On Socially Fair Low-Rank Approximation and Column Subset Selection

TL;DR

This paper investigates the computational complexity of socially fair low-rank approximation and column subset selection, providing both hardness results and efficient algorithms for specific cases, advancing fair machine learning methods.

Contribution

It establishes exponential time hardness for constant-factor fair low-rank approximation and introduces practical algorithms for cases with limited groups and polynomial-time bicriteria solutions.

Findings

Constant-factor approximation is NP-hard under certain hypotheses.

An algorithm with exponential time complexity for fixed groups and accuracy.

Polynomial-time bicriteria approximation algorithms are achievable.

Abstract

Low-rank approximation and column subset selection are two fundamental and related problems that are applied across a wealth of machine learning applications. In this paper, we study the question of socially fair low-rank approximation and socially fair column subset selection, where the goal is to minimize the loss over all sub-populations of the data. We show that surprisingly, 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 $2^{\text{poly}(k)}$ time 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. We then show that there exist bicriteria approximation algorithms for fair low-rank approximation and fair column subset selection that run in polynomial time.