Learning with distributional inverters

We generalize the “indirect learning” technique of Furst et al. (1991) to reduce from learning a concept class over a samplable distribution µ to learning the same concept class over the uniform distribution. The reduction succeeds when the sampler for µ is both contained in the target concept class and efficiently invertible in the sense of Impagliazzo and Luby (1989). We give two applications. • We show that AC0[q] is learnable over any succinctly-described product distribution. AC0[q] is the class of constant-depth Boolean circuits of polynomial size with AND, OR, NOT, and counting modulo q gates of unbounded fanins. Our algorithm runs in randomized quasi-polynomial time and uses membership queries. • If there is a strongly useful natural property in the sense of Razborov and Rudich (1997) — an efficient algorithm that can distinguish between random strings and strings of non-trivial circuit complexity — then general polynomial-sized Boolean circuits are learnable over any efficiently samplable distribution in randomized polynomial time, given membership queries to the target funuction.