Fast Algorithms for ℓp-Regression

• Published in: Journal of the ACM Vol. 71; no. 5; pp. 1 - 45
• Main Authors: Adil, Deeksha, Kyng, Rasmus, Peng, Richard, Sachdeva, Sushant
• Format: Journal Article
• Language: English
• Published: New York, NY ACM 05.10.2024; Association for Computing Machinery
• Subjects: Algorithms; Continuous optimization; Linear systems; Machine learning; Regression; Theory of computation; ℓ_p-regression; IRLS; iterative refinement
• ISSN: 0004-5411; 1557-735X
• DOI: 10.1145/3686794

The \(\ell _p\) -norm regression problem is a classic problem in optimization with wide ranging applications in machine learning and theoretical computer science. The goal is to compute \(\boldsymbol {\mathit {x}}^{\star } =\arg \min _{\boldsymbol {\mathit {A}}\boldsymbol {\mathit {x}}=\boldsymbol {\mathit {b}}}\Vert \boldsymbol {\mathit {x}}\Vert _p^p\) , where \(\boldsymbol {\mathit {x}}^{\star }\in \mathbb {R}^n,\boldsymbol {\mathit {A}}\in \mathbb {R}^{d\times n},\boldsymbol {\mathit {b}}\in \mathbb {R}^d\) and \(d\le n\) . Efficient high-accuracy algorithms for the problem have been challenging both in theory and practice and the state-of-the-art algorithms require \(poly(p)\cdot n^{\frac{1}{2}-\frac{1}{p}}\) linear system solves for \(p\ge 2\) . In this article, we provide new algorithms for \(\ell _p\) -regression (and a more general formulation of the problem) that obtain a high-accuracy solution in \(O(p n^{ {(p-2)}{(3p-2)}})\) linear system solves. We further propose a new inverse maintenance procedure that speeds-up our algorithm to \(\widetilde{O}(n^{\omega })\) total runtime, where \(O(n^{\omega })\) denotes the running time for multiplying \(n \times n\) matrices. Additionally, we give the first Iteratively Reweighted Least Squares (IRLS) algorithm that is guaranteed to converge to an optimum in a few iterations. Our IRLS algorithm has shown exceptional practical performance, beating the currently available implementations in MATLAB/CVX by 10–50×.