Exact Identification of Read-Once Formulas Using Fixed Points of Amplification Functions

@article{Goldman1993ExactIO,
  title={Exact Identification of Read-Once Formulas Using Fixed Points of Amplification Functions},
  author={Sally A. Goldman and Michael Kearns and Robert E. Schapire},
  journal={SIAM J. Comput.},
  year={1993},
  volume={22},
  pages={705-726}
}
In this paper a new technique is described for exactly identifying certain classes of read-once Boolean formulas. The method is based on sampling the input-output behavior of the target formula on a probability distribution that is determined by the fixed point of the formula’s amplification function (defined as the probability that a one is output by the formula when each input bit is one independently with probability p). By performing various statistical tests on easily sampled variants of… 

Read-Once Functions Revisited and the Readability Number of a Boolean Function

INTERPOLATING ARITHMETIC READ-ONCE FORMULAS

TLDR
A randomized (Las Vegas) parallel algorithm for the exact interpolation of arithmetic read-once formulas over sufficiently large fields and fields of size 3(n + 3n − 2), which complements some results from [N.H. Bshouty and R. Cleve].

Learning arithmetic read-once formulas

TLDR
An algorithm for exactly learning (or interpolating) arithmetic read-once formulas computing functions over a field and an algorithm that uses randomized membership queries (or substitutions) to identify such formulas over large finite fields and infinite fields are presented.

Locating Errors in Faulty Formulas

TLDR
Under a fault model, it is shown that there is an efficient algorithm that makes a linear number of probes to the blackbox implementation and determines if the blueprint and implementation are identical, and it is proved that if the implementation has a property called polynomial balance, then it is possible to do this efficiently.

On Using Extended Statistical Queries to Avoid Membership Queries

TLDR
It is shown that the characterization of the KM algorithm when applied to SQ-Dρ is tight in terms of learning parity functions, and a characterization for learnability with these extended statistical queries is developed.

Computational Aspects of Parallel Attribute-Efficient Learning

TLDR
This work addresses the problem of nonadaptive learning of Boolean functions with few relevant variables by membership queries by introducing algorithms where also the computational complexity is reasonable, rather than the query number only.

Attribute-Efficient and Non-adaptive Learning of Parities and DNF Expressions

  • V. Feldman
  • Computer Science
    J. Mach. Learn. Res.
  • 2007
TLDR
It is shown that attribute-efficient learning of parities with respect to the uniform distribution is equivalent to decoding high-rate random linear codes from low number of errors, a long-standing open problem in coding theory.

Oracles and queries that are sufficient for exact learning (extended abstract)

TLDR
There is a randomized polynomial-time algorithm that learns any class that is learnable from membership queries with unlimited computational power.

Efficiency and computational limitations of learning algorithms

TLDR
This thesis presents new positive and negative results concerning the learnability of several well-studied function classes in the Probably Approximately Correct (PAC) model of learning, and shows that agnostic learning of parities reduces to learning parities with random classification noise.