Unconditional Coresets for Regularized Loss Minimization

Alireza Samadian, Kirk Pruhs, Benjamin Moseley, Sungjin Im, Ryan Curtin
Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, PMLR 108:482-492, 2020.

Abstract

We design and mathematically analyze sampling-based algorithms for regularized loss minimization problems that are implementable in popular computational models for large data, in which the access to the data is restricted in some way. Our main result is that if the regularizer’s effect does not become negligible as the norm of the hypothesis scales, and as the data scales, then a uniform sample of modest size is with high probability a coreset. In the case that the loss function is either logistic regression or soft-margin support vector machines, and the regularizer is one of the common recommended choices, this result implies that a uniform sample of size $O(d \sqrt{n})$ is with high probability a coreset of $n$ points in $\Re^d$. We contrast this upper bound with two lower bounds. The first lower bound shows that our analysis of uniform sampling is tight; that is, a smaller uniform sample will likely not be a core set. The second lower bound shows that in some sense uniform sampling is close to optimal, as significantly smaller core sets do not generally exist.

Cite this Paper


BibTeX
@InProceedings{pmlr-v108-samadian20a, title = {Unconditional Coresets for Regularized Loss Minimization}, author = {Samadian, Alireza and Pruhs, Kirk and Moseley, Benjamin and Im, Sungjin and Curtin, Ryan}, booktitle = {Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics}, pages = {482--492}, year = {2020}, editor = {Silvia Chiappa and Roberto Calandra}, volume = {108}, series = {Proceedings of Machine Learning Research}, month = {26--28 Aug}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v108/samadian20a/samadian20a.pdf}, url = { http://proceedings.mlr.press/v108/samadian20a.html }, abstract = {We design and mathematically analyze sampling-based algorithms for regularized loss minimization problems that are implementable in popular computational models for large data, in which the access to the data is restricted in some way. Our main result is that if the regularizer’s effect does not become negligible as the norm of the hypothesis scales, and as the data scales, then a uniform sample of modest size is with high probability a coreset. In the case that the loss function is either logistic regression or soft-margin support vector machines, and the regularizer is one of the common recommended choices, this result implies that a uniform sample of size $O(d \sqrt{n})$ is with high probability a coreset of $n$ points in $\Re^d$. We contrast this upper bound with two lower bounds. The first lower bound shows that our analysis of uniform sampling is tight; that is, a smaller uniform sample will likely not be a core set. The second lower bound shows that in some sense uniform sampling is close to optimal, as significantly smaller core sets do not generally exist.} }
Endnote
%0 Conference Paper %T Unconditional Coresets for Regularized Loss Minimization %A Alireza Samadian %A Kirk Pruhs %A Benjamin Moseley %A Sungjin Im %A Ryan Curtin %B Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2020 %E Silvia Chiappa %E Roberto Calandra %F pmlr-v108-samadian20a %I PMLR %P 482--492 %U http://proceedings.mlr.press/v108/samadian20a.html %V 108 %X We design and mathematically analyze sampling-based algorithms for regularized loss minimization problems that are implementable in popular computational models for large data, in which the access to the data is restricted in some way. Our main result is that if the regularizer’s effect does not become negligible as the norm of the hypothesis scales, and as the data scales, then a uniform sample of modest size is with high probability a coreset. In the case that the loss function is either logistic regression or soft-margin support vector machines, and the regularizer is one of the common recommended choices, this result implies that a uniform sample of size $O(d \sqrt{n})$ is with high probability a coreset of $n$ points in $\Re^d$. We contrast this upper bound with two lower bounds. The first lower bound shows that our analysis of uniform sampling is tight; that is, a smaller uniform sample will likely not be a core set. The second lower bound shows that in some sense uniform sampling is close to optimal, as significantly smaller core sets do not generally exist.
APA
Samadian, A., Pruhs, K., Moseley, B., Im, S. & Curtin, R.. (2020). Unconditional Coresets for Regularized Loss Minimization. Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 108:482-492 Available from http://proceedings.mlr.press/v108/samadian20a.html .

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