Mathematical Perspectives on Neural Networks

By Paul Smolensky; Michael C. Mozer et al. | Go to book overview

for this belief is given in Clarke and Barron ( 1990, n.d.); Haussler, Kearns, and Schapere ( 1994); and Opper and Haussler ( 1991a, 1995). Necessary sample size estimates for decision rule spaces as general as those studied from the minimax perspective using uniform convergence, however, have not yet been tackled from the Bayesian perspective. Vapnik's recent (work this volume Chap. 20.) gives an alternative, related approach.

Finally, many other issues would need to be considered in a complete treatment of the problem of overfitting, including distribution specific bounds on sample complexity (Theorem 18.2 is actually distribution specific, since the random covering numbers are distribution specific, yet we only apply it here in a distribution independent setting), decision rule spaces with infinite pseudo- and metric dimensions [these include various classes of smooth functions and their relatives, see Dudley ( 1984, Chap. 7) and Quiroz ( 1989)] and non-i.i.d, sources of examples; see Nobel and Dembo ( 1990) and White ( 1990a; this volume, Chap. 19). Despite these shortcomings, we feel that the theory we give here provides useful insights into the nature of the problem of overfitting in learning, and because of its generality it will be a useful starting point for further research in this area.


ACKNOWLEDGMENTS

Support from the Office of Naval Research through Grants N00014-B6-K-0454 and N00014-91-J-1162 is gratefully acknowledged. A preliminary version of Part 1 of this chapter appeared in 1990, Proceedings of the 8th National Conference on Artificial Intelligence, San Mateo, CA: Morgan Kaufmann, pp. 1101-1108.

I would like to thank Dana Angluin, David Pollard, and Phil Long for their careful criticisms of an earlier draft of this paper, and their numerous suggestions for improvements. I also thank Naoki Abe, Anselm Blumer, Richard Dudley, and Michael Kearns for helpful comments on earlier drafts. I would also like to thank Ron Rivest, David Rumelhart, Andrzej Ehrenfeucht, and Nick Littlestone for stimulating discussions on these topics.


APPENDIX

A1. Metric Spaces, Covering Numbers, and Metric Dimension

A pseudometric on a set S is a function p from S × S into R+ such that for all x, y, z, ∈ S, x = y ⇒ ρ(x, y) = 0, ρ(x, y) = ρ(y, x) (symmetry) and ρ(x, z) ≤ ρ(x, y) + ρ(y, z) (triangle inequality). If in addition ρ(x, y) = 0 ⇒ x = y, then p is metric. (S, ρ) is a (pseudo) metric space. (S, ρ) is complete if every Cauchy sequence of points in S converges to a point in S; (S, ρ) is separable if it contains a countable dense subset, that is, a countable subset A such that for every

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