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 $\mu$ to learning the same concept class over the uniform distribution. The reduction succeeds when the sampler for $\mu$ 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 $\mathsf{AC}^0[q]$ is learnable over any succinctly-described product distribution. $\mathsf{AC}^0[q]$ 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 function.