# Statistical-Query Lower Bounds via Functional Gradients

### Abstract

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."

Publication
Neural Information Processing Systems (NeurIPS) 2020