A Strongly Polynomial Algorithm for Approximate Forster Transforms and its Application to Halfspace Learning
This provides a breakthrough in computational efficiency for halfspace learning, addressing a fundamental problem in machine learning theory with practical implications for robust classification.
The authors developed the first strongly polynomial time algorithm for computing approximate Forster transforms, which are used to regularize datasets into radial isotropic position. By applying this algorithm, they achieved the first strongly polynomial time algorithm for distribution-free PAC learning of halfspaces, extending to settings with random classification and Massart noise.
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.