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DOI: 10.4230/LIPIcs.ITCS.2017.30
URN: urn:nbn:de:0030-drops-81722
URL: http://drops.dagstuhl.de/opus/volltexte/2017/8172/
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Servedio, Rocco A. ; Tan, Li-Yang

What Circuit Classes Can Be Learned with Non-Trivial Savings?

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LIPIcs-ITCS-2017-30.pdf (0.6 MB)


Abstract

Despite decades of intensive research, efficient - or even sub-exponential time - distribution-free PAC learning algorithms are not known for many important Boolean function classes. In this work we suggest a new perspective on these learning problems, inspired by a surge of recent research in complexity theory, in which the goal is to determine whether and how much of a savings over a naive 2^n runtime can be achieved. We establish a range of exploratory results towards this end. In more detail, (1) We first observe that a simple approach building on known uniform-distribution learning results gives non-trivial distribution-free learning algorithms for several well-studied classes including AC0, arbitrary functions of a few linear threshold functions (LTFs), and AC0 augmented with mod_p gates. (2) Next we present an approach, based on the method of random restrictions from circuit complexity, which can be used to obtain several distribution-free learning algorithms that do not appear to be achievable by approach (1) above. The results achieved in this way include learning algorithms with non-trivial savings for LTF-of-AC0 circuits and improved savings for learning parity-of-AC0 circuits. (3) Finally, our third contribution is a generic technique for converting lower bounds proved using Neciporuk's method to learning algorithms with non-trivial savings. This technique, which is the most involved of our three approaches, yields distribution-free learning algorithms for a range of classes where previously even non-trivial uniform-distribution learning algorithms were not known; these classes include full-basis formulas, branching programs, span programs, etc. up to some fixed polynomial size.

BibTeX - Entry

@InProceedings{servedio_et_al:LIPIcs:2017:8172,
  author =	{Rocco A. Servedio and Li-Yang Tan},
  title =	{{What Circuit Classes Can Be Learned with Non-Trivial Savingsl}},
  booktitle =	{8th Innovations in Theoretical Computer Science Conference (ITCS 2017)},
  pages =	{30:1--30:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-029-3},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{67},
  editor =	{Christos H. Papadimitriou},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2017/8172},
  URN =		{urn:nbn:de:0030-drops-81722},
  doi =		{10.4230/LIPIcs.ITCS.2017.30},
  annote =	{Keywords: computational learning theory, circuit complexity, non-trivial savings}
}

Keywords: computational learning theory, circuit complexity, non-trivial savings
Seminar: 8th Innovations in Theoretical Computer Science Conference (ITCS 2017)
Issue Date: 2017
Date of publication: 24.11.2017


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