Fairness-aware Gaussian Graphical Regression Models with Application to Brain Co-expression QTL Studies

Glioblastoma multiforme (GBM) is a highly aggressive brain cancer with largely ineffective treatment. It is imperative to explore more effective therapies, such as gene-based treatments. For co-expression QTL studies of GBM, we develop fairness-aware Gaussian graphical regression models (Fair RegGGMs), which can determine how genetic variants modulate subject-level gene networks, and recover both population-level and subject-level gene graphs, while ensuring that the developed learning and inference neither propagate biases nor amplify disparities. We introduce pairwise graph disparity risk to quantify fairness and propose fairness-aware multi-task learning (Fair-MTL) via a cross-task group sparsity penalty, within-task element-wise sparsity penalty, and pairwise fairness regularization. We also develop a projected Fair-SAGE debiasing method for statistical inference. In Fair-MTL, we strive for a balance among group-/element-wise sparsity of graphical network structures, fairness across different subgroups, and statistical effectiveness of RegGGMs. For the ultrahigh-dimensional overparameterized models, an efficient nonsmooth fair multi-objective optimization algorithm (Fair-MOO) is developed. Both Fair-MOO and the projected Fair-SAGE dramatically reduce computational costs. Under a general dependence structure, the nonasymptotic l2 convergence rate of Fair-MTL, asymptotic normality of Fair-SAGE debiased estimator, and Pareto optimality of Fair-MOO algorithm are established. The simulation study and brain co-expression QTL analysis confirm the fairness and effectiveness of our Fair RegGGMs and provide valuable insights for the gene graph of GBM.