This paper is concerned with the study of constrained statistical learning problems, the unconstrained version of which are at the core of virtually all of modern information processing. Accounting for constraints, however, is paramount to incorporate prior knowledge and impose desired structural and statistical properties on the solutions. Still, solving constrained statistical problems remains challenging and guarantees scarce, leaving them to be tackled using regularized formulations. Though practical and effective, selecting regularization parameters so as to satisfy requirements is challenging, if at all possible, due to the lack of a straightforward relation between parameters and constraints. In this work, we propose to directly tackle the constrained statistical problem overcoming its infinite dimensionality, unknown distributions, and constraints by leveraging finite dimensional parameterizations, sample averages, and duality theory. Aside from making the problem tractable, these tools allow us to bound the empirical duality gap, i.e., the difference between our approximate tractable solutions and the actual solutions of the original statistical problem. We demonstrate the effectiveness and usefulness of this constrained formulation in a fair learning application.

本文提出了直接解决约束统计问题的方法，通过有限维参数化，样本平均和对偶理论，克服了其无限维度，未知分布和约束的限制。此方法可以在公平学习应用中体现出其有效性和有用性。

约束统计学习的实证对偶间隙