Linear predictor on linearly-generated data with missing values: non consistency and solutions
This addresses the challenge of handling missing values in predictive modeling, which is a common issue in real-world datasets, but the work is incremental as it builds on existing linear and neural network methods.
The paper tackles the problem of building predictors when data have missing values, showing that even for linearly-generated data, the optimal predictor may not be linear, and it proves generalization bounds for approximations and demonstrates that multilayer perceptrons can be consistent.
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.