On Socially Fair Regression and Low-Rank Approximation

Zhao Song, Ali Vakilian, David Woodruff, Samson Zhou

Primary Area: societal considerations including fairness, safety, privacy

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Keywords: fairness, randomized numerical linear algebra, regression, low-rank approximation

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Abstract: Regression and low-rank approximation are two fundamental problems that are applied across a wealth of machine learning applications. In this paper, we study the question of socially fair regression and socially fair low-rank approximation, where the goal is to minimize the loss over all sub-populations of the data. We show that surprisingly, socially fair regression and socially fair low-rank approximation exhibit drastically different complexities. Specifically, we show that while fair regression can be solved up to arbitrary accuracy in polynomial time for a wide variety of loss functions, even constant-factor approximation to fair low-rank approximation requires exponential time under certain standard complexity hypotheses. On the positive side, we give an algorithm for fair low-rank approximation that, for a constant number of groups and constant-factor accuracy, runs in $2^{\text{poly}(k)}$ rather than the na\"{i}ve $n^{\text{poly}(k)}$, which is a substantial improvement when the dataset has a large number $n$ of observations. Finally, we show that there exists a bicriteria approximation algorithm for fair low-rank approximation that runs in polynomial time.

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Submission Number: 6638