Adaptive Hard Thresholding for Near-optimal Consistent Robust Regression

Arun Sai Suggala, Kush Bhatia, Pradeep Ravikumar, Prateek Jain
Proceedings of the Thirty-Second Conference on Learning Theory, PMLR 99:2892-2897, 2019.

Abstract

We study the problem of robust linear regression with response variable corruptions. We consider the oblivious adversary model, where the adversary corrupts a fraction of the responses in complete ignorance of the data. We provide a nearly linear time estimator which consistently estimates the true regression vector, even with $1-o(1)$ fraction of corruptions. Existing results in this setting either don’t guarantee consistent estimates or can only handle a small fraction of corruptions. We also extend our estimator to robust sparse linear regression and show that similar guarantees hold in this setting. Finally, we apply our estimator to the problem of linear regression with heavy-tailed noise and show that our estimator consistently estimates the regression vector even when the noise has unbounded variance (e.g., Cauchy distribution), for which most existing results don’t even apply. Our estimator is based on a novel variant of outlier removal via hard thresholding in which the threshold is chosen adaptively and crucially relies on randomness to escape bad fixed points of the non-convex hard thresholding operation.

Cite this Paper

BibTeX
@InProceedings{pmlr-v99-suggala19a, title = {Adaptive Hard Thresholding for Near-optimal Consistent Robust Regression}, author = {Suggala, Arun Sai and Bhatia, Kush and Ravikumar, Pradeep and Jain, Prateek}, booktitle = {Proceedings of the Thirty-Second Conference on Learning Theory}, pages = {2892--2897}, year = {2019}, editor = {Beygelzimer, Alina and Hsu, Daniel}, volume = {99}, series = {Proceedings of Machine Learning Research}, month = {25--28 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v99/suggala19a/suggala19a.pdf}, url = {https://proceedings.mlr.press/v99/suggala19a.html}, abstract = {We study the problem of robust linear regression with response variable corruptions. We consider the oblivious adversary model, where the adversary corrupts a fraction of the responses in complete ignorance of the data. We provide a nearly linear time estimator which consistently estimates the true regression vector, even with $1-o(1)$ fraction of corruptions. Existing results in this setting either don’t guarantee consistent estimates or can only handle a small fraction of corruptions. We also extend our estimator to robust sparse linear regression and show that similar guarantees hold in this setting. Finally, we apply our estimator to the problem of linear regression with heavy-tailed noise and show that our estimator consistently estimates the regression vector even when the noise has unbounded variance (e.g., Cauchy distribution), for which most existing results don’t even apply. Our estimator is based on a novel variant of outlier removal via hard thresholding in which the threshold is chosen adaptively and crucially relies on randomness to escape bad fixed points of the non-convex hard thresholding operation.} }
Endnote
%0 Conference Paper %T Adaptive Hard Thresholding for Near-optimal Consistent Robust Regression %A Arun Sai Suggala %A Kush Bhatia %A Pradeep Ravikumar %A Prateek Jain %B Proceedings of the Thirty-Second Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2019 %E Alina Beygelzimer %E Daniel Hsu %F pmlr-v99-suggala19a %I PMLR %P 2892--2897 %U https://proceedings.mlr.press/v99/suggala19a.html %V 99 %X We study the problem of robust linear regression with response variable corruptions. We consider the oblivious adversary model, where the adversary corrupts a fraction of the responses in complete ignorance of the data. We provide a nearly linear time estimator which consistently estimates the true regression vector, even with $1-o(1)$ fraction of corruptions. Existing results in this setting either don’t guarantee consistent estimates or can only handle a small fraction of corruptions. We also extend our estimator to robust sparse linear regression and show that similar guarantees hold in this setting. Finally, we apply our estimator to the problem of linear regression with heavy-tailed noise and show that our estimator consistently estimates the regression vector even when the noise has unbounded variance (e.g., Cauchy distribution), for which most existing results don’t even apply. Our estimator is based on a novel variant of outlier removal via hard thresholding in which the threshold is chosen adaptively and crucially relies on randomness to escape bad fixed points of the non-convex hard thresholding operation.
APA
Suggala, A.S., Bhatia, K., Ravikumar, P. & Jain, P.. (2019). Adaptive Hard Thresholding for Near-optimal Consistent Robust Regression. Proceedings of the Thirty-Second Conference on Learning Theory, in Proceedings of Machine Learning Research 99:2892-2897 Available from https://proceedings.mlr.press/v99/suggala19a.html.

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