By Martin Hartley Jones
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Similarly, the weights to the output neuron can be chosen such that the output is one as soon as one of the M predicate neurons is one: M yop = sgn h=1 yh + M − 1 2 . 21) This perceptron will give y o = 1 only if x ∈ X + : it performs the desired mapping. The problem is the large number of predicate units, which is equal to the number of patterns in X+, which is maximally 2 N . Of course we can do the same trick for X − , and we will always take the minimal number of mask units, which is maximally 2 N −1 .
This determines the ‘expressive power’ of the network. For ‘smooth’ functions only a few number of hidden units are needed, for wildly fluctuating functions more hidden units will be needed. In the previous sections we discussed the learning rules such as back-propagation and the other gradient based learning algorithms, and the problem of finding the minimum error. In this section we particularly address the effect of the number of learning samples and the effect of the number of hidden units.
3 is is clear that the approximation of the network is not perfect. The resulting approximation error is influenced by: 1. The learning algorithm and number of iterations. This determines how good the error on the training set is minimized. 8. HOW GOOD ARE MULTI-LAYER FEED-FORWARD NETWORKS? 43 2. The number of learning samples. This determines how good the training samples represent the actual function. 3. The number of hidden units. This determines the ‘expressive power’ of the network. For ‘smooth’ functions only a few number of hidden units are needed, for wildly fluctuating functions more hidden units will be needed.