Model selection and minimax estimation in generalized linear models

TL;DR

This paper develops a new model selection framework for high-dimensional generalized linear models using penalized likelihood, providing theoretical guarantees and minimax optimality for sparse settings.

Contribution

It introduces a class of penalized likelihood criteria with complexity penalties, establishing nonasymptotic bounds and minimax optimality in sparse GLMs.

Findings

Derived nonasymptotic upper bounds for expected Kullback-Leibler divergence.

Established minimax lower bounds for sparse GLMs.

Proposed estimators are asymptotically minimax and adaptive to sparsity.

Abstract

We consider model selection in generalized linear models (GLM) for high-dimensional data and propose a wide class of model selection criteria based on penalized maximum likelihood with a complexity penalty on the model size. We derive a general nonasymptotic upper bound for the expected Kullback-Leibler divergence between the true distribution of the data and that generated by a selected model, and establish the corresponding minimax lower bounds for sparse GLM. For the properly chosen (nonlinear) penalty, the resulting penalized maximum likelihood estimator is shown to be asymptotically minimax and adaptive to the unknown sparsity. We discuss also possible extensions of the proposed approach to model selection in GLM under additional structural constraints and aggregation.