Generalization Error Bounds for Multiclass Sparse Linear Classifiers

TL;DR

This paper develops minimax optimal generalization error bounds for high-dimensional multiclass sparse linear classifiers using penalized likelihood methods that incorporate various sparsity structures.

Contribution

It introduces a computationally feasible feature selection method with convex penalties for different sparsity notions in multiclass logistic regression, achieving minimax bounds.

Findings

Achieves minimax generalization error bounds for multiclass classifiers.

Develops a flexible penalized likelihood approach for various sparsity structures.

Demonstrates the method's adaptability to different sparsity assumptions.

Abstract

We consider high-dimensional multiclass classification by sparse multinomial logistic regression. Unlike binary classification, in the multiclass setup one can think about an entire spectrum of possible notions of sparsity associated with different structural assumptions on the regression coefficients matrix. We propose a computationally feasible feature selection procedure based on penalized maximum likelihood with convex penalties capturing a specific type of sparsity at hand. In particular, we consider global sparsity, double row-wise sparsity, and low-rank sparsity, and show that with the properly chosen tuning parameters the derived plug-in classifiers attain the minimax generalization error bounds (in terms of misclassification excess risk) within the corresponding classes of multiclass sparse linear classifiers. The developed approach is general and can be adapted to other types of sparsity as well.