Robust Semi-supervised Least Squares Classification by Implicit Constraints

TL;DR

This paper proposes the implicitly constrained least squares (ICLS) classifier, a semi-supervised method that improves classification performance by leveraging implicit assumptions without adding explicit constraints, formulated as a quadratic programming problem.

Contribution

The paper introduces ICLS, a semi-supervised least squares classifier that uses implicit constraints and can be efficiently optimized with gradient descent, offering performance guarantees in certain settings.

Findings

ICLS never performs worse than supervised classifier in 1D case

ICLS improves squared loss in multidimensional settings

ICLS reduces expected classification error

Abstract

We introduce the implicitly constrained least squares (ICLS) classifier, a novel semi-supervised version of the least squares classifier. This classifier minimizes the squared loss on the labeled data among the set of parameters implied by all possible labelings of the unlabeled data. Unlike other discriminative semi-supervised methods, this approach does not introduce explicit additional assumptions into the objective function, but leverages implicit assumptions already present in the choice of the supervised least squares classifier. This method can be formulated as a quadratic programming problem and its solution can be found using a simple gradient descent procedure. We prove that, in a limited 1-dimensional setting, this approach never leads to performance worse than the supervised classifier. Experimental results show that also in the general multidimensional case performance improvements can be expected, both in terms of the squared loss that is intrinsic to the classifier, as well as in terms of the expected classification error.