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This paper is concerned with the study of constrained statistical learningproblems, the unconstrained version of which are at the core of virtually allof modern information processing. Accounting for constraints, however, isparamount to incorporate prior knowledge and impose desired structural andstatistical properties on the solutions. Still, solving constrained statisticalproblems remains challenging and guarantees scarce, leaving them to be tackledusing regularized formulations. Though practical and effective, selectingregularization parameters so as to satisfy requirements is challenging, if atall possible, due to the lack of a straightforward relation between parametersand constraints. In this work, we propose to directly tackle the constrainedstatistical problem overcoming its infinite dimensionality, unknowndistributions, and constraints by leveraging finite dimensionalparameterizations, sample averages, and duality theory. Aside from making theproblem tractable, these tools allow us to bound the empirical duality gap,i.e., the difference between our approximate tractable solutions and the actualsolutions of the original statistical problem. We demonstrate the effectivenessand usefulness of this constrained formulation in a fair learning application.

The empirical duality gap of constrained statistical learning