Householder Sketch for Accurate and Accelerated Least-Mean-Squares Solvers
Jyotikrishna Dass, Rabi Mahapatra
Abstract
Least-Mean-Squares (\textsc{LMS}) solvers comprise a class of fundamental optimization problems such as linear regression, and regularized regressions such as Ridge, LASSO, and Elastic-Net. Data summarization techniques for big data generate summaries called coresets and sketches to speed up model learning under streaming and distributed settings. For example, \citep{nips2019} design a fast and accurate Caratheodory set on input data to boost the performance of existing \textsc{LMS} solvers. In retrospect, we explore classical Householder transformation as a candidate for sketching and accurately solving LMS problems. We find it to be a simpler, memory-efficient, and faster alternative that always existed to the above strong baseline. We also present a scalable algorithm based on the construction of distributed Householder sketches to solve \textsc{LMS} problem across multiple worker nodes. We perform thorough empirical analysis with large synthetic and real datasets to evaluate the performance of Householder sketch and compare with \citep{nips2019}. Our results show Householder sketch speeds up existing \textsc{LMS} solvers in the scikit-learn library up to $100$x-$400$x. Also, it is $10$x-$100$x faster than the above baseline with similar numerical stability. The distributed algorithm demonstrates linear scalability with a near-negligible communication overhead.
BibTeX
@InProceedings{pmlr-v139-dass21a,
title = {Householder Sketch for Accurate and Accelerated Least-Mean-Squares Solvers},
author = {Dass, Jyotikrishna and Mahapatra, Rabi},
booktitle = {Proceedings of the 38th International Conference on Machine Learning},
pages = {2467--2477},
year = {2021},
editor = {Meila, Marina and Zhang, Tong},
volume = {139},
series = {Proceedings of Machine Learning Research},
month = {18--24 Jul},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v139/dass21a/dass21a.pdf},
url = {https://proceedings.mlr.press/v139/dass21a.html},
abstract = {Least-Mean-Squares (\textsc{LMS}) solvers comprise a class of fundamental optimization problems such as linear regression, and regularized regressions such as Ridge, LASSO, and Elastic-Net. Data summarization techniques for big data generate summaries called coresets and sketches to speed up model learning under streaming and distributed settings. For example, \citep{nips2019} design a fast and accurate Caratheodory set on input data to boost the performance of existing \textsc{LMS} solvers. In retrospect, we explore classical Householder transformation as a candidate for sketching and accurately solving LMS problems. We find it to be a simpler, memory-efficient, and faster alternative that always existed to the above strong baseline. We also present a scalable algorithm based on the construction of distributed Householder sketches to solve \textsc{LMS} problem across multiple worker nodes. We perform thorough empirical analysis with large synthetic and real datasets to evaluate the performance of Householder sketch and compare with \citep{nips2019}. Our results show Householder sketch speeds up existing \textsc{LMS} solvers in the scikit-learn library up to $100$x-$400$x. Also, it is $10$x-$100$x faster than the above baseline with similar numerical stability. The distributed algorithm demonstrates linear scalability with a near-negligible communication overhead.}
}