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 $f_t(x) = g_t(langle x, thetarangle)$ 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.