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
(nlogb)
-way Majority of
O(
log(nlogb)
loglog(nlogb)
)
-dimensional rectangles.
- Union of poly
(log(nlogb))
many
O(
log
2
(nlogb)
(loglog(nlogb)logloglog(nlogb)
)
2
)
-dimensional rectangles.
- poly
(nlogb)
-way Majority of poly
(nlogb)
-Or of disjoint
O(
log(nlogb)
loglog(nlogb)
)
-dimensional rectangles.
Our main algorithmic tool is an extension of Jackson's boosting- and Fourier-based Harmonic Sieve algorithm [Jackson 1997] to the domain
[b
]
n
, building on work of [Akavia, Goldwasser, Safra 2003]. Other ingredients used to obtain the results stated above are techniques from exact learning [Beimel, Kushilevitz 1998] and ideas from recent work on learning augmented
A
C
0
circuits [Jackson, Klivans, Servedio 2002] and on representing Boolean functions as thresholds of parities [Klivans, Servedio 2001].