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The following theorem bounds the VC dimension of the G-composition of C, based on the VC dimension of C and the structure of G

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Theorem 74 VC-dimension of directed acyclic layered networks (See Kearns and Vazirani 1994) Let G be a layered directed acyclic graph with n input nodes and s 2 2 internal nodes, each having at most r inputs Let C be a concept class over 8Y of VC dimension d, corresponding to the set of functions that can be described by each of the s internal nodes Let CG be the G-composition of C, corresponding to the set of functions that can be represented by G Then VC(CG)5 2dslog(es), where e is the base of the natural logarithm

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Note this bound on the VC dimension of the network G grows linearly with the VC dimension d of its individual units and log times linear in s, the number of threshold units in the network Suppose we consider acyclic layered networks whose individual nodes are perceptrons Recall from 4 that an r input perceptron uses linear decision surfaces to represent boolean functions over %' As noted in Section 7421, the VC dimension of linear decision surfaces over is r + 1 Therefore, a single perceptron with r inputs has VC dimension r + 1 We can use this fact, together with the above theorem, to bound the VC dimension of acyclic layered networks containing s perceptrons, each with r inputs, as

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We can now bound the number m of training examples sufficient to learn perceptrons to within (with probability at least (1 - 6)) any target concept from C, error E Substituting the above expression for the network VC dimension into Equation (77), we have

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As illustrated by this perceptron network example, the above theorem is interesting because it provides a general method for bounding the VC dimension of layered, acyclic networks of units, based on the network structure and the VC dimension of the individual units Unfortunately the above result does not directly apply to networks trained using BACKPROPAGATION, for two reasons First, this result applies to networks of perceptrons rather than networks of sigmoid units to which the BACKPROPAGATION algorithm applies Nevertheless, notice that the VC dimension of sigmoid units will be at least as great as that of perceptrons, because a sigmoid unit can approximate a perceptron to arbitrary accuracy by using sufficiently large weights Therefore, the above bound on m will be at least as large for acyclic layered networks of sigmoid units The second shortcoming of the above result is that it fails to account for the fact that BACKPROPAGATION

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