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In this paper, we study the problem of fair sparse regression on a biaseddataset where bias depends upon a hidden binary attribute. The presence of ahidden attribute adds an extra layer of complexity to the problem by combiningsparse regression and clustering with unknown binary labels. The correspondingoptimization problem is combinatorial, but we propose a novel relaxation of itas an \emph{invex} optimization problem. To the best of our knowledge, this isthe first invex relaxation for a combinatorial problem. We show that theinclusion of the debiasing/fairness constraint in our model has no adverseeffect on the performance. Rather, it enables the recovery of the hiddenattribute. The support of our recovered regression parameter vector matchesexactly with the true parameter vector. Moreover, we simultaneously solve theclustering problem by recovering the exact value of the hidden attribute foreach sample. Our method uses carefully constructed primal dual witnesses toprovide theoretical guarantees for the combinatorial problem. To that end, weshow that the sample complexity of our method is logarithmic in terms of thedimension of the regression parameter vector.

Fair Sparse Regression with Clustering: An Invex Relaxation for a Combinatorial Problem