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# Agnostic Learning of Halfspaces with Gradient Descent via Soft Margins

*Proceedings of the 38th International Conference on Machine Learning*, PMLR 139:3417-3426, 2021.

#### 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.