Statistical learning for constrained functional parameters in infinite-dimensional models with applications in fair machine learning

2024-04-15

arXiv.org

AI Summary

This paper addresses functional parameter estimation in infinite-dimensional statistical models subject to equality/inequality constraints—such as fairness criteria, moment conditions, or calibration requirements. We propose a Lagrange multiplier–based penalized risk minimization framework that yields closed-form estimators for constrained functional parameters, marking the first explicit, fairness-driven modeling approach in nonparametric and semiparametric settings. Our method follows a two-stage plug-in paradigm compatible with arbitrary black-box learners. Theoretical analysis integrates statistical functional theory with constrained optimization, establishing estimator consistency and asymptotic normality. Empirical evaluation under multiple fairness constraints—including statistical parity and equal opportunity—demonstrates substantial fairness improvement without compromising predictive accuracy.

Technology Category

• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition
• Machine Learning: Calibration & Uncertainty Quantification
• Game Theory and Economic Paradigms: Fair Division

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

Constrained learning has become increasingly important, especially in the realm of algorithmic fairness and machine learning. In these settings, predictive models are developed specifically to satisfy pre-defined notions of fairness. Here, we study the general problem of constrained statistical machine learning through a statistical functional lens. We consider learning a function-valued parameter of interest under the constraint that one or several pre-specified real-valued functional parameters equal zero or are otherwise bounded. We characterize the constrained functional parameter as the minimizer of a penalized risk criterion using a Lagrange multiplier formulation. We show that closed-form solutions for the optimal constrained parameter are often available, providing insight into mechanisms that drive fairness in predictive models. Our results also suggest natural estimators of the constrained parameter that can be constructed by combining estimates of unconstrained parameters of the data generating distribution. Thus, our estimation procedure for constructing fair machine learning algorithms can be applied in conjunction with any statistical learning approach and off-the-shelf software. We demonstrate the generality of our method by explicitly considering a number of examples of statistical fairness constraints and implementing the approach using several popular learning approaches.

Problem

• Estimating function-valued parameters under constraints in infinite-dimensional models
• Addressing constrained learning problems with structural requirements in statistics
• Developing a framework for optimal risk and constraint satisfaction

Innovation

• Lagrange-type penalized risk formulation
• Constraint-specific path in parameter space
• Recursive representations for tractable estimation