Learning Conditional Averages

TL;DR

This paper introduces the problem of learning conditional averages within the PAC framework, extending traditional PAC learning to predict neighborhood-based averages, with a complete characterization of learnability and tight sample complexity bounds.

Contribution

It provides a full characterization of when conditional averages are learnable, based on novel combinatorial parameters related to the concept class and neighborhoods.

Findings

Conditional averages generalize PAC learning.

Learnability depends on joint finiteness of new combinatorial parameters.

Sample complexity bounds are tight up to logarithmic factors.

Abstract

We introduce the problem of learning conditional averages in the PAC framework. The learner receives a sample labeled by an unknown target concept from a known concept class, as in standard PAC learning. However, instead of learning the target concept itself, the goal is to predict, for each instance, the average label over its neighborhood -- an arbitrary subset of points that contains the instance. In the degenerate case where all neighborhoods are singletons, the problem reduces exactly to classic PAC learning. More generally, it extends PAC learning to a setting that captures learning tasks arising in several domains, including explainability, fairness, and recommendation systems. Our main contribution is a complete characterization of when conditional averages are learnable, together with sample complexity bounds that are tight up to logarithmic factors. The characterization hinges on the joint finiteness of two novel combinatorial parameters, which depend on both the concept class and the neighborhood system, and are closely related to the independence number of the associated neighborhood graph.