数未知的情形，消除了先前工作（Mhammedi和Koolen，2020）中的范围比率问题。具体而言，我们首先证明，在问题的连续时间类比中（环境由任意连续半鞅建模），目标中的同时自适应性可较易实现。随后，我们的关键创新在于一种新的离散化论证，该论证在离散时间对抗性设定中保留了此类自适应性。这从算法与分析角度对(Harvey et al., 2023)中的非梯度自适应离散化论证进行了精细化改进，可能具有独立的研究价值。