Modified Taguchi techniques of robust design optimization are used in an innovative method of training artificial neural networks — for example the network shown in the figure. As in other neural-network-training methods, the synaptic weights (strengths of connections between neurons) are adjusted iteratively in an effort to reduce a cost function, which is usually the sum of squared errors between the actual network outputs and the prescribed (correct) network outputs for training sets of inputs. However, this method offers advantages over older methods, as explained below.

Heretofore, the most popular neural-network-training methods have been based, variously, on gradient-descent (GD) techniques or genetic algorithms (GAs). GD techniques are local search/ optimization techniques; in a typical case, a GD technique leads to a local minimum of the cost function, which may not be the desired global minimum. Moreover, GD learning is slow, typically requiring thousands of iterations. GA learning involves global searches but is also slow, typically requiring hundreds of generations with hundreds of individuals in the population. The present innovative method involves global searches and enables a neural network to learn much faster (equivalently, in fewer iterations) than do GD- and GA-based methods.

This Representative Neural Network contains seven neurons connected via 13 synaptic weights Wi. The output of one typical node is given Y1 = f(W1X1+W4X2+W7X3), where f(x) is a sigmoidal nonlinear function that could be, for example, (1+e¯x) ¯1.

In the present method, each of the N synaptic weights is treated as one of N design parameters to be optimized by Taguchi techniques. At each iteration, a number, M, of trial network designs are considered. In each design, each parameter is set at one of three distinct arbitrary levels. The various designs considered at each iteration are thus represented by M sets of N parameters each, each set being a distinct combination of the three levels. The ensemble of all M trial designs considered at each iteration can thus be represented by an N ×M rectangular array. For example, the table shows an array of M= 27 trial sets of N = 13 parameters each for the neural network illustrated in the figure.

An Array of Parameters comprising 27 sets of synaptic weights represents the trial designs of the network on the first iteration. The costs for the sets of training inputs and outputs are then analyzed according to the procedure described in the text to shift and narrow the search ranges of the parameters on the next iteration.

The cost is computed for each trial design. For each parameter, one computes three different cumulative costs, each comprising the sum of all costs for one of the three parameter levels. The three cumulative costs are compared. Then the search range for the parameter is narrowed and is shifted toward whichever level gave the lowest cumulative cost; that is, three shifted, more-closely-spaced levels of the parameter are chosen for the trial designs on the next iteration.

The process is repeated until the parameters converge on values that make the cost acceptably small. Typically, the search range for each parameter can be narrowed by half on each iteration, so that the intervals containing the solution can shrink by a factor of about 103 in only ten iterations. Thus, learning time is shortened considerably, relative to GD- and GA-based methods.

This work was done by Julian O. Blosiu and Adrian Stoica of Caltech forNASA's Jet Propulsion Laboratory. For further information, access the Technical Support Package (TSP) free on-line at www.techbriefs.com under the Mathematics and Information Sciences category, or circle no. 115 on the TSP Order Card in this issue to receive a copy by mail ($5 charge).

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Training of neural networks by modified Taguchi techniques

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This article first appeared in the March, 1998 issue of NASA Tech Briefs Magazine.

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