© 2017 Neural information processing systems foundation. All rights reserved. We study a variant of online linear optimization where the player receives a hint about the loss function at the beginning of each round. The hint is given in the form of a vector that is weakly correlated with the loss vector on that round. We show that the player can benefit from such a hint if the set of feasible actions is sufficiently round. Specifically, if the set is strongly convex, the hint can be used to guarantee a regret of O(log(T)), and if the set is q-uniformly convex for q ∈ (2, 3), the hint can be used to guarantee a regret of o(√T). In contrast, we establish Ω(VT) lower bounds on regret when the set of feasible actions is a polyhedron.