Degree-𝑑 chow parameters robustly determine degree-𝑑 PTFs (and algorithmic applications)

The degree-d Chow parameters of a Boolean function are its degree at most d Fourier coefficients. It is well-known that degree-d Chow parameters uniquely characterize degree-d polynomial threshold functions (PTFs) within the space of all bounded functions. In this paper, we prove a robust version of this theorem: For f any Boolean degree-d PTF and g any bounded function, if the degree-d Chow parameters of f are close to the degree-d Chow parameters of g in ℓ2-norm, then f is close to g in ℓ1-distance. Notably, our bound relating the two distances is independent of the dimension. That is, we show that Boolean degree-d PTFs are robustly identifiable from their degree-d Chow parameters. No non-trivial bound was previously known for d >1. Our robust identifiability result gives the following algorithmic applications: First, we show that Boolean degree-d PTFs can be efficiently approximately reconstructed from approximations to their degree-d Chow parameters. This immediately implies that degree-d PTFs are efficiently learnable in the uniform distribution d-RFA model. As a byproduct of our approach, we also obtain the first low integer-weight approximations of degree-d PTFs, for d>1. As our second application, our robust identifiability result gives the first efficient algorithm, with dimension-independent error guarantees, for malicious learning of Boolean degree-d PTFs under the uniform distribution. The proof of our robust identifiability result involves several new technical ingredients, including the following structural result for degree-d multivariate polynomials with very poor anti-concentration: If p is a degree-d polynomial where p(x) is very close to 0 on a large number of points in { ± 1 }n, then there exists a degree-d hypersurface that exactly passes though almost all of these points. We leverage this structural result to show that if the degree-d Chow distance between f and g is small, then we can find many degree-d polynomials that vanish on their disagreement region, and in particular enough that forces the ℓ1-distance between f and g to also be small. To implement this proof strategy, we require additional technical ideas. In particular, in the d=2 case we show that for any large vector space of degree-2 polynomials with a large number of common zeroes, there exists a linear function that vanishes on almost all of these zeroes. The degree-d degree generalization of this statement is significantly more complex, and can be viewed as an effective version of Hilbert’s Basis Theorem for our setting.