The PAC learning of rectangles has been studied because they have been found experimentally to yield excellent hypotheses for several applied learning problems. Also, pseudorandom sets for rectangles have been actively studied recently because (i) they are a subproblem common to the derandomization of depth-2 (DNF) circuits and derandomizing Randomized Logspace, and (ii) they approximate the distribution of n independent multivalued random variables. We present improved upper bounds for a class of such problems of "approximating " high-dimensional rectangles that arise in PAC learning and pseudorandomness. Key words and phrases. Rectangles, machine learning, PAC learning, derandomization, pseudorandomness, multipleinstance learning, explicit constructions, Ramsey graphs, random graphs, sample complexity, approximations of distributions. 1
|
1395
|
A theory of the learnable
– Valiant
- 1984
|
|
1338
|
The probabilistic method
– Alon, Spencer
- 2000
|
|
685
|
On the uniform convergence of relative frequencies of events to their probabilities. Theory of Probability and its Applications, 16(2): 264–280
– Vapnik, Chervonenkis
- 1971
|
|
545
|
Learnability and the VapnikChervonenkis dimension
– Blumer, Ehrenfeucht, et al.
- 1989
|
|
525
|
Learning quickly when irrelevant attributes abound: a new linear-threshold algorithm
– Littlestone
- 1993
|
|
490
|
Universal Classes of Hash Functions
– Carter, Wegman
- 1979
|
|
340
|
Convergence of Stochastic Processes
– Pollard
- 1984
|
|
321
|
Computer Systems that Learn
– Weiss, Kulikowski
- 1991
|
|
276
|
A simple parallel algorithm for the maximal independent set problem
– Luby
- 1986
|
|
214
|
Small-bias probability spaces: efficient constructions and applications
– Naor, Naor
- 1993
|
|
211
|
Efficient Noise-Tolerant Learning from Statistical Queries
– Kearns
- 1998
|
|
209
|
Simple construction of almost k-wise independent random variables. Random Structures and Algorithms
– Alon, Goldreich, et al.
- 1992
|
|
208
|
Ramanujan graphs
– Lubotzky, Phillips, et al.
- 1988
|
|
180
|
A general lower bound on the number of examples needed for learning
– Ehrenfeucht, Haussler, et al.
- 1989
|
|
180
|
Computational limitations on learning from examples
– Pitt, Valiant
- 1988
|
|
162
|
A fast and simple randomized parallel algorithm for the maximal independent set problem
– Alon, Babai, et al.
- 1986
|
|
151
|
On the learnability of Boolean formulae
– Kearns, Li, et al.
- 1987
|
|
118
|
Solving the Multiple-Instance Problem with Axis-Parallel Rectangles
– Dietterich, Lathrop, et al.
|
|
100
|
Intersection theorems with geometric consequences, Combinatorica 1
– Frankl, Wilson
- 1981
|
|
78
|
Some remarks on the theory of graphs
– Erdős
- 1947
|
|
59
|
Predicting f0� 1g-functions on randomly drawn points
– Haussler, Littlestone, et al.
- 1990
|
|
45
|
On learning from multi-instance examples: Empirical evaluation of a theoretical approach
– Auer
- 1997
|
|
40
|
Explicit Ramsey graphs and orthonormal labelings, The Electronic
– Alon
- 1994
|
|
35
|
Prediction preserving reducibility
– Warmuth
- 1990
|
|
31
|
A note on learning from multiple-instance examples
– Blum, Kalai
- 1998
|
|
28
|
Approximations of general independent distributions
– Even, Goldreich, et al.
- 1992
|
|
28
|
Pseudorandomness for Network Algorithms
– Impagliazzo, Nisan, et al.
- 1994
|
|
23
|
Efficient construction of a small hitting set for combinatorial rectangles in high dimension
– Linial, Luby, et al.
- 1997
|
|
23
|
Extracting randomness: How and why
– Nisan
- 1996
|
|
22
|
PAC learning axis-aligned rectangles with respect to product distributions from multiple-instance examples
– Long, Tan
- 1998
|
|
18
|
Improved algorithms via approximations of probability distributions
– Chari, Rohatgi, et al.
- 1994
|
|
18
|
Bounding Ramsey numbers through large deviation inequalities, Random Structures and Algorithms 7
– Krivelevich
- 1995
|
|
16
|
Asymptotic lower bounds for Ramsey functions
– Spencer
- 1977
|
|
12
|
Discrepancy sets and pseudorandom generators for combinatorial rectangles
– Armoni, Saks, et al.
- 1996
|
|
9
|
An Introduction to Probability and its Applications
– Feller
- 1968
|
|
8
|
An Introduction to Probability and its Applications, volume 1
– Feller
- 1968
|
|
7
|
Constructing ramsey graphs from small probabiltiy spaces. Unpublished. Available from the author’s homepage on http://www. wisdom.weizmann.ac.il/˜naor
– Naor
|
|
6
|
Computer systems that
– Weiss, Kulikowski
- 1991
|
|
4
|
A constructive lower bound for Ramsey numbers
– Frankl
- 1977
|
|
2
|
A note on Ramsey's theorem
– Abbott
- 1972
|
|
