Part of Advances in Neural Information Processing Systems 33 (NeurIPS 2020)
Surbhi Goel, Aravind Gollakota, Adam Klivans
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/ϵ)b must use at least 2ncϵ queries for some constants b,c>0, where n is the dimension and ϵ is the accuracy parameter. Our results rule out {\em 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.