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Achieving Fairness with a Simple Ridge Penalty

Marco Scutari, Francesca Panero, Manuel Proissl

TL;DR

The paper tackles fairness in regression by bounding the share of outcome variance explained by sensitive attributes, using $R^2_S(\boldsymbol{\alpha},\boldsymbol{\beta})$ to enforce a user-specified fairness level $r$. It introduces FRRM (and its GLM extension FGRRM), a two-stage, penalty-driven approach that places a ridge penalty on the sensitive-attribute coefficients $\boldsymbol{\alpha}$ and estimates $\boldsymbol{\beta}$ from decorrelated predictors $\widehat{\mathbf{U}}$, yielding mostly closed-form solutions and a simple one-dimensional root-finding step for the penalty. This modular framework is extendable to kernels and multiple definitions of fairness (statistical parity, equality of opportunity, individual fairness) and separates model selection from estimation for ease of interpretation. Empirically, FRRM and FGRRM achieve better predictive accuracy and goodness-of-fit than nonconvex NCLM and Zafar baselines at the same fairness level, while being more robust and simpler to implement. The work also highlights biases in prior evaluations and argues for a practical, extensible approach to fair regression in diverse data settings.

Abstract

In this paper we present a general framework for estimating regression models subject to a user-defined level of fairness. We enforce fairness as a model selection step in which we choose the value of a ridge penalty to control the effect of sensitive attributes. We then estimate the parameters of the model conditional on the chosen penalty value. Our proposal is mathematically simple, with a solution that is partly in closed form, and produces estimates of the regression coefficients that are intuitive to interpret as a function of the level of fairness. Furthermore, it is easily extended to generalised linear models, kernelised regression models and other penalties; and it can accommodate multiple definitions of fairness. We compare our approach with the regression model from Komiyama et al. (2018), which implements a provably-optimal linear regression model; and with the fair models from Zafar et al. (2019). We evaluate these approaches empirically on six different data sets, and we find that our proposal provides better goodness of fit and better predictive accuracy for the same level of fairness. In addition, we highlight a source of bias in the original experimental evaluation in Komiyama et al. (2018).

Achieving Fairness with a Simple Ridge Penalty

TL;DR

The paper tackles fairness in regression by bounding the share of outcome variance explained by sensitive attributes, using to enforce a user-specified fairness level . It introduces FRRM (and its GLM extension FGRRM), a two-stage, penalty-driven approach that places a ridge penalty on the sensitive-attribute coefficients and estimates from decorrelated predictors , yielding mostly closed-form solutions and a simple one-dimensional root-finding step for the penalty. This modular framework is extendable to kernels and multiple definitions of fairness (statistical parity, equality of opportunity, individual fairness) and separates model selection from estimation for ease of interpretation. Empirically, FRRM and FGRRM achieve better predictive accuracy and goodness-of-fit than nonconvex NCLM and Zafar baselines at the same fairness level, while being more robust and simpler to implement. The work also highlights biases in prior evaluations and argues for a practical, extensible approach to fair regression in diverse data settings.

Abstract

In this paper we present a general framework for estimating regression models subject to a user-defined level of fairness. We enforce fairness as a model selection step in which we choose the value of a ridge penalty to control the effect of sensitive attributes. We then estimate the parameters of the model conditional on the chosen penalty value. Our proposal is mathematically simple, with a solution that is partly in closed form, and produces estimates of the regression coefficients that are intuitive to interpret as a function of the level of fairness. Furthermore, it is easily extended to generalised linear models, kernelised regression models and other penalties; and it can accommodate multiple definitions of fairness. We compare our approach with the regression model from Komiyama et al. (2018), which implements a provably-optimal linear regression model; and with the fair models from Zafar et al. (2019). We evaluate these approaches empirically on six different data sets, and we find that our proposal provides better goodness of fit and better predictive accuracy for the same level of fairness. In addition, we highlight a source of bias in the original experimental evaluation in Komiyama et al. (2018).

Paper Structure

This paper contains 4 sections, 14 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A Nonconvex Optimisation Approach to Fairness
  3. The Optimisation Problem
  4. Avoiding Bias in the Auxiliary Model

Figures (1)

  • Figure 1: Bias introduced by penalised regression in $R^2_\mathbf{S}(\boldsymbol{\alpha}, \boldsymbol{\beta})$. The shaded area corresponds to the range of optimal values for $\lambda$ used in computing $\widehat{\mathbf{B}}_{\lambda}$.