Learning Combinatorial Functions from Pairwise Comparisons

TL;DR

This paper introduces algorithms for learning combinatorial functions using only pairwise comparisons, addressing practical challenges where exact labels are hard to obtain, with applications across economics, social networks, and machine learning.

Contribution

It proposes novel algorithms for learning a wide range of combinatorial functions solely from pairwise comparison data, expanding the scope of learnable functions.

Findings

Algorithms successfully learn various combinatorial functions from pairwise comparisons.

The methods apply to functions like submodular, XOS, and coverage functions.

The approach is effective in settings where cardinal labels are inaccessible or unreliable.

Abstract

A large body of work in machine learning has focused on the problem of learning a close approximation to an underlying combinatorial function, given a small set of labeled examples. However, for real-valued functions, cardinal labels might not be accessible, or it may be difficult for an expert to consistently assign real-valued labels over the entire set of examples. For instance, it is notoriously hard for consumers to reliably assign values to bundles of merchandise. Instead, it might be much easier for a consumer to report which of two bundles she likes better. With this motivation in mind, we consider an alternative learning model, wherein the algorithm must learn the underlying function up to pairwise comparisons, from pairwise comparisons. In this model, we present a series of novel algorithms that learn over a wide variety of combinatorial function classes. These range from graph functions to broad classes of valuation functions that are fundamentally important in microeconomic theory, the analysis of social networks, and machine learning, such as coverage, submodular, XOS, and subadditive functions, as well as functions with sparse Fourier support.