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Note that just as in the bound from Equation (72), the number of required training examples m grows logarithmically in 118 It now grows log times linear in 116, rather than linearly Significantly, the In I HI term in the earlier bound has now been replaced by the alternative measure of hypothesis space complexity, VC(H) log2 I H I) (recall VC(H) I Equation (77) provides an upper bound on the number of training examples sufficientto probably approximately learn any target concept in C, for any desired t and a It is also possible to obtain a lower bound, as summarized in the following theorem (see Ehrenfeucht et al 1989)

It s easy to overlook Pixie as a lightweight and not very useful utility on the face of it, it simply provides a magnified view of the display. Actually, though, it is a remarkably useful tool, and I use it in most of my projects. Here is a quick tour (I have to admit that there is no tour other than a quick tour for Pixie even the help-file content is only two lines long!). Start up Pixie (/Developer/Applications/Graphics Tools), and you see the magnified view of your current mouse location (see Figure 13 8; in this figure the mouse pointer is over the three Manage... buttons in the Journal Entry window):

Theorem 73 Lower bound on sample complexity Consider any concept class C such that V C ( C ) 2 2, any learner L, and any 0 < E < $, and 0 < S < & Then there exists a distribution 2 and target concept in C such that if L observes fewer 3 examples than

This theorem states that if the number of training examples is too few, then no learner can PAC-learn every target concept in any nontrivial C Thus, this theorem provides a lower bound on the number of training examples necessary for successful learning, complementing the earlier upper bound that gives a suficient number Notice this lower bound is determined by the complexity of the concept class C , whereas our earlier upper bounds were determined by H (why )+ This lower bound shows that the upper bound of the inequality in Equation (77) is fairly tight Both bounds are logarithmic in 116 and linear in V C ( H ) The only difference in the order of these two bounds is the extra log(l/c) dependence in the upper bound

If that were all Pixie did, it would be a little dull. However, there is more to it: you can change the size of the inspection window and set the magnification (Use the Magnify menu, or + and -, or use 1- 9). You can also lock the window using horizontal ( X), vertical ( Y), or both ( L) axes. Open up the Preferences window to see a wider range of options (see Figure 13 9).

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Given the discussion of artificial neural network learning in 4, it is interesting to consider how we might calculate the VC dimension of a network of interconnected units such as the feedforward networks trained by the BACKPROPAGATION procedure This section presents a general result that allows computing the VC dimension of layered acyclic networks, based on the structure of the network and the VC dimension of its individual units This VC dimension can then be used to bound the number of training examples sufficient to probably approximately correctly learn a feedforward network to desired values of c and 6 This section may be skipped on a first reading without loss of continuity Consider a network, G, of units, which forms a layered directed acyclic graph A directed acyclic graph is one for which the edges have a direction (eg, the units have inputs and outputs), and in which there are no directed cycles A layered graph is one whose nodes can be partitioned into layers such that all directed edges from nodes at layer 1 go to nodes at layer 1 + 1 The layered feedforward neural networks discussed throughout 4 are examples of such layered directed acyclic graphs It turns out that we can bound the VC dimension of such networks based on their graph structure and the VC dimension of the primitive units from which they are constructed To formalize this, we must first define a few more terms Let n be the number of inputs to the network G, and let us assume that there is just one output node Let each internal unit Niof G (ie, each node that is not an input) have at most r inputs and implement a boolean-valued function ci : 8'' + (0, 1) from some function class C For example, if the internal nodes are perceptrons, then C will be the class of linear threshold functions defined over 8' We can now define the G-composition of C to be the class of all functions that can be implemented by the network G assuming individual units in G take on functions from the class C In brief, the G-composition of C is the hypothesis space representable by the network G.