# Learning by extended statistical queries and its relation to PAC learning

## Abstract

PAC learning from examples is factored so that (i) the membership queries are used to evaluate empirically “statistical queries” — certain expectations of functionals involving the unknown target. (ii) approximate value of these statistical queries are used to compute an output — an approximation of the target.

Kearns' original formulation of statistical queries [we use the abbreviation *SQ*], is extended here to include as *SQ* functionals of arbitrary range and order higher than one — second order being the most useful addition. This enables us to capture more ground for casting efficient PAC learning algorithms in *SQ* form: The important Kushilevitz-Mansour Fourier - based algorithm has an *SQ* rendition, as well as its derivatives, e.g. Jackson's recent DNF learning.

Efficient evaluation of extended SQ by membership queries, if possible at all, becomes quite intricate. We show, however, that it is usually robust under classification noise.

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## References

- [0 ]Javed A. Aslam and Scott E. Decatur,
*Simulating statistical queries in the presence of classification noise*, unpublished manuscript, 1993.Google Scholar - [1 ]Avrim Blum, Merrick Furst, Jeffrey Jackson, Michael Kearns, Yishay Mansour, Steven Rudich,
*Weakly learning DNF and characterizing statistical query learning using Fourier analysis. Proceedings of the 26th Annual ACM Symposium on Theory of Computation*, 253–262, 1994.Google Scholar - [2 ]Jeffrey Jackson,
*An Efficient Membership-Query Algorithm for Learning DNF with Respect to the Uniform Distribution*, Proceedings of the 35th Annual IEEE Symposium on the Foundation of Computer Science, 42–53, 1994.Google Scholar - [3 ]Michael J. Kearns,
*Efficient noise-tolerant learning form statistical queries*. In*Proceedings of the 25th Annual ACM Symposium on Theory of Computing*pages 392–401, 1993.Google Scholar - [4 ]Eyal Kushilevitz and Yishay Mansour,
*Learning decision trees using the Fourier transform. SIAM Journal con Computing*,**22**(6) 1331–1348, 1993.Google Scholar - [5 ]Thimothy L. H. Watkins, Albrecht Rau and Michael Biehl,
*The Statistical Mechanics of Learning a Rule*, Reviews of Modern Physics**65**, p.499, 1993.Google Scholar - [6 ]
- [7 ]Leonid Khachiyan,
*Complexity of Polytope Volume Computation*, In:*New Trends in Discrete and Computation Geometry*, Janos Pach (Ed.), Springer Verlag, 1993.Google Scholar