Learning with Statistical Equality Constraints

2025-11-18

AI Summary

Existing methods for enforcing equality constraints—such as inter-group statistical parity—in machine learning struggle with inefficient hyperparameter tuning and lack theoretical generalization guarantees, particularly when such constraints cannot be adequately modeled via weighted penalty terms. Method: We propose the first statistical learning generalization theory specifically for equality-constrained learning, establishing a generalization error bound for constrained empirical risk minimization. Building upon this theory, we design a novel algorithmic framework that solves a sequence of unconstrained subproblems, integrating sample-based approximation, rich parametrization, and progressive constraint satisfaction. Contribution/Results: Experiments demonstrate that our approach significantly outperforms standard weighted-penalty baselines on fairness-aware learning tasks, while enabling learning under strong semantic constraints—e.g., exact statistical equalities—that are inexpressible within conventional penalization paradigms. The framework provides both theoretical grounding and practical efficacy for equality-constrained statistical learning.

Technology Category

• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition
• Machine Learning: Learning Theory
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments

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.

Problem

• Developing generalization theory for statistical learning with equality constraints
• Proposing practical algorithm for fair learning and boundary value problems
• Addressing limitations of weighted penalty methods in constrained machine learning

Innovation

• Generalization theory for equality-constrained statistical learning
• Sequence of unconstrained empirical learning problems
• Rich parametrizations approximating equality-constrained solutions