Dealing with mixed hard/soft constraints via support constraint machines

A learning paradigm is presented, which extends the classical framework of learning from examples by including hard pointwise constraints, i.e., constraints that cannot be violated. In applications, hard pointwise constraints may encode very precise prior knowledge coming from rules, applied, e.g., to a large collection of unsupervised examples. The classical learning framework corresponds to soft pointwise constraints, which can be violated at the cost of some penalization. The functional structure of the optimal solution is derived in terms of a set of “support constraints”, which generalize the classical concept of “support vectors”. They are at the basis of a novel learning parading, that we called “Support Constraint Machines”. A case study and a numerical example are presented.