Statistical learning for constrained functional parameters in infinite-dimensional models with applications in fair machine learning (2404.09847v1)

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.

• The paper introduces a statistical framework for constrained functional parameter estimation that ensures fairness in predictive models.
• It leverages a Lagrange multiplier method to solve infinite-dimensional optimization problems under common risk criteria such as MSE and cross-entropy.
• Simulations and a COMPAS dataset application validate the estimator's consistency and practical utility in controlling fairness constraints.

Statistical Learning for Constrained Functional Parameters in Infinite-Dimensional Models

Introduction

Statistical machine learning is critical for making decisions based on data across various domains. An emerging concern is ensuring these decisions do not exacerbate societal disparities, leading to an increased focus on algorithmic fairness. This research addresses designing predictive models within explicit fairness constraints, offering a general framework that dovetail naturally with existing machine learning approaches.

Problem Formulation

The core challenge is creating a predictive model that meets predefined fairness constraints. Traditionally, attempts to ensure fairness have been sequestered into pre-processing, in-processing, and post-processing strategies, each with limitations regarding flexibility and applicability. The proposed method transcends these categorizations by treating the constrained learning problem through a statistical functional lens, leading to a broader application spectrum.

Statistical Estimation Problem

The estimation problem is initiated by assuming access to data drawn from a distribution within an infinite-dimensional statistical model. The objective is to learn a function-valued parameter satisfying certain fairness constraints, which are specified through real-valued functional parameters. This approach employs a penalty involving a Lagrange multiplier to optimize the constrained parameter, which contrasts with existing methods that tailor solutions to specific fairness definitions or data contexts.

Methods

The methodological contribution of this paper lies in its foundational framework tailored for constrained optimization, leveraging general results in functional analysis. This facilitates applying our framework across a range of constrained learning problems. Notably, for mean squared error and cross-entropy risk criteria, the development elucidates that solutions to several fairness constraints can be characterized in a closed form. The paper delineates the estimation procedure, accentuating its model-agnostic nature, thus enabling integration with any statistical learning approach.

Examples and Applications

The paper further illustrates the versatility of the proposed framework by applying it to different fairness constraints, including average total effect and natural direct effect, showcasing its adaptability to both the mean squared error and cross-entropy risk criteria. Through these examples, the paper expounds on obtaining estimators of the constrained functional parameter, demonstrating the application's practicality and the estimator's consistency under the defined constraints.

Simulations

Simulation studies affirm the method's efficacy, demonstrating convergence of the risk of the estimator to the optimal constrained risk and appropriateness of constraint control under both equality and inequality constraints.

Real-World Application

An application to the COMPAS dataset, a benchmark in fairness studies, further evidences the practical utility of the proposed method. Through this application, the paper not only substantiates the methodological claims but also illustrates the potential for real-world impact, especially in critical decision-making domains where fairness is paramount.

Conclusion and Future Directions

This research extends the frontier of statistical machine learning by offering a robust and flexible framework for generating fair predictive models under constraints. Its broad applicability and compatibility with prevalent machine learning approaches make it a valuable tool in ensuring algorithmic fairness. Future work could explore extensions to multiple and complex constraints, enhancing the wide array of problems where this approach can be applied, further bridging the gap between technical feasibility and ethical imperatives in machine learning.