Fast Algorithms for <i> ℓ <sub>p</sub> </i> -Regression
Bibliographic record
Abstract
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×.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.011 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.003 | 0.003 |
| Open science | 0.003 | 0.004 |
| Research integrity | 0.002 | 0.005 |
| Insufficient payload (model declined to judge) | 0.018 | 0.015 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".