A Strongly Polynomial Algorithm for Approximate Forster Transforms and its Application to Halfspace Learning

• URL: http://arxiv.org/abs/2212.03008v1
• Date: Tue, 6 Dec 2022 14:32:02 GMT
• Title: A Strongly Polynomial Algorithm for Approximate Forster Transforms and its Application to Halfspace Learning
• Authors: Ilias Diakonikolas and Christos Tzamos and Daniel M. Kane
• Abstract summary: The Forster transform is a method of regularizing a dataset by placing it in em radial isotropic position while maintaining some of its essential properties.
• Score: 56.86097988183311
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The Forster transform is a method of regularizing a dataset by placing it in {\em radial isotropic position} while maintaining some of its essential properties. Forster transforms have played a key role in a diverse range of settings spanning computer science and functional analysis. Prior work had given {\em weakly} polynomial time algorithms for computing Forster transforms, when they exist. Our main result is the first {\em strongly polynomial time} algorithm to compute an approximate Forster transform of a given dataset or certify that no such transformation exists. By leveraging our strongly polynomial Forster algorithm, we obtain the first strongly polynomial time algorithm for {\em distribution-free} PAC learning of halfspaces. This learning result is surprising because {\em proper} PAC learning of halfspaces is {\em equivalent} to linear programming. Our learning approach extends to give a strongly polynomial halfspace learner in the presence of random classification noise and, more generally, Massart noise.

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