Learning Unions of ω(1)-Dimensional Rectangles

Abstract

We consider the problem of learning unions of rectangles over the domain [b]ⁿ, in the uniform distribution membership query learning setting, where both b and n are “large”. We obtain poly(n, logb)-time algorithms for the following classes:

– poly (n logb)-Majority of $O(\frac{\log(n \log b)} {\log \log(n \log b)})$-dimensional rectangles.

–Unions of poly(log(n logb)) many rectangles with dimension

$O(\frac{\log^2 (n \log b)} {(\log \log(n \log b) \log \log \log (n \log b))^2})$.

– poly (n logb)-Majority of poly (n logb)-Or of disjoint rectangles

with dimension $O(\frac{\log(n \log b)} {\log \log(n \log b)})$

Our main algorithmic tool is an extension of Jackson’s boosting- and Fourier-based Harmonic Sieve algorithm [13] to the domain [b]ⁿ, building on work of Akavia et al. [1]. Other ingredients used to obtain the results stated above are techniques from exact learning [4] and ideas from recent work on learning augmented AC ⁰ circuits [14] and on representing Boolean functions as thresholds of parities [16].