  Missing values are unavoidable in many applications of machine learning and
present challenges both during training and at test time. When variables are
missing in recurring patterns, fitting separate pattern submodels have been
proposed as a solution. However, fitting models independently does not make
efficient use of all available data. Conversely, fitting a single shared model
to the full data set relies on imputation which often leads to biased results
when missingness depends on unobserved factors. We propose an alternative
approach, called sharing pattern submodels, which i) makes predictions that are
robust to missing values at test time, ii) maintains or improves the predictive
power of pattern submodels, and iii) has a short description, enabling improved
interpretability. Parameter sharing is enforced through sparsity-inducing
regularization which we prove leads to consistent estimation. Finally, we give
conditions for when a sharing model is optimal, even when both missingness and
the target outcome depend on unobserved variables. Classification and
regression experiments on synthetic and real-world data sets demonstrate that
our models achieve a favorable tradeoff between pattern specialization and
information sharing.
