Before revision: 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 n \log(  diam  \mathcal K))  where n is the number of interactions, d the dimension and diam(\mathcal K) is the diameter of the constraint set .
After revision: 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 that is homogeneous over time . We provide a short information-theoretic proof that the minimax regret is at most O(d n \log( diam \mathcal K)) where n is the number of interactions, d the dimension and diam(\mathcal K) is the diameter of the constraint set .