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

Authors Rocco A. Servedio, Li-Yang Tan

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Rocco A. Servedio
Li-Yang Tan

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Rocco A. Servedio and Li-Yang Tan. What Circuit Classes Can Be Learned with Non-Trivial Savings?. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 30:1-30:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017) https://doi.org/10.4230/LIPIcs.ITCS.2017.30

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.

Subject Classification

Keywords
  • computational learning theory
  • circuit complexity
  • non-trivial savings

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