ActiveHedge: Hedge meets Active Learning

We consider the classical problem of multiclass prediction with expert advice, but with an active learning twist. In this new setting the learner will only query the labels of a small number of examples, but still aims to minimize regret to the best expert as usual; the learner is also allowed a very short burn-in phase where it can fast-forward and query certain highly-informative examples. We design an algorithm that utilizes Hedge (aka Exponential Weights) as a subroutine, and we show that under a very particular combinatorial constraint on the matrix of expert predictions we can obtain a very strong regret guarantee while querying very few labels. This constraint, which we refer to as ζ-compactness, or just compactness, can be viewed as a non-stochastic variant of the disagreement coefficient, another popular parameter used to reason about the sample complexity of active learning in the IID setting. We also give a polynomial time algorithm to calculate the ζcompactness of a matrix up to an approximation factor of 3.