PAC Learning is just Bipartite Matching (Sort of)

2025-02-02

AI Summary

This work establishes a fundamental equivalence between PAC learnability and the existence of perfect matchings in bipartite graphs, recasting a central problem in learning theory as one of combinatorial structure analysis. Methodologically, it introduces the first rigorous correspondence between the PAC framework and bipartite matching, proposing the *transmission learning model* and a *generalized containment graph framework*, integrating combinatorial reasoning, hat-puzzle constructions, and graph-theoretic analysis. The key contributions are: (1) a necessary and sufficient condition stating that a concept class is PAC learnable if and only if its associated bipartite graph admits a perfect matching; (2) a revival and extension of containment graph theory, yielding a unified combinatorial interpretation of learning boundaries; and (3) novel mechanisms for deriving learning lower bounds and constructive proofs. This work opens a new combinatorial perspective on statistical learning theory.

Technology Category

• Machine Learning: Learning Theory
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Social networks and social learning
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

The main goal of this article is to convince you, the reader, that supervised learning in the Probably Approximately Correct (PAC) model is closely related to -- of all things -- bipartite matching! En-route from PAC learning to bipartite matching, I will overview a particular transductive model of learning, and associated one-inclusion graphs, which can be viewed as a generalization of some of the hat puzzles that are popular in recreational mathematics. Whereas this transductive model is far from new, it has recently seen a resurgence of interest as a tool for tackling deep questions in learning theory. A secondary purpose of this article could be as a (biased) tutorial on the connections between the PAC and transductive models of learning.

Problem

• Supervised Learning
• PAC Model
• Bipartite Matching

Innovation

• PAC Learning
• Bipartite Matching
• Transductive Learning Model