Attribute-Efficient PAC Learning of Low-Degree Polynomial Threshold Functions with Nasty Noise

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/∊)^O(d⁾ and under the Gaussian marginal distribution, PAC learns the class up to error rate ∊ with O(^K_∊2⁴_d^d · log^5d n) samples even when an η ≤ O(∊^d) 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.