PAC-Bayesian Generalization Bound on Confusion Matrix for Multi-Class Classification

TL;DR

This paper introduces the first PAC-Bayesian generalization bounds for multi-class classifiers using confusion matrices, providing a richer performance measure than traditional scalar metrics.

Contribution

It develops novel PAC-Bayesian bounds for the confusion matrix in multi-class classification, leveraging recent matrix concentration inequalities.

Findings

Bounds relate true and empirical confusion risks.

Bounds depend on class-specific training sample sizes.

First PAC-Bayes bounds based on confusion matrices.

Abstract

In this work, we propose a PAC-Bayes bound for the generalization risk of the Gibbs classifier in the multi-class classification framework. The novelty of our work is the critical use of the confusion matrix of a classifier as an error measure; this puts our contribution in the line of work aiming at dealing with performance measure that are richer than mere scalar criterion such as the misclassification rate. Thanks to very recent and beautiful results on matrix concentration inequalities, we derive two bounds showing that the true confusion risk of the Gibbs classifier is upper-bounded by its empirical risk plus a term depending on the number of training examples in each class. To the best of our knowledge, this is the first PAC-Bayes bounds based on confusion matrices.