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arXiv · 2112.12340

Learning with distributional inverters

Eric Binnendyk·Marco Carmosino·Antonina Kolokolova·Ramyaa Ramyaa·Manuel Sabin
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Abstract

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 & 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 & 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

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Eric Binnendyk, Marco Carmosino, Antonina Kolokolova, Ramyaa Ramyaa, Manuel Sabin. 2021-12-23. Learning with distributional inverters. https://arxiv.org/abs/2112.12340

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