Improved Regret for Zeroth-Order Adversarial Bandit Convex Optimisation

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$d$isthedimensionand$n$ 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.