  We give the first statistical-query lower bounds for agnostically learning
any non-polynomial activation with respect to Gaussian marginals (e.g., ReLU,
sigmoid, sign). For the specific problem of ReLU regression (equivalently,
agnostically learning a ReLU), we show that any statistical-query algorithm
with tolerance $n^{-(1/\epsilon)^b}$ must use at least $2^{n^c} \epsilon$
queries for some constant $b, c > 0$, where $n$ is the dimension and $\epsilon$
is the accuracy parameter. Our results rule out general (as opposed to
correlational) SQ learning algorithms, which is unusual for real-valued
learning problems. Our techniques involve a gradient boosting procedure for
"amplifying" recent lower bounds due to Diakonikolas et al. (COLT 2020) and
Goel et al. (ICML 2020) on the SQ dimension of functions computed by two-layer
neural networks. The crucial new ingredient is the use of a nonstandard convex
functional during the boosting procedure. This also yields a best-possible
reduction between two commonly studied models of learning: agnostic learning
and probabilistic concepts.
