Learning Constant-Depth Circuits

Keywords and Synonyms

Learning AC ⁰ circuits          

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 oracle EX(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 ${ \epsilon, \delta > 0 }$, given ϵ and δ the learning algorithm should output an h which with probability at least ${ 1 - \delta }$ has ${ \Pr_{x \in U} [h(x) \neq f(x)] \leq \epsilon }$.

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