The concept class of low-degree polynomial threshold functions (PTFs) plays a
fundamental role in machine learning. In this paper, we study PAC learning of
K-sparse degree-d PTFs on Rn, where any such concept depends
only on K out of n attributes of the input. Our main contribution is a new
algorithm that runs in time (nd/ϵ)O(d) and under the Gaussian
marginal distribution, PAC learns the class up to error rate ϵ with
O(ϵ2dK4d⋅log5dn) samples even when an η≤O(ϵd) fraction of them are corrupted by the nasty noise of
Bshouty et al. (2002), possibly the strongest corruption model. Prior to this
work, attribute-efficient robust algorithms are established only for the
special case of sparse homogeneous halfspaces. Our key ingredients are: 1) a
structural result that translates the attribute sparsity to a sparsity pattern
of the Chow vector under the basis of Hermite polynomials, and 2) a novel
attribute-efficient robust Chow vector estimation algorithm which uses
exclusively a restricted Frobenius norm to either certify a good approximation
or to validate a sparsity-induced degree-2d polynomial as a filter to detect
corrupted samples.Comment: ICML 202