Improved Regret for Zeroth-Order Adversarial Bandit Convex Optimisation

TL;DR

This paper improves the theoretical upper bound on the minimax regret for zeroth-order adversarial bandit convex optimization, reducing the dependence on dimension and logarithmic factors.

Contribution

It introduces an improved exploratory distribution for convex functions, leading to a tighter regret bound compared to previous work.

Findings

Reduced the regret bound from $O(d^{9.5} ext{log}(n)^{7.5})$ to $O(d^{2.5} ext{sqrt}(n) ext{log}(n))$

Demonstrated the effectiveness of the new exploratory distribution in convex bandit optimization

Provided a novel proof technique based on information-theoretic analysis

Abstract

We prove that the information-theoretic upper bound on the minimax regret for zeroth-order adversarial bandit convex optimisation is at most $O(d^{2.5} \sqrt{n} \log(n))$, where $d$ is the dimension and $n$ is the number of interactions. This improves on $O(d^{9.5} \sqrt{n} \log(n)^{7.5}$ by Bubeck et al. (2017). The proof is based on identifying an improved exploratory distribution for convex functions.