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.

翻译：在本文中, 我们研究偏向数据集上偏差偏差偏差的微缩回归问题, 偏差取决于隐藏的二进制属性。 隐藏属性的存在通过将稀释回归和组群与未知的二进制标签结合起来, 增加了问题的复杂性层。 相应的优化问题是组合式的, 但我们建议重新放松它, 把它当作一个优化问题。 据我们所知, 这是首个组合问题 。 我们显示, 将偏差/ 公平限制纳入模型不会对性能产生不利影响 。 相反, 它能恢复隐藏属性。 我们回收回归参数矢量的支持与真实的参数矢量完全吻合。 此外, 我们同时通过恢复每个样本隐藏属性的准确值来解决聚合问题。 我们的方法使用精心构建的原始双目证人来解决组合问题。 我们提供理论保障, 只要样本数量在回归参数矢量方面是多式的, 我们就能维持多久。