Linear Classifiers Under Infinite Imbalance

TL;DR

This paper analyzes the asymptotic behavior of linear classifiers in binary classification when one class's sample size becomes infinitely large, revealing divergence in intercepts and finite limits in coefficients depending on weight functions.

Contribution

It extends prior work by characterizing the limits of linear classifiers under infinite class imbalance for a broad class of loss functions, including logistic regression.

Findings

Intercept diverges under infinite imbalance.

Coefficient vectors have finite limits depending on weight function tail growth.

Application to credit risk shows relevance in high-sensitivity and high-specificity regions.

Abstract

We study the behavior of linear discriminant functions for binary classification in the infinite-imbalance limit, where the sample size of one class grows without bound while the sample size of the other remains fixed. The coefficients of the classifier minimize an empirical loss specified through a weight function. We show that for a broad class of weight functions, the intercept diverges but the rest of the coefficient vector has a finite almost sure limit under infinite imbalance, extending prior work on logistic regression. The limit depends on the left-tail growth rate of the weight function, for which we distinguish two cases: subexponential and exponential. The limiting coefficient vectors reflect robustness or conservatism properties in the sense that they optimize against certain worst-case alternatives. In the subexponential case, the limit is equivalent to an implicit choice of upsampling distribution for the minority class. We apply these ideas in a credit risk setting, with particular emphasis on performance in the high-sensitivity and high-specificity regions.