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 row-wise 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.

In this paper we consider high-dimensional multiclass classification by sparse multinomial logistic regression. We propose first 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 is a phase transition between small and large number of classes. The bounds can be reduced under the additional low noise condition. To find a penalized maximum likelihood solution with a complexity penalty requires, however, a combinatorial search over all possible models. To design 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 minimax order.

This paper is a theoretical study of a practical problem: the automatic generation of Java Fluent APIs from their specification. We explain why the problem's core lies with the expressive power of Java generics. Our main result is that automatic generation is possible whenever the specification is an instance of the set of deterministic context-free languages, a set which contains most "practical" languages. Other contributions include a collection of techniques and idioms of the limited meta-programming possible with Java generics, and an empirical measurement demonstrating that the runtime of the "javac" compiler of Java may be exponential in the program's length, even for programs composed of a handful of lines and which do not rely on overly complex use of generics.