Multi-Task Learning with Covariate-Overlap Regularization

Multi-task learning improves data efficiency by sharing information across related tasks, but indiscriminate sharing can be harmful when their covariate distributions and response relationships differ. We propose COVariate-ovERlap regularized multi-task learning (COVER) to address covariate and posterior heterogeneity. The model combines a common component function with a shared neural representation and low-dimensional task-specific coefficients. Taskwise second-moment matrices summarize covariate heterogeneity and determine the strength of coefficient integration in each representation direction. We derive a covariate-overlap penalty by minimizing the total squared change in two task predictors when their coefficients are replaced by one auxiliary coefficient. An equivalent auxiliary formulation supports end-to-end training without matrix inversion. An exact fixed-representation bias--variance decomposition quantifies how covariate overlap controls variance reduction and how posterior heterogeneity determines shrinkage bias. Global and localized end-to-end oracle inequalities account for jointly learning the neural functions and estimating the overlap matrices from the same observations. We give explicit neural-network rates and sharpen the stochastic prediction term when the regularized oracle risk and overlap-estimation error are small. Simulations across diverse heterogeneity settings show competitive performance against deep-learning and statistical data-integration methods, with the largest gains under joint heterogeneity. In a GTEx central-nervous-system analysis, COVER achieves the lowest response-averaged prediction error among the compared methods and reveals tissue-pair integration patterns.