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Figure The figure shows data clouds situated a t the points ( O , O ) , ( l O , l O ) , (10, -lo), (-10, l o ) , and (-10, -10). At each location the initial and final decision boundaries are shown for a n MLP model. The lines intersecting between the two classes are always the initial and final decision boundaries for those classes. In addition, the initial and final lines for each d a t a set are in the same continuous/dashed line style. For a simplex minimization procedure, the MLP models have a high bias for all examples except those sit,uat,eda t (0,O).
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a movement of the weights to put the decision boundary through the origin, both cause the p value to drop. However, for a gradient-descent algorithm, the move towards zero will predominate as this is the direction of steepest descent. Simplex, on the other hand, may jump directly towards the region a t infinity. Now as the magnitude of the weights increases p becomes stuck as the derivative is close to 0 (bringing gradient based minimization schemes to a halt) and p changes little per unit change in the weights. Figure 12.16 shows this diagrammatically. An interesting question is whether the simplex algorithm can jump to the point at effective infinity from the given starting positions. The answer is that it can if the initial step size is set large enough5. The algorithm moves from initial weights of = (-1.098 2.208 1.098 -2.208 to (-128.255 198.221 - I = 77.224 -228.463 at which point the outputs of the MLP are 0 and 1 to machine precision. Remarkably the R weights were not changed at all, remaining a t
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Figure T h e figure shows d a t a clouds situated a t t h e points ( O , O ) , (10, -lo), (-10, lo), and (-10, -10). At each location the initial and final decision boundaries are shown for an MLP model. T h e lines intersecting between the two classes are always the initial and final decision boundaries for those classes. In addition, the initial and final lines for each d a t a set are in the same continuous/dashed line style. For a conjugate gradients minimization procedure, the MLP models have a high bias for all examples except those situated at (0,O).
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Figure For a given slope of the activation function, the vertical orientation must give lower p value than the oblique orientation.
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It is clear from an examination of the shape of the $ function, and from these examples, that if the data are not translated to be near the origin then a significant bias may be introduced into the MLP classifier. It is clear that both centering the
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The figure shows the iterations of the simplex algorithm as it moves to its Figure filial position as shown in Figure 12.10.
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Figure The figure shows the iterations of the simplex algorithm as it moves to its film1 position. The starting values were the starting values for Figure 12.13 rriultiplied by 10.
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Figure T h e figure shows the iterations of the simplex algorithm it moves to it,s final position. The starting values were the starting values for Figure 12.13 divided by 10.
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Figure As the magnitude of the weights increases the difference in p for t,he two strategies: increasing t,he magnitude of the weights; and moving the decision boundary close t,o the origin, becomes less pronounced. When the weight,s are effectively infinite there is no difference bet,ween these strategies.
CONCLUSION
data and scaling to moderate maximum values both have a beneficial effect on the final decision boundaries. The strategies arising from this investigation are obvious. What is more surprising is how this has escaped notice in the literature. We offer two possible reasons for this. One is than MLPs have very frequently been tested on problems much harder than the test example presented here. For hard high dimensional problems the resulting bias may be hard to see. In addition the flexible nature of the MLP classifier may mean that for some problems the results achieved with an MLP classifier are usable despite the bias. The other is that the bias can be avoided by practice of normalizing the columns of so that each variable is in the range [-1,1]. This is advisable if one is using a single parameter X with, say, a weight decay penalty term, and ensures that all weights in the MLP (on all layers) have a multiplicand in the same range. The use of penalty terms and this scaling has undoubtedly insulated some fitted MLP models from the bias described here.