An homotopy method for $\ell_p$ regression provably beyond self-concordance and in input-sparsity time

TL;DR

This paper introduces an efficient homotopy method for solving _p regression problems, achieving input-sparsity time and surpassing traditional self-concordance limitations, thus advancing optimization techniques for _p norms.

Contribution

The authors develop a novel homotopy algorithm for _p regression that operates in input-sparsity time and demonstrates limitations of existing interior point methods.

Findings

Achieves _p regression in input-sparsity time.

Surpasses previous methods for p = 1, 2, = .

Shows limitations of self-concordant barriers for _p balls.

Abstract

We consider the problem of linear regression where the $\ell_2^n$ norm loss (i.e., the usual least squares loss) is replaced by the $\ell_p^n$ norm. We show how to solve such problems up to machine precision in $O^*(n^{|1/2 - 1/p|})$ (dense) matrix-vector products and $O^*(1)$ matrix inversions, or alternatively in $O^*(n^{|1/2 - 1/p|})$ calls to a (sparse) linear system solver. This improves the state of the art for any $p\not\in \{1,2,+\infty\}$. Furthermore we also propose a randomized algorithm solving such problems in {\em input sparsity time}, i.e., $O^*(Z + \mathrm{poly}(d))$ where $Z$ is the size of the input and $d$ is the number of variables. Such a result was only known for $p=2$. Finally we prove that these results lie outside the scope of the Nesterov-Nemirovski's theory of interior point methods by showing that any symmetric self-concordant barrier on the $\ell_p^n$ unit ball has self-concordance parameter $\tilde{\Omega}(n)$.