Fast, Provably convergent IRLS Algorithm for p-norm Linear Regression

TL;DR

This paper introduces p-IRLS, a novel IRLS algorithm that guarantees geometric convergence for all p in [2, ∞), significantly improving efficiency and reliability in solving ℓ_p-regression problems across various applications.

Contribution

The paper presents the first IRLS algorithm with proven geometric convergence for all p ≥ 2, addressing longstanding divergence issues for p > 3 and enhancing practical performance.

Findings

Achieves geometric convergence for p ≥ 2

Outperforms standard solvers by 10-50x in experiments

Faster in high-accuracy regimes than existing methods

Abstract

Linear regression in $\ell_p$-norm is a canonical optimization problem that arises in several applications, including sparse recovery, semi-supervised learning, and signal processing. Generic convex optimization algorithms for solving $\ell_p$-regression are slow in practice. Iteratively Reweighted Least Squares (IRLS) is an easy to implement family of algorithms for solving these problems that has been studied for over 50 years. However, these algorithms often diverge for p > 3, and since the work of Osborne (1985), it has been an open problem whether there is an IRLS algorithm that is guaranteed to converge rapidly for p > 3. We propose p-IRLS, the first IRLS algorithm that provably converges geometrically for any $p \in [2,\infty).$ Our algorithm is simple to implement and is guaranteed to find a $(1+\varepsilon)$-approximate solution in $O(p^{3.5} m^{\frac{p-2}{2(p-1)}} \log \frac{m}{\varepsilon}) \le O_p(\sqrt{m} \log \frac{m}{\varepsilon} )$ iterations. Our experiments demonstrate that it performs even better than our theoretical bounds, beats the standard Matlab/CVX implementation for solving these problems by 10--50x, and is the fastest among available implementations in the high-accuracy regime.