Function Learning from Interpolation

In this paper, we study a statistical property of classes of real-valued functions that we call approximation from interpolated examples. We derive a characterization of function classes that have this property, in terms of their ‘fat-shattering function’, a notion that has proved useful in computational learning theory. The property is central to a problem of learning real-valued functions from random examples in which we require satisfactory performance from every algorithm that returns a function which approximately interpolates the training examples.

[1]  A. Kolmogorov,et al.  Entropy and "-capacity of sets in func-tional spaces , 1961 .

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

[3]  D. Pollard.  Convergence of stochastic processes , 1984 .

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

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

[6]  Robert E. Schapire,et al.  Efficient distribution-free learning of probabilistic concepts , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[7]  R. Dudley,et al.  Uniform and universal Glivenko-Cantelli classes , 1991 .

[8]  Martin Anthony,et al.  Computational learning theory: an introduction , 1992 .

[9]  Shai Ben-David,et al.  Characterizations of learnability for classes of {O, …, n}-valued functions , 1992, COLT '92.

[10]  David Haussler,et al.  Decision Theoretic Generalizations of the PAC Model for Neural Net and Other Learning Applications , 1992, Inf. Comput..

[11]  Balas K. Natarajan,et al.  Occam's razor for functions , 1993, COLT '93.

[12]  Philip M. Long,et al.  Fat-shattering and the learnability of real-valued functions , 1994, COLT '94.

[13]  Philip M. Long,et al.  Characterizations of Learnability for Classes of {0, ..., n}-Valued Functions , 1995, J. Comput. Syst. Sci..

[14]  Noga Alon,et al.  Scale-sensitive dimensions, uniform convergence, and learnability , 1997, JACM.