Multitask semi–supervised learning with constraints and constraint exceptions

Many applications require to jointly learn a set of related functions for which some a–priori mutual constraints are known. In particular, we consider a multitask learning problem in which a set of constraints among the different tasks are know to hold in most cases. Basically, beside a set of supervised examples provided to learn each task, we assume that some background knowledge is available in the form of functions that define the admissible configurations of the task function outputs for almost each input. We exploit a semi–supervised approach in which a potentially large set of unlabeled examples is used to enforce the constraints on a large region of the input space by means of a proper penalty function. However, since the constraints are known to be subject to exceptions and the inputs corresponding to these exceptions are not known a–priori, we propose to embed a selection criterion in the penalty function that reduces the constraint effect on those points that are likely to yield an exception. We report some experiments on multi–view object recognition showing the benefits of the proposed selection mechanism with respect to an uniform enforcement of the constraints.