High-dimensional Multi-Task Learning using Multivariate Regression and Generalized Fiducial Inference

Over the past decades, the Multi-Task Learning (MTL) problem has attracted much attention in the artificial intelligence and machine learning communities. However, most published work in this area focuses on point estimation; i.e., estimating model parameters and/or making predictions. This paper studies another important aspect of the MTL problem: uncertainty quantification for model choices and predictions. To be more specific, this paper approaches the MTL problem with multivariate regression and develops a novel method for deriving a probability density function on the space of all potential regression models. With this density function, point estimates, as well as confidence and prediction ellipsoids, can be obtained for quantities of interest, such as future observations. The proposed method, termed GMTask, is based on the generalized fiducial inference (GFI) framework and is shown to enjoy desirable theoretical properties. Its promising empirical properties are illustrated via a sequence of numerical experiments and applications to two real data sets.