  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.
