Linear predictor on linearly-generated data with missing values: non consistency and solutions

Marine Le Morvan (1, 2) , Nicolas Prost (3, 4) , Julie Josse (3, 4) , Erwan Scornet (3) , Gaël Varoquaux (1, 5)

1 PARIETAL - Modelling brain structure, function and variability based on high-field MRI data
2 IJCLab - Laboratoire de Physique des 2 Infinis Irène Joliot-Curie
3 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
4 XPOP - Modélisation en pharmacologie de population
5 MILA - Montreal Institute for Learning Algorithms [Montréal]

We consider building predictors when the data have missing values. We study the seemingly-simple case where the target to predict is a linear function of the fully-observed data and we show that, in the presence of missing values, the optimal predictor may not be linear. In the particular Gaussian case, it can be written as a linear function of multiway interactions between the observed data and the various missing-value indicators. Due to its intrinsic complexity, we study a simple approximation and prove generalization bounds with finite samples, highlighting regimes for which each method performs best. We then show that multilayer perceptrons with ReLU activation functions can be consistent, and can explore good trade-offs between the true model and approximations. Our study highlights the interesting family of models that are beneficial to fit with missing values depending on the amount of data available.

Domaines: Autres [stat.ML]; Intelligence artificielle [cs.AI]

Marine Le Morvan, Nicolas Prost, Julie Josse, Erwan Scornet, Gaël Varoquaux. Linear predictor on linearly-generated data with missing values: non consistency and solutions. AISTATS 2020 - International Conference on Artificial Intelligence and Statistics, Aug 2020, Online, France. pp.3165-3174. ⟨hal-02464569v2⟩