Minimax Regret for Bandit Convex Optimisation of Ridge Functions

TL;DR

This paper studies adversarial bandit convex optimization where functions are ridge functions, providing an information-theoretic bound on the minimax regret that depends on dimension, number of rounds, and set diameter.

Contribution

It introduces a novel analysis of bandit convex optimization with ridge functions, establishing a regret bound using an information-theoretic approach.

Findings

Minimax regret is bounded by O(d√n log(n diam(K)))

The analysis applies to functions of the form g_t(⟨x, θ⟩) with unknown θ

Provides a short, elegant proof of regret bounds in this setting

Abstract

We analyse adversarial bandit convex optimisation with an adversary that is restricted to playing functions of the form $f_t(x) = g_t(\langle x, \theta\rangle)$ for convex $g_t : \mathbb R \to \mathbb R$ and unknown $\theta \in \mathbb R^d$ that is homogeneous over time. We provide a short information-theoretic proof that the minimax regret is at most $O(d \sqrt{n} \log(n \operatorname{diam}(\mathcal K)))$ where $n$ is the number of interactions, $d$ the dimension and $\operatorname{diam}(\mathcal K)$ is the diameter of the constraint set.