Learning Constant-Depth Circuits (1993; Linial, Mansour, Nisan)

Problem Definition

This problem deals with learning “simple” Boolean functions $f :\{ 0,1\}^{n} \rightarrow \{-1,1\}$ from uniform random labeled examples. In the basic uniform-distribution PAC framework, the learning algorithm is given access to a uniform random example oracleEX(f, U) which, when queried, provides a labeled random example (x, f(x)) where x is drawn from the uniform distribution U over the Boolean cube {0, 1}ⁿ. Successive calls to the EX(f, U) oracle yield independent uniform random examples. The goal of the learning algorithm is to output a representation of a hypothesis function $h :\{ 0,1\}^{n} \rightarrow \{-1,1\}$ which with high probability has high accuracy; formally, for any ε, δ > 0, given ε and δ the learning algorithm should output an h which with probability at least 1 − δ has $\Pr _{x\in U}[h(x)\neq f(x)] \leq \epsilon.$

Many variants of the basic framework described above have been considered. In the distribution-independentPAC learning model, the...