Fair Kernel Regression Through Cross-Covariance Operators

Abstract

Ensuring fairness in machine learning models is a difficult problem from both a formulation and implementation perspective. One sensible criterion for achieving fairness is Equalised Odds, which requires that subjects in protected and unprotected groups have equal true and false positive rates. However, practical implementation is challenging. This work proposes two ways to address this issue through the conditional independence operator. First, given the output values, it is used as a fairness measure of independence between model predictions and sensitive variables. Second, it is used as a regularisation term in the problem formulation, which seeks optimal models that balance performance and fairness concerning the sensitive variables. To illustrate the potential of our approach, we consider different scenarios. First, we use the Gaussian model to provide new insights into the problem formulation and numerical results on its convergence. Second, we present the formulation using the conditional cross-covariance operator. We anticipate that a closed-form solution is possible in the general problem formulation, including in the case of a kernel formulation setting. Third, we introduce a normalised criterion of the conditional independence operator. All formulations are posed under the risk minimisation principle, which leads to theoretical results on the performance. Additionally, insights are provided into using these operators under a Gaussian Process setting. Our methods are compared to state-of-the-art methods in terms of performance and fairness metrics on a representative set of real problems. The results obtained with our proposed methodology show promising performance-fairness curves. Furthermore, we discuss the usefulness of linear weights in the fair model to describe the behaviour of the features when enforcing fairness over a particular set of input features.