Improving Adaptive Online Learning Using Refined Discretization

TL;DR

This paper introduces a new adaptive online learning algorithm that achieves optimal regret bounds without prior knowledge of environment parameters, using a novel continuous-to-discrete time discretization approach.

Contribution

The paper presents a discretization technique that preserves adaptivity from continuous to discrete online learning, enabling an algorithm with optimal regret bounds and parameter independence.

Findings

Achieves $O(\sqrt{V_T})$ regret without logarithmic factors.

Does not require a priori Lipschitz constant estimates.

Uses a novel discretization preserving continuous-time adaptivity.

Abstract

We study unconstrained Online Linear Optimization with Lipschitz losses. Motivated by the pursuit of instance optimality, we propose a new algorithm that simultaneously achieves ($i$) the AdaGrad-style second order gradient adaptivity; and ($ii$) the comparator norm adaptivity also known as "parameter freeness" in the literature. In particular, - our algorithm does not employ the impractical doubling trick, and does not require an a priori estimate of the time-uniform Lipschitz constant; - the associated regret bound has the optimal $O(\sqrt{V_T})$ dependence on the gradient variance $V_T$, without the typical logarithmic multiplicative factor; - the leading constant in the regret bound is "almost" optimal. Central to these results is a continuous time approach to online learning. 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.