What circuit classes can be learned with non-trivial savings?
R. Servedio and L.-Y. Tan.
8th Innovations in Theoretical Computer Science Conference (ITCS), 2017, pp. 23:1--23:22.


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,

  • 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.
  • 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.
  • 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.
  • pdf

    Back to main papers page