Minimax Regret for Bandit Convex Optimisation of Ridge Functions

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