Nearly Optimal Algorithms with Sublinear Computational Complexity for Online Kernel Regression

Junfan Li, Shizhong Liao
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:19743-19766, 2023.

Abstract

The trade-off between regret and computational cost is a fundamental problem for online kernel regression, and previous algorithms worked on the trade-off can not keep optimal regret bounds at a sublinear computational complexity. In this paper, we propose two new algorithms, AOGD-ALD and NONS-ALD, which can keep nearly optimal regret bounds at a sublinear computational complexity, and give sufficient conditions under which our algorithms work. Both algorithms dynamically maintain a group of nearly orthogonal basis used to approximate the kernel mapping, and keep nearly optimal regret bounds by controlling the approximate error. The number of basis depends on the approximate error and the decay rate of eigenvalues of the kernel matrix. If the eigenvalues decay exponentially, then AOGD-ALD and NONS-ALD separately achieves a regret of $O(\sqrt{L(f)})$ and $O(\mathrm{d}_{\mathrm{eff}}(\mu)\ln{T})$ at a computational complexity in $O(\ln^2{T})$. If the eigenvalues decay polynomially with degree $p\geq 1$, then our algorithms keep the same regret bounds at a computational complexity in $o(T)$ in the case of $p>4$ and $p\geq 10$, respectively. $L(f)$ is the cumulative losses of $f$ and $\mathrm{d}_{\mathrm{eff}}(\mu)$ is the effective dimension of the problem. The two regret bounds are nearly optimal and are not comparable.

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-li23r, title = {Nearly Optimal Algorithms with Sublinear Computational Complexity for Online Kernel Regression}, author = {Li, Junfan and Liao, Shizhong}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {19743--19766}, 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/li23r/li23r.pdf}, url = {https://proceedings.mlr.press/v202/li23r.html}, abstract = {The trade-off between regret and computational cost is a fundamental problem for online kernel regression, and previous algorithms worked on the trade-off can not keep optimal regret bounds at a sublinear computational complexity. In this paper, we propose two new algorithms, AOGD-ALD and NONS-ALD, which can keep nearly optimal regret bounds at a sublinear computational complexity, and give sufficient conditions under which our algorithms work. Both algorithms dynamically maintain a group of nearly orthogonal basis used to approximate the kernel mapping, and keep nearly optimal regret bounds by controlling the approximate error. The number of basis depends on the approximate error and the decay rate of eigenvalues of the kernel matrix. If the eigenvalues decay exponentially, then AOGD-ALD and NONS-ALD separately achieves a regret of $O(\sqrt{L(f)})$ and $O(\mathrm{d}_{\mathrm{eff}}(\mu)\ln{T})$ at a computational complexity in $O(\ln^2{T})$. If the eigenvalues decay polynomially with degree $p\geq 1$, then our algorithms keep the same regret bounds at a computational complexity in $o(T)$ in the case of $p>4$ and $p\geq 10$, respectively. $L(f)$ is the cumulative losses of $f$ and $\mathrm{d}_{\mathrm{eff}}(\mu)$ is the effective dimension of the problem. The two regret bounds are nearly optimal and are not comparable.} }
Endnote
%0 Conference Paper %T Nearly Optimal Algorithms with Sublinear Computational Complexity for Online Kernel Regression %A Junfan Li %A Shizhong Liao %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-li23r %I PMLR %P 19743--19766 %U https://proceedings.mlr.press/v202/li23r.html %V 202 %X The trade-off between regret and computational cost is a fundamental problem for online kernel regression, and previous algorithms worked on the trade-off can not keep optimal regret bounds at a sublinear computational complexity. In this paper, we propose two new algorithms, AOGD-ALD and NONS-ALD, which can keep nearly optimal regret bounds at a sublinear computational complexity, and give sufficient conditions under which our algorithms work. Both algorithms dynamically maintain a group of nearly orthogonal basis used to approximate the kernel mapping, and keep nearly optimal regret bounds by controlling the approximate error. The number of basis depends on the approximate error and the decay rate of eigenvalues of the kernel matrix. If the eigenvalues decay exponentially, then AOGD-ALD and NONS-ALD separately achieves a regret of $O(\sqrt{L(f)})$ and $O(\mathrm{d}_{\mathrm{eff}}(\mu)\ln{T})$ at a computational complexity in $O(\ln^2{T})$. If the eigenvalues decay polynomially with degree $p\geq 1$, then our algorithms keep the same regret bounds at a computational complexity in $o(T)$ in the case of $p>4$ and $p\geq 10$, respectively. $L(f)$ is the cumulative losses of $f$ and $\mathrm{d}_{\mathrm{eff}}(\mu)$ is the effective dimension of the problem. The two regret bounds are nearly optimal and are not comparable.
APA
Li, J. & Liao, S.. (2023). Nearly Optimal Algorithms with Sublinear Computational Complexity for Online Kernel Regression. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:19743-19766 Available from https://proceedings.mlr.press/v202/li23r.html.

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