Randomized Iterative Solver as Iterative Refinement: A Simple Fix Towards Backward Stability

Ruihan Xu, Yiping Lu
Proceedings of The 28th International Conference on Artificial Intelligence and Statistics, PMLR 258:82-90, 2025.

Abstract

Iterative sketching and sketch-and-precondition are well-established randomized algorithms for solving large-scale over-determined linear least-squares problems. In this paper, we introduce a new perspective that interprets Iterative Sketching and Sketching-and-Precondition as forms of Iterative Refinement. We also examine the numerical stability of two distinct refinement strategies: iterative refinement and recursive refinement, which progressively improve the accuracy of a sketched linear solver. Building on this insight, we propose a novel algorithm, Sketched Iterative and Recursive Refinement (SIRR), which combines both refinement methods. SIRR demonstrates a four order of magnitude improvement in backward error compared to iterative sketching, achieved simply by reorganizing the computational order, ensuring that the computed solution exactly solves a modified least-squares system where the coefficient matrix deviates only slightly from the original matrix. To the best of our knowledge, SIRR is the first asymptotically fast, single-stage randomized least-squares solver that achieves both forward and backward stability.

Cite this Paper

BibTeX
@InProceedings{pmlr-v258-xu25a, title = {Randomized Iterative Solver as Iterative Refinement: A Simple Fix Towards Backward Stability}, author = {Xu, Ruihan and Lu, Yiping}, booktitle = {Proceedings of The 28th International Conference on Artificial Intelligence and Statistics}, pages = {82--90}, year = {2025}, editor = {Li, Yingzhen and Mandt, Stephan and Agrawal, Shipra and Khan, Emtiyaz}, volume = {258}, series = {Proceedings of Machine Learning Research}, month = {03--05 May}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v258/main/assets/xu25a/xu25a.pdf}, url = {https://proceedings.mlr.press/v258/xu25a.html}, abstract = {Iterative sketching and sketch-and-precondition are well-established randomized algorithms for solving large-scale over-determined linear least-squares problems. In this paper, we introduce a new perspective that interprets Iterative Sketching and Sketching-and-Precondition as forms of Iterative Refinement. We also examine the numerical stability of two distinct refinement strategies: iterative refinement and recursive refinement, which progressively improve the accuracy of a sketched linear solver. Building on this insight, we propose a novel algorithm, Sketched Iterative and Recursive Refinement (SIRR), which combines both refinement methods. SIRR demonstrates a four order of magnitude improvement in backward error compared to iterative sketching, achieved simply by reorganizing the computational order, ensuring that the computed solution exactly solves a modified least-squares system where the coefficient matrix deviates only slightly from the original matrix. To the best of our knowledge, SIRR is the first asymptotically fast, single-stage randomized least-squares solver that achieves both forward and backward stability.} }
Endnote
%0 Conference Paper %T Randomized Iterative Solver as Iterative Refinement: A Simple Fix Towards Backward Stability %A Ruihan Xu %A Yiping Lu %B Proceedings of The 28th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2025 %E Yingzhen Li %E Stephan Mandt %E Shipra Agrawal %E Emtiyaz Khan %F pmlr-v258-xu25a %I PMLR %P 82--90 %U https://proceedings.mlr.press/v258/xu25a.html %V 258 %X Iterative sketching and sketch-and-precondition are well-established randomized algorithms for solving large-scale over-determined linear least-squares problems. In this paper, we introduce a new perspective that interprets Iterative Sketching and Sketching-and-Precondition as forms of Iterative Refinement. We also examine the numerical stability of two distinct refinement strategies: iterative refinement and recursive refinement, which progressively improve the accuracy of a sketched linear solver. Building on this insight, we propose a novel algorithm, Sketched Iterative and Recursive Refinement (SIRR), which combines both refinement methods. SIRR demonstrates a four order of magnitude improvement in backward error compared to iterative sketching, achieved simply by reorganizing the computational order, ensuring that the computed solution exactly solves a modified least-squares system where the coefficient matrix deviates only slightly from the original matrix. To the best of our knowledge, SIRR is the first asymptotically fast, single-stage randomized least-squares solver that achieves both forward and backward stability.
APA
Xu, R. & Lu, Y.. (2025). Randomized Iterative Solver as Iterative Refinement: A Simple Fix Towards Backward Stability. Proceedings of The 28th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 258:82-90 Available from https://proceedings.mlr.press/v258/xu25a.html.

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