# Model selection and minimax estimation in generalized linear models

arXiv: https://arxiv.org/abs/1409.8491 · HTML: https://mlsanity.com/paper/1409.8491/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Sep 2014.

## Built on by (most cited first, 7 of 7)

- [High-dimensional Bayesian Tobit regression for censored response with Horseshoe prior](https://mlsanity.com/paper/2505.08288/) (2025)
- [Heterogeneous Overdispersed Count Data Regressions via Double Penalized Estimations](https://mlsanity.com/paper/2110.03552/) (2021)
- [Structured Low-Rank Tensors for Generalized Linear Models](https://mlsanity.com/paper/2308.02922/) (2023)
- [Finite-Sample Performance of Gradient Descent in Logistic Regression with Gaussian Design](https://mlsanity.com/paper/2606.21683/) (2026)
- [High-dimensional prediction for count response via sparse exponential weights](https://mlsanity.com/paper/2410.15381/) (2024)
- [Kullback-Leibler excess risk bounds for exponential weighted aggregation in Generalized linear models](https://mlsanity.com/paper/2504.10171/) (2025)
- [High-dimensional online learning via asynchronous decomposition: Non-divergent results, dynamic regularization, and beyond](https://mlsanity.com/paper/2603.20696/) (2026)

## Abstract (arXiv)

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.