Forster Decomposition and Learning Halfspaces with Noise
This solves a distribution-independent PAC learning problem for halfspaces with noise, addressing a bottleneck in sample complexity that was previously dependent on bit complexity.
The paper tackles the problem of learning halfspaces with Massart noise by introducing a Forster decomposition that efficiently transforms distributions, resulting in the first polynomial-time algorithm with strongly polynomial sample complexity independent of example bit complexity.
A Forster transform is an operation that turns a distribution into one with good anti-concentration properties. While a Forster transform does not always exist, we show that any distribution can be efficiently decomposed as a disjoint mixture of few distributions for which a Forster transform exists and can be computed efficiently. As the main application of this result, we obtain the first polynomial-time algorithm for distribution-independent PAC learning of halfspaces in the Massart noise model with strongly polynomial sample complexity, i.e., independent of the bit complexity of the examples. Previous algorithms for this learning problem incurred sample complexity scaling polynomially with the bit complexity, even though such a dependence is not information-theoretically necessary.