The Polynomial Method is Universal for Distribution-Free Correlational SQ Learning

TL;DR

This paper demonstrates that the polynomial method provides a universal and optimal approach for distribution-free correlational SQ learning of Boolean functions, establishing tight lower bounds based on degree measures.

Contribution

It generalizes prior lower bounds to all Boolean function classes using simple, self-contained proofs linking degree measures to CSQ lower bounds.

Findings

Lower bounds on threshold or approximate degree imply CSQ lower bounds.

The polynomial method is shown to be optimal for distribution-free CSQ learning.

Results unify and extend prior bounds in PAC and agnostic models.

Abstract

We consider the problem of distribution-free learning for Boolean function classes in the PAC and agnostic models. Generalizing a beautiful work of Malach and Shalev-Shwartz (2022) that gave tight correlational SQ (CSQ) lower bounds for learning DNF formulas, we give new proofs that lower bounds on the threshold or approximate degree of any function class directly imply CSQ lower bounds for PAC or agnostic learning respectively. While such bounds implicitly follow by combining prior results by Feldman (2008, 2012) and Sherstov (2008, 2011), to our knowledge the precise statements we give had not appeared in this form before. Moreover, our proofs are simple and largely self-contained. These lower bounds match corresponding positive results using upper bounds on the threshold or approximate degree in the SQ model for PAC or agnostic learning, and in this sense these results show that the polynomial method is a universal, best-possible approach for distribution-free CSQ learning.