Learning with Statistical Equality Constraints

TL;DR

This paper develops a generalization theory for equality-constrained learning, enabling practical algorithms that improve fairness, boundary value problems, and classifier interpolation by effectively handling equality constraints.

Contribution

It introduces a novel generalization framework for equality-constrained learning and proposes a practical algorithm based on unconstrained problems, addressing limitations of existing methods.

Findings

Effective in fair learning scenarios

Improves boundary value problem solutions

Enables new classifier interpolation methods

Abstract

As machine learning applications grow increasingly ubiquitous and complex, they face an increasing set of requirements beyond accuracy. The prevalent approach to handle this challenge is to aggregate a weighted combination of requirement violation penalties into the training objective. To be effective, this approach requires careful tuning of these hyperparameters (weights), involving trial-and-error and cross-validation, which becomes ineffective even for a moderate number of requirements. These issues are exacerbated when the requirements involve parities or equalities, as is the case in fairness and boundary value problems. An alternative technique uses constrained optimization to formulate these learning problems. Yet, existing approximation and generalization guarantees do not apply to problems involving equality constraints. In this work, we derive a generalization theory for equality-constrained statistical learning problems, showing that their solutions can be approximated using samples and rich parametrizations. Using these results, we propose a practical algorithm based on solving a sequence of unconstrained, empirical learning problems. We showcase its effectiveness and the new formulations enabled by equality constraints in fair learning, interpolating classifiers, and boundary value problems.