Improving Adaptive Online Learning Using Refined Discretization

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.