Efficient Algorithms for Learning and Compressing Monophonic Halfspaces in Graphs

TL;DR

This paper introduces efficient algorithms for learning and compressing monophonic halfspaces in graphs, enabling polynomial-time empirical risk minimization and proper learning, contrasting with the NP-hardness of geodesic halfspaces.

Contribution

It provides a novel 2-satisfiability based decomposition theorem and efficient algorithms for learning monophonic halfspaces, addressing open questions and improving over prior NP-hard results.

Findings

Polynomial-time algorithm for empirical risk minimization.

Proper and stable sample compression scheme.

Contrast with NP-hard geodesic halfspaces.

Abstract

Abstract notions of convexity over the vertices of a graph, and corresponding notions of halfspaces, have recently gained attention from the machine learning community. In this work we study monophonic halfspaces, a notion of graph halfspaces defined through closure under induced paths. Our main result is a $2$-satisfiability based decomposition theorem, which allows one to represent monophonic halfspaces as a disjoint union of certain vertex subsets. Using this decomposition, we achieve efficient and (nearly) optimal algorithms for various learning problems, such as teaching, active, and online learning. Most notably, we obtain a polynomial-time algorithm for empirical risk minimization. Independently of the decomposition theorem, we obtain an efficient, stable, and proper sample compression scheme. This makes monophonic halfspaces efficiently learnable with proper learners and linear error rate $1/\varepsilon$ in the realizable PAC setting. Our results answer open questions from the literature, and show a stark contrast with geodesic halfspaces, for which most of the said learning problems are NP-hard.