Fast, Provably convergent IRLS Algorithm for pnorm Linear Regression
Abstract
Linear regression in $\ell_p$norm is a canonical optimization problem that arises in several applications, including sparse recovery, semisupervised 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 pIRLS, 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 $O(p^{3.5} m^{\frac{p2}{2(p1)}} \log \frac{m}{\varepsilon}) \le O_p(\sqrt{m} \log \frac{m}{\varepsilon} )$ iterations. Our experiments demonstrate that it performs even better than our theoretical bounds, beats the standard Matlab/CVX implementation for solving these problems by 1050x, and is the fastest among available implementations in the highaccuracy regime.
 Publication:

arXiv eprints
 Pub Date:
 July 2019
 arXiv:
 arXiv:1907.07167
 Bibcode:
 2019arXiv190707167A
 Keywords:

 Computer Science  Data Structures and Algorithms;
 Computer Science  Machine Learning;
 Mathematics  Numerical Analysis;
 Mathematics  Optimization and Control
 EPrint:
 Code for this work is available at https://github.com/utorontotheory/pIRLS