A.13 DISCUSSION A N D CONCLUSION

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While these three experiments can hardly be called an exhaustive survey of the area, they do allow a number of observations: steepest descent is very slow to converge and is probably not a feasible algorithm in practice; however, the addition of a line search improves the performance of the algorithm considerably;

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simplex does remarkably well for an algorithm that uses neither derivatives nor a line search5. It is not as fast as the conjugate gradient and quai-Newton algorithms in some cases, but it does not fail in any of the three experiments. As it doesn t need derivatives (and is still reasonably fast), it has been found useful in developing and debugging new network architectures; it is reported (Smagt, 1994) that steepest descent (algorithm 1) is less sensitive to local minima than many more sophisticated algorithms (algorithms 2-6), and this also appears t o be the case with the simplex algorithm (see example

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An examination of the neural network literature on function minimization reveals a definite trend. Earlier works considered a range of ad hoc methods and implementations of more standard methods taken from the numerical analysis literature. In addition, there was a blurring of the distinction between the model being fitted and the fitting procedure (e.g., see Benediktsson et al. (1995) where reference is made to a conjugate gradient backpropagation network ). In actual fact, the model and the fitting algorithm are totally different questions. The trend in later works has been to consider the question of fitting the model one rightly belonging in the domain of numerical analysis (there is not much that is special about a function made up of sums of sigmoids). The best advice that can be offered is to adopt the current best unconstrained general function minimization routine from the numerical analysis literature. This appears to be the BFGS quasi-Newton method, or conjugate gradients with the Polak-Ribihre algorithm with Beale restarts, and a good line-search algorithm. However, in the case where derivatives are not available, or less sensitivity to local minima is required, then the simplex algorithm has been found to be very useful.

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The algorithm itself could be said t o incorporate a line search; however, compared to the line search used in the other algorithms, it is very rudimentary.

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DISCUSSION AND CONCLUSION

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Figure A.2 A schematic representation of the action of the simplex algorithm. See Section A.10 for details.

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Table A . l The seven minimization schemes implemented in the software of Dunne and Campbell (1994). See Smagt (1994) for details of 1 to 6 and Nelder and Mead (1965), Rowan (1990), O Neill (1971) or Press et al. (1992) for details of 7.

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conjugate gradients (Fletcher-Reeves algorithm with radient restarts scaled conjugate gradients Mdler (1993) quasi-Newton (BFGS algorithm)

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FUNCTION MINIMIZATION

T h e results from example 1. T h e value of p is plotted against the number of Figure A.3 function evaluations (times 10) for the seven algorithms.

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f Figure A.4 T h e results from example 2. The value of p is plotted against the number o function evaluations (times 10) for the seven algorithms.

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DISCUSSION AND CONCLUSION

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steepe~i descent a lhne search ( 72 seconds ) am gradmnfs (Flewher AeevesI( 4341 08 seconds I sca/~conlylale gradenis ( 2458 75 seconds ) qu~~#-Newlon ( 3366 22 seconds ) Simplex ( 2507 45 seconds 1

Figure A.5 The results from example 3. The value of p is plot,ted against the number of funct,ion evaluations (times 10) for five of the the seven algorithms. The curves for steepest descent and conjugate gradients (Polak-Ribibre algorithm with Beale restarts) are outside the range of the plot.

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APPENDIX B MAXIMUM VALUES OF THE INFLUENCE CURVE

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Restricting our attention to the case of two classes, so that the decision boundary is the surface such that mlp(z) = 0.5, we show that the maximum value of IC(x,w ) occurs when the point x is misclassified. We show this explicitly only for the weight in an MLP of size 1.1.1, so that the 'Y matrix consists of two weights 210 and and the R matrix also consists of two weights, wo and w1. The fitted model is then

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However, a similar argument is possible for each of the weights in the MLP. In addition, the argument may be extended as follows: for a weight E 'Y where h or