A PTAS for Agnostically Learning Halfspaces

Amit Daniely
Proceedings of The 28th Conference on Learning Theory, PMLR 40:484-502, 2015.

Abstract

We present a PTAS for agnostically learning halfspaces w.r.t. the uniform distribution on the d dimensional sphere. Namely, we show that for every μ>0 there is an algorithm that runs in time \mathrmpoly\left(d,\frac1ε\right), and is guaranteed to return a classifier with error at most (1+μ)\mathrmopt+ε, where \mathrmopt is the error of the best halfspace classifier. This improves on Awasthi, Balcan and Long (STOC 2014) who showed an algorithm with an (unspecified) constant approximation ratio. Our algorithm combines the classical technique of polynomial regression, together with the new localization technique of Awasthi et. al.

Cite this Paper

BibTeX
@InProceedings{pmlr-v40-Daniely15, title = {A PTAS for Agnostically Learning Halfspaces}, author = {Daniely, Amit}, booktitle = {Proceedings of The 28th Conference on Learning Theory}, pages = {484--502}, year = {2015}, editor = {Grünwald, Peter and Hazan, Elad and Kale, Satyen}, volume = {40}, series = {Proceedings of Machine Learning Research}, address = {Paris, France}, month = {03--06 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v40/Daniely15.pdf}, url = {https://proceedings.mlr.press/v40/Daniely15.html}, abstract = {We present a PTAS for agnostically learning halfspaces w.r.t. the uniform distribution on the d dimensional sphere. Namely, we show that for every μ>0 there is an algorithm that runs in time \mathrmpoly\left(d,\frac1ε\right), and is guaranteed to return a classifier with error at most (1+μ)\mathrmopt+ε, where \mathrmopt is the error of the best halfspace classifier. This improves on Awasthi, Balcan and Long (STOC 2014) who showed an algorithm with an (unspecified) constant approximation ratio. Our algorithm combines the classical technique of polynomial regression, together with the new localization technique of Awasthi et. al.} }
Endnote
%0 Conference Paper %T A PTAS for Agnostically Learning Halfspaces %A Amit Daniely %B Proceedings of The 28th Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2015 %E Peter Grünwald %E Elad Hazan %E Satyen Kale %F pmlr-v40-Daniely15 %I PMLR %P 484--502 %U https://proceedings.mlr.press/v40/Daniely15.html %V 40 %X We present a PTAS for agnostically learning halfspaces w.r.t. the uniform distribution on the d dimensional sphere. Namely, we show that for every μ>0 there is an algorithm that runs in time \mathrmpoly\left(d,\frac1ε\right), and is guaranteed to return a classifier with error at most (1+μ)\mathrmopt+ε, where \mathrmopt is the error of the best halfspace classifier. This improves on Awasthi, Balcan and Long (STOC 2014) who showed an algorithm with an (unspecified) constant approximation ratio. Our algorithm combines the classical technique of polynomial regression, together with the new localization technique of Awasthi et. al.
RIS
TY - CPAPER TI - A PTAS for Agnostically Learning Halfspaces AU - Amit Daniely BT - Proceedings of The 28th Conference on Learning Theory DA - 2015/06/26 ED - Peter Grünwald ED - Elad Hazan ED - Satyen Kale ID - pmlr-v40-Daniely15 PB - PMLR DP - Proceedings of Machine Learning Research VL - 40 SP - 484 EP - 502 L1 - http://proceedings.mlr.press/v40/Daniely15.pdf UR - https://proceedings.mlr.press/v40/Daniely15.html AB - We present a PTAS for agnostically learning halfspaces w.r.t. the uniform distribution on the d dimensional sphere. Namely, we show that for every μ>0 there is an algorithm that runs in time \mathrmpoly\left(d,\frac1ε\right), and is guaranteed to return a classifier with error at most (1+μ)\mathrmopt+ε, where \mathrmopt is the error of the best halfspace classifier. This improves on Awasthi, Balcan and Long (STOC 2014) who showed an algorithm with an (unspecified) constant approximation ratio. Our algorithm combines the classical technique of polynomial regression, together with the new localization technique of Awasthi et. al. ER -
APA
Daniely, A.. (2015). A PTAS for Agnostically Learning Halfspaces. Proceedings of The 28th Conference on Learning Theory, in Proceedings of Machine Learning Research 40:484-502 Available from https://proceedings.mlr.press/v40/Daniely15.html.

Related Material