On Consistency of Graph-based Semi-supervised Learning

TL;DR

This paper investigates the statistical consistency of graph-based semi-supervised learning methods, proving conditions under which the estimator is consistent and providing counterexamples where it is not, supported by numerical experiments.

Contribution

It establishes the first proof of estimator consistency in graph-based semi-supervised learning under a non-parametric framework, including cases with and without response enforcement.

Findings

Proves consistency when scores are enforced to match observed responses.

Provides a counterexample showing potential inconsistency without response enforcement.

Supports theoretical results with numerical studies.

Abstract

Graph-based semi-supervised learning is one of the most popular methods in machine learning. Some of its theoretical properties such as bounds for the generalization error and the convergence of the graph Laplacian regularizer have been studied in computer science and statistics literatures. However, a fundamental statistical property, the consistency of the estimator from this method has not been proved. In this article, we study the consistency problem under a non-parametric framework. We prove the consistency of graph-based learning in the case that the estimated scores are enforced to be equal to the observed responses for the labeled data. The sample sizes of both labeled and unlabeled data are allowed to grow in this result. When the estimated scores are not required to be equal to the observed responses, a tuning parameter is used to balance the loss function and the graph Laplacian regularizer. We give a counterexample demonstrating that the estimator for this case can be inconsistent. The theoretical findings are supported by numerical studies.