  In this paper, we study the problem of fair sparse regression on a biased
dataset where bias depends upon a hidden binary attribute. The presence of a
hidden attribute adds an extra layer of complexity to the problem by combining
sparse regression and clustering with unknown binary labels. The corresponding
optimization problem is combinatorial but we propose a novel relaxation of it
as an \emph{invex} optimization problem. To the best of our knowledge, this is
the first invex relaxation for a combinatorial problem. We show that the
inclusion of the debiasing/fairness constraint in our model has no adverse
effect on the performance. Rather, it enables the recovery of the hidden
attribute. The support of our recovered regression parameter vector matches
exactly with the true parameter vector. Moreover, we simultaneously solve the
clustering problem by recovering the exact value of the hidden attribute for
each sample. Our method uses carefully constructed primal dual witnesses to
solve the combinatorial problem. We provide theoretical guarantees which hold
as long as the number of samples is polynomial in terms of the dimension of the
regression parameter vector.
