Sample and Computationally Efficient Learning Algorithms under S-Concave Distributions
This work addresses the problem of sample-efficient learning for researchers and practitioners dealing with non-log-concave distributions, though it is incremental as it builds on existing learning frameworks.
The paper tackles the challenge of learning under s-concave distributions, which are broad and include fat-tailed cases like Pareto and t-distributions, by introducing new convex geometry tools to analyze properties such as disagreement probabilities and coefficients, leading to generalizations of prior results for active and passive learning of halfspaces.
We provide new results for noise-tolerant and sample-efficient learning algorithms under $s$-concave distributions. 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 optimization, but much remains unknown about the geometry of this class of distributions and their applications in the context of learning. The challenge is that unlike the commonly used distributions in learning (uniform or more generally log-concave distributions), this broader class is not closed under the marginalization operator and many such distributions are fat-tailed. In this work, we introduce new convex geometry tools to study the properties of $s$-concave distributions and use these properties to provide bounds on 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 generalize prior results for margin-based active learning, disagreement-based active learning, and passive learning of intersections of halfspaces. Our analysis of geometric properties of $s$-concave distributions might be of independent interest to optimization more broadly.