In case of generic convex losses, we use the more complex parameterless algorithm AdaNormalHedge. The following theorem states a slightly more general bound that holds for anyη-exp-concave loss function (for completeness,theproofisgiveninAppendixD). Nownotethatalthough the algorithm is actually initialized withw1,i = 1, Lemma 1 shows that the regret remains the same if we assume the algorithm is initialized withwE1. Suppose that Algorithm 5 is run using predictions and updates provided by AdaNormalHedge. Asinourlocally-adaptive setting node experts are local learners,byi,t should be viewed as the prediction of the local online learning algorithm sitting at nodeiof the tree.