We develop unified information-theoretic machinery for deriving lower bounds for passive and active learning schemes. Our bounds involve the so-called Alexander's capacity function. The supremum of this function has been recently rediscovered by Hanneke in the context of active learning under the name of "disagreement coefficient." For passive learning, our lower bounds match the upper bounds of Giné and Koltchinskii up to constants and generalize analogous results of Massart and Nédélec. For active learning, we provide first known lower bounds based on the capacity function rather than the disagreement coefficient.

Adversarial Optimization (AO) provides a reliable, practical way to match two implicitly defined distributions, one of which is usually represented by a sample of real data, and the other is defined by a generator. Typically, AO involves training of a high-capacity model on each step of the optimization. In this work, we consider computationally heavy generators, for which training of high-capacity models is associated with substantial computational costs. To address this problem, we introduce a novel family of divergences, which varies the capacity of the underlying model, and allows for a significant acceleration with respect to the number of samples drawn from the generator. We demonstrate the performance of the proposed divergences on several tasks, including tuning parameters of a physics simulator, namely, Pythia event generator.

We develop unified information-theoretic machinery for deriving lower bounds for passive and active learning schemes. Our bounds involve the so-called Alexander's capacity function. The supremum of this function has been recently rediscovered by Hanneke in the context of active learning under the name of "disagreement coefficient." For passive learning, our lower bounds match the upper bounds of Gine and Koltchinskii up to constants and generalize analogous results of Massart and Nedelec. For active learning, we provide first known lower bounds based on the capacity function rather than the disagreement coefficient.