Approximating hyper-rectangles: learning and pseudo-random sets

The PAC learning of rectangles has been studied because they have been found experimentally to yield excellent hypotheses for severaf applied learning problems. Also, pseudorandom sets for rectangles have been actively studied recently because (i) they are a subpmblem common to the derandomization of depth-2 (DIW) circuits and derandotnizing 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” highdlmensional rectangles that arise in PAC learning and pseudorandomness.

[1]  M. Naor Constructing Ramsey graphs from small probability spaces , 1992 .

[2]  N. Littlestone Learning Quickly When Irrelevant Attributes Abound: A New Linear-Threshold Algorithm , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[3]  Leslie G. Valiant,et al.  A general lower bound on the number of examples needed for learning , 1988, COLT '88.

[4]  Larry Carter,et al.  Universal Classes of Hash Functions , 1979, J. Comput. Syst. Sci..

[5]  Noam Nisan,et al.  Approximations of general independent distributions , 1992, STOC '92.

[6]  Noga Alon,et al.  Explicit Ramsey graphs and orthonormal labelings , 1994, Electron. J. Comb..

[7]  J. Gates Introduction to Probability and its Applications , 1992 .

[8]  Leslie G. Valiant,et al.  On the learnability of Boolean formulae , 1987, STOC.

[9]  Leslie G. Valiant,et al.  A theory of the learnable , 1984, STOC '84.

[10]  Leonard Pitt,et al.  Prediction-Preserving Reducibility , 1990, J. Comput. Syst. Sci..

[11]  Noam Nisan,et al.  Extracting randomness: how and why. A survey , 1996, Proceedings of Computational Complexity (Formerly Structure in Complexity Theory).

[12]  Philip M. Long,et al.  PAC Learning Axis-Aligned Rectangles with Respect to Product Distributions from Multiple-Instance Examples , 1996, COLT.

[13]  Michael Luby,et al.  A simple parallel algorithm for the maximal independent set problem , 1985, STOC '85.

[14]  David Haussler,et al.  Learnability and the Vapnik-Chervonenkis dimension , 1989, JACM.

[15]  Leslie G. Valiant,et al.  Computational limitations on learning from examples , 1988, JACM.

[16]  Peter Frankl,et al.  Intersection theorems with geometric consequences , 1981, Comb..

[17]  Peter Auer,et al.  On Learning From Multi-Instance Examples: Empirical Evaluation of a Theoretical Approach , 1997, ICML.

[18]  Michael Krivelevich,et al.  Bounding Ramsey Numbers through Large Deviation Inequalities , 1995, Random Struct. Algorithms.

[19]  Michael E. Saks,et al.  Discrepancy sets and pseudorandom generators for combinatorial rectangles , 1996, Proceedings of 37th Conference on Foundations of Computer Science.

[20]  David Haussler,et al.  Predicting {0,1}-functions on randomly drawn points , 1988, COLT '88.

[21]  M. Murty Ramanujan Graphs , 1965 .

[22]  Sholom M. Weiss,et al.  Computer Systems That Learn , 1990 .

[23]  Michael E. Saks,et al.  Efficient construction of a small hitting set for combinatorial rectangles in high dimension , 1993, Comb..

[24]  Thomas G. Dietterich,et al.  Solving the Multiple Instance Problem with Axis-Parallel Rectangles , 1997, Artif. Intell..

[25]  Moni Naor,et al.  Small-bias probability spaces: efficient constructions and applications , 1990, STOC '90.

[26]  Aaas News,et al.  Book Reviews , 1893, Buffalo Medical and Surgical Journal.

[27]  J. Lamperti ON CONVERGENCE OF STOCHASTIC PROCESSES , 1962 .

[28]  Noga Alon,et al.  Simple Construction of Almost k-wise Independent Random Variables , 1992, Random Struct. Algorithms.

[29]  Noga Alon,et al.  A Fast and Simple Randomized Parallel Algorithm for the Maximal Independent Set Problem , 1985, J. Algorithms.

[30]  Noam Nisan,et al.  Pseudorandomness for network algorithms , 1994, STOC '94.

[31]  Noam Nisan Extracting randomness: how and why , 1996 .

[32]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[33]  Vladimir Vapnik,et al.  Chervonenkis: On the uniform convergence of relative frequencies of events to their probabilities , 1971 .

[34]  Michael Kearns,et al.  Efficient noise-tolerant learning from statistical queries , 1993, STOC.

[35]  Noam Nisan,et al.  Velickovic approximations of general independent distributions , 1992, Symposium on the Theory of Computing.

[36]  Moni Naor,et al.  Small-Bias Probability Spaces: Efficient Constructions and Applications , 1993, SIAM J. Comput..

[37]  Noga Alon,et al.  Simple construction of almost k-wise independent random variables , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[38]  P. Erdös Some remarks on the theory of graphs , 1947 .

[39]  Aravind Srinivasan,et al.  Improved algorithms via approximations of probability distributions (extended abstract) , 1994, STOC '94.

[40]  G. G. Stokes "J." , 1890, The New Yale Book of Quotations.

[41]  Philip M. Long,et al.  PAC Learning Axis-aligned Rectangles with Respect to Product Distributions from Multiple-Instance Examples , 1996, COLT '96.

[42]  H. L. Abbott A Note on Ramsey's Theorem , 1972, Canadian Mathematical Bulletin.

[43]  Joel H. Spencer,et al.  Asymptotic lower bounds for Ramsey functions , 1977, Discret. Math..