Agnostic Learning of Halfspaces with Gradient Descent via Soft Margins
Spencer Frei, Yuan Cao, Quanquan Gu
Abstract
We analyze the properties of gradient descent on convex surrogates for the zero-one loss for the agnostic learning of halfspaces. We show that when a quantity we refer to as the \textit{soft margin} is well-behaved—a condition satisfied by log-concave isotropic distributions among others—minimizers of convex surrogates for the zero-one loss are approximate minimizers for the zero-one loss itself. As standard convex optimization arguments lead to efficient guarantees for minimizing convex surrogates of the zero-one loss, our methods allow for the first positive guarantees for the classification error of halfspaces learned by gradient descent using the binary cross-entropy or hinge loss in the presence of agnostic label noise.
BibTeX
@InProceedings{pmlr-v139-frei21a,
title = {Agnostic Learning of Halfspaces with Gradient Descent via Soft Margins},
author = {Frei, Spencer and Cao, Yuan and Gu, Quanquan},
booktitle = {Proceedings of the 38th International Conference on Machine Learning},
pages = {3417--3426},
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/frei21a/frei21a.pdf},
url = {https://proceedings.mlr.press/v139/frei21a.html},
abstract = {We analyze the properties of gradient descent on convex surrogates for the zero-one loss for the agnostic learning of halfspaces. We show that when a quantity we refer to as the \textit{soft margin} is well-behaved—a condition satisfied by log-concave isotropic distributions among others—minimizers of convex surrogates for the zero-one loss are approximate minimizers for the zero-one loss itself. As standard convex optimization arguments lead to efficient guarantees for minimizing convex surrogates of the zero-one loss, our methods allow for the first positive guarantees for the classification error of halfspaces learned by gradient descent using the binary cross-entropy or hinge loss in the presence of agnostic label noise.}
}