Title: Optimal learning via local entropies and sample compression,
Abstract: The aim of this paper is to provide several novel upper bounds on the excess
risk with a primal focus on classification problems. We suggest two approaches
and the obtained bounds are represented via the distribution dependent local
entropies of the classes or the sizes of specific sample com- pression schemes.
We show that in some cases, our guarantees are optimal up to constant factors
and outperform previously known results. As an application of our results, we
provide a new tight PAC bound for the hard-margin SVM, an extended analysis of
certain empirical risk minimizers under log-concave distributions, a new
variant of an online to batch conversion, and distribution dependent localized
bounds in the aggregation framework. We also develop techniques that allow to
replace empirical covering number or covering numbers with bracketing by the
coverings with respect to the distribution of the data. The proofs for the
sample compression schemes are based on the moment method combined with the
analysis of voting algorithms.