Generalization Error Bounds for Multiclass Sparse Linear Classifiers
This work provides theoretical guarantees for multiclass sparse linear classifiers, which is incremental but important for statistical learning applications.
The authors tackled high-dimensional multiclass classification by developing a feature selection procedure using penalized maximum likelihood with convex penalties for various sparsity types, achieving minimax generalization error bounds for misclassification excess risk.
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.