Linear Classifiers Under Infinite Imbalance

53 Pages Posted: 11 Jun 2021 Last revised: 12 May 2023

See all articles by Paul Glasserman

Paul Glasserman

Columbia Business School

Mike Li

Columbia University - Columbia Business School

Date Written: October 2022

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.

Keywords: classification, statistics, data imbalance, credit risk

JEL Classification: C38, C18, C44, G21

Suggested Citation

Glasserman, Paul and Li, Mike, Linear Classifiers Under Infinite Imbalance (October 2022). Columbia Business School Research Paper , Available at SSRN: https://ssrn.com/abstract=3863653 or http://dx.doi.org/10.2139/ssrn.3863653

Paul Glasserman

Columbia Business School ( email )

New York, NY
United States

Mike Li (Contact Author)

Columbia University - Columbia Business School ( email )

New York, NY
United States

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