Computing Lewis Weights to High Precision

TL;DR

This paper introduces an efficient algorithm for high-precision computation of $oldsymbol{ ext{l}}_p$ Lewis weights for matrices, enabling faster solutions for related optimization problems.

Contribution

It provides the first polylogarithmic-depth polynomial-work algorithm for computing $oldsymbol{ ext{l}}_p$ Lewis weights for all constant $p > 0$, extending previous results.

Findings

Achieves $ ilde{O}_p( ext{log}(1/ ext{epsilon}))$ iteration complexity

Computes $ ext{l}_p$ Lewis weights for all $p > 0$ with high precision

Enables efficient computation of nearly optimal self-concordant barriers

Abstract

We present an algorithm for computing approximate $\ell_p$ Lewis weights to high precision. Given a full-rank $\mathbf{A} \in \mathbb{R}^{m \times n}$ with $m \geq n$ and a scalar $p>2$, our algorithm computes $\epsilon$-approximate $\ell_p$ Lewis weights of $\mathbf{A}$ in $\widetilde{O}_p(\log(1/\epsilon))$ iterations; the cost of each iteration is linear in the input size plus the cost of computing the leverage scores of $\mathbf{D}\mathbf{A}$ for diagonal $\mathbf{D} \in \mathbb{R}^{m \times m}$. Prior to our work, such a computational complexity was known only for $p \in (0, 4)$ [CohenPeng2015], and combined with this result, our work yields the first polylogarithmic-depth polynomial-work algorithm for the problem of computing $\ell_p$ Lewis weights to high precision for all constant $p > 0$. An important consequence of this result is also the first polylogarithmic-depth polynomial-work algorithm for computing a nearly optimal self-concordant barrier for a polytope.