Learning Unions of omega(1)-Dimensional Rectangles

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, log b)-time algorithms for the following classes: • poly(n log b)-way MAJORITY of O( log(n log b) log log(n log b))-dimensional rectangles. • Union of poly(log(n log b)) many O( log (n log b) (log log(n log b) log log log(n log b))2 )-dimensional rectangles. • poly(n log b)-way MAJORITY of poly(n log b)-OR of disjoint O( log(n log b) log log(n log b)) dimensional rectangles. Our main algorithmic tool is an extension of Jackson’s boostingand Fourier-based Harmonic Sieve algorithm [12] 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 [3] and ideas from recent work on learning augmented AC circuits [13] and on representing Boolean functions as thresholds of parities [15].