Abstract

We consider multi-task learning in the setting of multiple linear regression, and where some relevant features could be shared across the tasks. Recent research has studied the use of norm block-regularizations with for such blocksparse structured problems, establishing strong guarantees on recovery even under high-dimensional scaling where the number of features scale with the number of observations. However, these papers also caution that the performance of such block-regularized methods are very dependent on the extent to which the features are shared across tasks. Indeed they show [8] that if the extent of overlap is less than a threshold, or even if parameter values in the shared features are highly uneven, then block regularization could actually perform worse than simple separate elementwise regularization. Since these caveats depend on the unknown true parameters, we might not know when and which method to apply. Even otherwise, we are far away from a realistic multi-task setting: not only do the set of relevant features have to be exactly the same across tasks, but their values have to as well.