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Now, heading for the given precondition,

Figure 7.4 Values of the strain components in an In 0.2 Ga 0.8 As quantum well surrounded by Al 0.4 Ga 0.6 As barriers.

we choose n = d+l. } n = d+l A m = Q A r ^ Q A P = Qx(d+l) + (r - Q) { arithmetic }

the Generalized Prediction Error (GPE) which gives a limit on the generalization error as a function of the training error, training set size, the number of effective parameters, and the effective noise variance [Moody 1992, Moody 1994] Murata, Yoshizawa and Amari derived a similar Network Information Criterion [Murata et al 1991, Murata et al 1994a, Murata et al 1994b] Using a different approach, ie Vapnik's Bernoulli theorem, Depenau and M011er derived a bound as a function of training error, the VC-dimension and training set size [Depenau and M011er 1994] These research results give, sometimes overly pessimistic, limits that help to clarify the behavior of generalization and its relationship with architecture, training set size and training error Another important issue in the study of generalization is that of overfitting Overfitting means that the NN learns too much detail, effectively memorizing training patterns.

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Knowing the strain components makes it possible to determine the relative change in the lattice volume (or dilation) using equation (7.3) which now reads:

We conclude that a loop body consisting of the simple assignment r,d := r-Q,d+l maintains the invariant and makes progress to the termination condition. The algorithm we have thus obtained is shown in Figure 15.1. Exercise 15.5 (Elementary remainder algorithm for non-positive P). Derive an algorithm to compute the remainder and quotient when P ^ 0. (Hint: add r < Q to the invariant property (15.3) and weaken the termination condition (15.4) to 0 ^ r.) D

For quantum wells with the layers based on the hexagonal crystal geometry grown in the [0001] direction a similar approach can be applied, modifying Hooke's law as:

This normally happens when the network complexity does not match the size of the training set, ie the number of adjustable weights (free parameters) is larger than the number of independent patterns If this is the case, the weights learn individual patterns, and even capture noise This overfitting phenomenon is the consequence of training on a finite data set, minimizing the empirical error function given in equation (72), which differs from the true risk function given in equation (71) Amari et al developed a statistical theory of overtraining in the asymptotic case of large training set sizes [Amari et al 1995, Amari et al 1996] They analytically determine the ratio in which patterns should be divided into training and test sets to obtain optimal generalization performance and to avoid overfitting.

The algorithm we have developed computes both a remainder and a quotient. Often, however, only the remainder is required, and the quotient is not needed. If this is the case, the algorithm can be simplified, at the expense of making the specification and invariant properties slightly more complicated. We note that, in Figure 15.1, assignments to d play no role in the computation of r. If only the remainder is required, these assignments can be discarded. The

which implies the solutions for the strain components are:

A 0<Q }

or in matrix form:

Overfitting effects under large, medium and small training set sizes have been investigated analytically by Amari et al [Amari et al 1995] and Muller et al [Muller et al 1995]..

r,d := P,0 ;

Similarly as for the cubic crystal geometry, for epitaxial layers based on hexagonal materials the relative change of the volume can be expressed as:

{ Invariant: O^r Bound function:

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