Generalization Error Bounds for Multiclass Sparse Linear Classifiers

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.

Multiclass classification by sparse multinomial logistic regression

In this paper we consider high-dimensional multiclass classification by sparse multinomial logistic regression. We propose a feature selection procedure based on penalized maximum likelihood with a complexity penalty on the model size and derive the nonasymptotic bounds for misclassification excess risk of the resulting classifier. We establish also their tightness by deriving the corresponding minimax lower bounds. In particular, we show that there exist two regimes corresponding to small and large number of classes. The bounds can be reduced under the additional low noise condition. Implementation of any complexity penalty based procedure, however, requires a combinatorial search over all possible models. To find a feature selection procedure computationally feasible for high-dimensional data, we propose multinomial logistic group Lasso and Slope classifiers and show that they also achieve the optimal order in the minimax sense.