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the MLP model may be extremely sensitive to a single outlying point. While this has been noted in the literature (Ripley, 1994a), we give a clear graphical demonstration of this fact; an MLP model with an excessive number of hidden layer units will not have any robustness property, and will readily overfit the data; there is a considerable difference between the influence curves (ICs) and the (SCs).This is contrary to the experience with other models (Campbell, 1978) and is due to the iterative fitting procedure with a redescending estimator and to the multiple minima of the error surface. We suggest that the shape of the SC is likely to prove important as the shape of the IC in any question of robustness. We also argue, on numerical evidence, that the beneficial effects of using initial weights with small magnitudes, and of early stopping, can be understood in terms of the shape of the IC.
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The IC is a major theoretical tool in the study of robustness and is also the basis of robustness; bound the IC and you have a robust estimator. However, the derivation of the IC depends on taking the limit as the sample size goes to infinity (Hampel et al., 1986, $2); hence we might expect some problems in using it as an applied tool when dealing with particular (finite) data sets. For example, the influence of one point may mask the influence of other points, so that the fitted model may have a sensitivity to the position of certain points that is quite different from that predicted by the IC. While the gradient gives the direction of steepest descent in weight space
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at W O , i t does not indicate how far to step in that direction to find a minimum, as this is also determined by the non-local shape of the surface. In addition, however far we should step in order t o get t o a minimum, the actual length of the step taken from the fitted model will also depend on the numerical minimization routine employed . Hence even for a single step, the function d p / a w does not reliably tell us how the parameters will change. Moreover, these same considerations will apply with each step of the iteration process. For these two reasons, we consider a finite sample version of the IC, namely, the sensitivity curve (SC). If we have an estimator that takes a sample of size and a given sample we can plot .. a function of Then, if we replace F by F N - ~ and by 1 / N in the definition of the IC (equation 8.1, p. 124), we get the sensitivity curve
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We then have three tools for examining the behavior of an MLP on a finite sample:
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A major source of variation, even between different implementation of the same numerical minimization routine, is the line search for the function minimum in the chosen direction. Minimization algorithms may use: an exact; an approximate; or no line search routine.
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the derivative - can be plotted a t the current model during the iterative dp 5 fitting process - this is referred to the "empirical the decision boundary (where mlp(z) = 0.5) can be plotted a t some point in the minimization process or for successive iterations; the SC can be calculated and plotted. However, we will see, each of these three methods (as well as the final model) are dependent on the starting values and the minimization routine. Hence we are only exploring a subset of the possible solutions to these three simple experiments.
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