On Learning Embedded Midbit Functions

Abstract

A midbit function on l binary inputs x ₁, … , x l outputs the middle bit in the binary representation of x ₁ + ⋯ + x l. We consider the problem of PAC learning embeddedmidbit functions, where the set S ⊂ {x ₁, . . . , x _n} of relevant variables on which the midbit depends is unknown to the learner.

To motivate this problem, we first show that a polynomial time learning algorithm for the class of embedded midbit functions would immediately yield a fairly efficient (quasipolynomial time) PAC learning algorithm for the entire complexity class ACC. We then give two different subexponential learning algorithms, each of which learns embedded midbit functions under any probability distribution in 2√ ^{n log n} log n time. Finally, we give a polynomial time algorithm for learning embedded midbit functions under the uniform distribution.