Sample and Computationally Efficient Learning Algorithms under S-Concave Distributions

We provide new results for noise-tolerant and sample-efficient learning algorithms under s-concavedistributions. The new class of s-concave distributions is a broad and natural generalization of log-concavity, and includes many important additional distributions, e.g., the Pareto distribution and t-distribution. This class has been studied in the context of efficient sampling, integration, and optimiza-tion, but much remains unknown about the geometry of this class of distributions and their applications inthe context of learning. The challenge is that unlike the commonly used distributions in learning (uniformor more generally log-concave distributions), this broader class is not closed under the marginalizationoperator and many such distributions are fat-tailed. In this work, we introduce new convex geome-try tools to study the properties of s-concave distributions and use these properties to provide boundson quantities of interest to learning including the probability of disagreement between two halfspaces,disagreement outside a band, and the disagreement coefficient. We use these results to significantly gen-eralize prior results for margin-based active learning, disagreement-based active learning, and passivelearning of intersections of halfspaces. Our analysis of geometric properties of s-concave distributionsmight be of independent interest to optimization more broadly.