Fair Envelope Regression: Predictive-Sensitive Subspace Decomposition for Fairness-Utility Trade-offs

Fair regression with multivariate sensitive attributes requires controlling sensitive association without unnecessarily discarding predictive signal. We study this problem in linear regression with a continuous response and vector-valued sensitive attributes, using the proportion of prediction variance attributable to the sensitive attributes as the fairness criterion. We introduce Fair Envelope Regression Models (FERM), which uses envelope methodology to organize feature variation according to its relevance to the response and its association with the sensitive attributes. Under a compatibility condition, this yields four orthogonal components: predictive-only, shared predictive-sensitive, sensitive-only, and residual variation. FERM excludes sensitive-associated variation that is immaterial for prediction and penalizes only the shared predictive-sensitive component, providing a tunable path between a sensitive-invariant predictor and an unconstrained predictive model. Under the envelope model and regularity conditions, we establish consistency, asymptotic fairness at the fully constrained endpoint, reduced asymptotic estimation variance from removing response-immaterial directions, and a closed-form characterization of the resulting fairness–utility interpolation. Simulations and real-data experiments show that FERM can improve predictive utility at comparable levels of the chosen unfairness measure relative to closely matched fair-regression baselines. Beyond prediction performance, the decomposition makes explicit which predictive directions also carry sensitive-associated information, providing subspace-level auditability of the fairness-utility trade-off.