Journal
IEEE TRANSACTIONS ON INFORMATION THEORY
Volume 62, Issue 6, Pages 3721-3730Publisher
IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC
DOI: 10.1109/TIT.2016.2555812
Keywords
Complexity penalty; generalized linear models; Kullback-Leibler risk; minimax estimator; model selection; sparsity
Funding
- Israel Science Foundation [ISF-820/13]
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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 Kullback-Leibler risk of the resulting estimators and establish the corresponding minimax lower bounds for the 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 also discuss possible extensions of the proposed approach to model selection in the GLM under additional structural constraints and aggregation.
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