# Mathematical Perspectives on Neural Networks

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

continuity we have that dλn = [dλnik] has norm |dλn(θ)| ≤ 3/4 on some neighborhood

of when n is sufficiently large.

Let η + ̂n (3δ/2) be the neighborhood of θ + ̂n (ω) with radius 3δ/2. Because

we have ηn(δ) ⊂ η + ̂n (3δ/2) ⊂ ηn (3δ/2) for all n sufficiently large. For large n, θ + ̂n (ω) is a zero of , and hence a fixed point of λn. Thus for all n sufficiently large, (i) ηn(δ) ⊂ η + ̂n(3δ/2); (ii) |dλn(θ)| ≤ 3/4 〈 1 for all θ in η + ̂n(3δ/2) ⊂ ηn(2δ); and (iii) θ + ̂n (ω) is a fixed point of λn. It follows from that the Gauss-Newton iteration is numerically stable in ηn(δ). Because ω ∈ F5, P0 (F5) = 1, the result follows.

Proof of Corollary 19.3. We argue explicitly only for the case in which the conditions of Theorem 19.13 hold. As

for all n sufficiently large, θ + ̃n(ω′, ω) ∈ ηn(δ0). The conditions of Theorem 19.14 are satisfied for starting value θ + ̃n (ω′, ω) for all n sufficiently large almost surely, so the desired conclusion follows.

ACKNOWLEDGMENT

Support from the IBM Corporation and the National Science Foundation under grant SES-8921382 is gratefully acknowledged. The author wishes to thank Max Stinchcombe for helpful discussions.

REFERENCES

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Baba, N. ( 1989). "A new approach for finding the global minimum of error function of neural networks", Neural Networks, 2, 367-374.

Baldi, P., & Chauvin, Y. ( 1991). "Temporal evolution of generalization during learning in linear networks". Neural Computation, 3, 589-603.

Bahadur, R. R., & Raghavachari, M. ( 1970). "Some asymptotic properties of likelihood ratios on general spaces". In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 1 (pp. 129-152). Berkeley, CA: University of California Press.

Barton, A. ( 1990). "Complexity regularization with application to artificial neural networks" (Tech. Rep. 57), University of Ilinois Urbana-Champaign, Department of Statistics.

Barron, A. ( 1993). "Universal approximation bounds for superpositions of a sigmoidal function," IEEE Transactions on Information Theory, 39, 930-945.

Bartle, R. ( 1966). Integration and measure. New York: Wiley.

Beran, R. ( 1977). "Minimum Hellinger distance estimates for parametric model". Annals of Statistics, 5, 445-463.

Billingsley, P. ( 1968). Convergence of probability measures. New York: Wiley.

Billingsley, P. ( 1979). Probability and measure, New York: Wiley.

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