Agnostic Learning of Halfspaces with Gradient Descent via Soft Margins

• URL: http://arxiv.org/abs/2010.00539v2
• Date: Sun, 14 Feb 2021 02:20:24 GMT
• Title: Agnostic Learning of Halfspaces with Gradient Descent via Soft Margins
• Authors: Spencer Frei and Yuan Cao and Quanquan Gu
• Abstract summary: We show that gradient descent finds halfspaces with classification error $tilde O(mathsfOPT1/2) + varepsilon$ in $mathrmpoly(d,1/varepsilon)$ time and sample complexity.
• Score: 92.7662890047311
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We analyze the properties of gradient descent on convex surrogates for the zero-one loss for the agnostic learning of linear halfspaces. If $\mathsf{OPT}$ is the best classification error achieved by a halfspace, by appealing to the notion of soft margins we are able to show that gradient descent finds halfspaces with classification error $\tilde O(\mathsf{OPT}^{1/2}) + \varepsilon$ in $\mathrm{poly}(d,1/\varepsilon)$ time and sample complexity for a broad class of distributions that includes log-concave isotropic distributions as a subclass. Along the way we answer a question recently posed by Ji et al. (2020) on how the tail behavior of a loss function can affect sample complexity and runtime guarantees for gradient descent.

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