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

arXiv: https://arxiv.org/abs/1907.07167 · HTML: https://mlsanity.com/paper/1907.07167/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Jul 2019.

## Built on by (most cited first, 10 of 10)

- [Complementary Composite Minimization, Small Gradients in General Norms, and Applications](https://mlsanity.com/paper/2101.11041/) (2021)
- [Non-Uniform Smoothness for Gradient Descent](https://mlsanity.com/paper/2311.08615/) (2023)
- [Optimal Excess Risk Bounds for Empirical Risk Minimization on p-Norm Linear Regression](https://mlsanity.com/paper/2310.12437/) (2023)
- [Optimal Algorithms for Stochastic Complementary Composite Minimization](https://mlsanity.com/paper/2211.01758/) (2022)
- [Recovering Simultaneously Structured Data via Non-Convex Iteratively Reweighted Least Squares](https://mlsanity.com/paper/2306.04961/) (2023)
- [PROMPT: Parallel Iterative Algorithm for ℓp norm linear regression via Majorization Minimization with an application to semi-supervised graph learning](https://mlsanity.com/paper/2110.12190/) (2021)
- [Flag Aggregator: Scalable Distributed Training under Failures and Augmented Losses using Convex Optimization](https://mlsanity.com/paper/2302.05865/) (2023)
- [Sharper Bounds for ℓp Sensitivity Sampling](https://mlsanity.com/paper/2306.00732/) (2023)
- [Computing Approximate ℓp Sensitivities](https://mlsanity.com/paper/2311.04158/) (2023)
- [Planner-Admissible Graph-PDE Value Extensions for Sparse Goal-Conditioned Planning](https://mlsanity.com/paper/2605.19185/) (2026)

## Abstract (arXiv)

Linear regression in ℓp-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 ℓp-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 ∈ [2,∞). Our algorithm is simple to implement and is guaranteed to find a (1+varepsilon)-approximate solution in O(p3.5 m^fracp-22(p-1) log fracmvarepsilon) ≤ Op(√m log fracmvarepsilon ) 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.