Sketch-and-Project Meets Newton Method: Global $O(1/k^2)$ Convergence with Low-Rank Updates

Slavomir Hanzely
Proceedings of The 28th International Conference on Artificial Intelligence and Statistics, PMLR 258:3205-3213, 2025.

Abstract

In this paper, we propose the first sketch-and-project Newton method with the fast $O(1/k^2$) global convergence rate for self-concordant functions. Our method, SGN, can be viewed in three ways: i) as a sketch-and-project algorithm projecting updates of the Newton method, ii) as a cubically regularized Newton method in the sketched subspaces, and iii) as a damped Newton method in the sketched subspaces. SGN inherits the best of all three worlds: the cheap iteration costs of the sketch-and-project methods, the state-of-the-art $O(1/k^2)$ global convergence rate of the full-rank Newton-like methods, and the algorithm simplicity of the damped Newton methods. Finally, we demonstrate its comparable empirical performance to the baseline algorithms.

Cite this Paper

BibTeX
@InProceedings{pmlr-v258-hanzely25a, title = {Sketch-and-Project Meets Newton Method: Global $O(1/k^2)$ Convergence with Low-Rank Updates}, author = {Hanzely, Slavomir}, booktitle = {Proceedings of The 28th International Conference on Artificial Intelligence and Statistics}, pages = {3205--3213}, 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/hanzely25a/hanzely25a.pdf}, url = {https://proceedings.mlr.press/v258/hanzely25a.html}, abstract = {In this paper, we propose the first sketch-and-project Newton method with the fast $O(1/k^2$) global convergence rate for self-concordant functions. Our method, SGN, can be viewed in three ways: i) as a sketch-and-project algorithm projecting updates of the Newton method, ii) as a cubically regularized Newton method in the sketched subspaces, and iii) as a damped Newton method in the sketched subspaces. SGN inherits the best of all three worlds: the cheap iteration costs of the sketch-and-project methods, the state-of-the-art $O(1/k^2)$ global convergence rate of the full-rank Newton-like methods, and the algorithm simplicity of the damped Newton methods. Finally, we demonstrate its comparable empirical performance to the baseline algorithms.} }
Endnote
%0 Conference Paper %T Sketch-and-Project Meets Newton Method: Global $O(1/k^2)$ Convergence with Low-Rank Updates %A Slavomir Hanzely %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-hanzely25a %I PMLR %P 3205--3213 %U https://proceedings.mlr.press/v258/hanzely25a.html %V 258 %X In this paper, we propose the first sketch-and-project Newton method with the fast $O(1/k^2$) global convergence rate for self-concordant functions. Our method, SGN, can be viewed in three ways: i) as a sketch-and-project algorithm projecting updates of the Newton method, ii) as a cubically regularized Newton method in the sketched subspaces, and iii) as a damped Newton method in the sketched subspaces. SGN inherits the best of all three worlds: the cheap iteration costs of the sketch-and-project methods, the state-of-the-art $O(1/k^2)$ global convergence rate of the full-rank Newton-like methods, and the algorithm simplicity of the damped Newton methods. Finally, we demonstrate its comparable empirical performance to the baseline algorithms.
APA
Hanzely, S.. (2025). Sketch-and-Project Meets Newton Method: Global $O(1/k^2)$ Convergence with Low-Rank Updates. Proceedings of The 28th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 258:3205-3213 Available from https://proceedings.mlr.press/v258/hanzely25a.html.

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