Favorite Theorems: Learning from Natural Proofs

October Edition

I had a tough choice for my final favorite theorem from the decade 2015-2024. Runners up include Pseudodeterministic Primes and Hardness of Partial MCSP.  But instead in memory of the recently departed Steven Rudich, this month's favorite theorem draws a strong connection between Razborov and Rudich's Natural Proofs and learning theory.

In 1993, Linial, Mansour and Nisan showed that any function f computed by a constant-depth polynomial-size AND-OR-NOT circuits (AC⁰), has a quasipolynomial-time PAC learning algorithm that, after seeing a quasipolynomial number of uniformly randomly chosen inputs x and their value f(x), can with high probability predict f on other randomly chosen inputs. Their proof works by looking a the Fourier basis of f, showing that learning the low-degree coefficients gives a good approximation to f. Their paper was one of the first to apply Fourier transformations to computational complexity. Adam Klivans did a guest post on this paper back in 2004.

Now, suppose we allow unbounded parity gates to the circuit. The Fourier argument falls apart and learning such circuits remained open for over a quarter century until the Carmosino et al. paper. Their paper uses a different technique, building on the proof of the Nisan-Wigderson pseudorandom generator function. They show that if a circuit class has a natural proof of hardness, the constructivity and largeness properties of the natural proof can break a pseudorandom generator for that class, which can then be used to create a learning algorithm. 

The authors then applied the Razborov and Rudich naturalization of Razborov and Smolensky's lower bounds for AC⁰ with parity gates to get a quasipolynomial learning algorithm for that circuit class.