Online Newton Method for Bandit Convex Optimisation

TL;DR

This paper presents an efficient zeroth-order bandit convex optimization algorithm with provable regret bounds in both adversarial and stochastic settings, advancing the theoretical understanding of high-dimensional online learning.

Contribution

It introduces a novel online Newton method tailored for bandit convex optimization, achieving improved regret bounds and computational efficiency.

Findings

Regret bound of $d^{3.5} \sqrt{n} ext{polylog}(n, d)$ in adversarial setting

Improved regret bound of $M d^{2} \sqrt{n} ext{polylog}(n, d)$ in stochastic setting

Algorithm is computationally efficient for high-dimensional problems

Abstract

We introduce a computationally efficient algorithm for zeroth-order bandit convex optimisation and prove that in the adversarial setting its regret is at most $d^{3.5} \sqrt{n} \mathrm{polylog}(n, d)$ with high probability where $d$ is the dimension and $n$ is the time horizon. In the stochastic setting the bound improves to $M d^{2} \sqrt{n} \mathrm{polylog}(n, d)$ where $M \in [d^{-1/2}, d^{-1 / 4}]$ is a constant that depends on the geometry of the constraint set and the desired computational properties.