Polynomial Preconditioning for Gradient Methods

Nikita Doikov, Anton Rodomanov
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:8162-8187, 2023.

Abstract

We study first-order methods with preconditioning for solving structured convex optimization problems. We propose a new family of preconditioners generated by the symmetric polynomials. They provide the first-order optimization methods with a provable improvement of the condition number, cutting the gaps between highest eigenvalues, without explicit knowledge of the actual spectrum. We give a stochastic interpretation of this preconditioning in terms of the coordinate volume sampling and compare it with other classical approaches, including the Chebyshev polynomials. We show how to incorporate a polynomial preconditioning into the Gradient and Fast Gradient Methods and establish their better global complexity bounds. Finally, we propose a simple adaptive search procedure that automatically ensures the best polynomial preconditioning for the Gradient Method, minimizing the objective along a low-dimensional Krylov subspace. Numerical experiments confirm the efficiency of our preconditioning strategies for solving various machine learning problems.

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-doikov23b, title = {Polynomial Preconditioning for Gradient Methods}, author = {Doikov, Nikita and Rodomanov, Anton}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {8162--8187}, year = {2023}, editor = {Krause, Andreas and Brunskill, Emma and Cho, Kyunghyun and Engelhardt, Barbara and Sabato, Sivan and Scarlett, Jonathan}, volume = {202}, series = {Proceedings of Machine Learning Research}, month = {23--29 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v202/doikov23b/doikov23b.pdf}, url = {https://proceedings.mlr.press/v202/doikov23b.html}, abstract = {We study first-order methods with preconditioning for solving structured convex optimization problems. We propose a new family of preconditioners generated by the symmetric polynomials. They provide the first-order optimization methods with a provable improvement of the condition number, cutting the gaps between highest eigenvalues, without explicit knowledge of the actual spectrum. We give a stochastic interpretation of this preconditioning in terms of the coordinate volume sampling and compare it with other classical approaches, including the Chebyshev polynomials. We show how to incorporate a polynomial preconditioning into the Gradient and Fast Gradient Methods and establish their better global complexity bounds. Finally, we propose a simple adaptive search procedure that automatically ensures the best polynomial preconditioning for the Gradient Method, minimizing the objective along a low-dimensional Krylov subspace. Numerical experiments confirm the efficiency of our preconditioning strategies for solving various machine learning problems.} }
Endnote
%0 Conference Paper %T Polynomial Preconditioning for Gradient Methods %A Nikita Doikov %A Anton Rodomanov %B Proceedings of the 40th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2023 %E Andreas Krause %E Emma Brunskill %E Kyunghyun Cho %E Barbara Engelhardt %E Sivan Sabato %E Jonathan Scarlett %F pmlr-v202-doikov23b %I PMLR %P 8162--8187 %U https://proceedings.mlr.press/v202/doikov23b.html %V 202 %X We study first-order methods with preconditioning for solving structured convex optimization problems. We propose a new family of preconditioners generated by the symmetric polynomials. They provide the first-order optimization methods with a provable improvement of the condition number, cutting the gaps between highest eigenvalues, without explicit knowledge of the actual spectrum. We give a stochastic interpretation of this preconditioning in terms of the coordinate volume sampling and compare it with other classical approaches, including the Chebyshev polynomials. We show how to incorporate a polynomial preconditioning into the Gradient and Fast Gradient Methods and establish their better global complexity bounds. Finally, we propose a simple adaptive search procedure that automatically ensures the best polynomial preconditioning for the Gradient Method, minimizing the objective along a low-dimensional Krylov subspace. Numerical experiments confirm the efficiency of our preconditioning strategies for solving various machine learning problems.
APA
Doikov, N. & Rodomanov, A.. (2023). Polynomial Preconditioning for Gradient Methods. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:8162-8187 Available from https://proceedings.mlr.press/v202/doikov23b.html.

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