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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.

FooBar is allocated, initialized, and added to myArray There are two references to myFooBar, so its retainCount is two However, deallocating myFooBar makes both references invalid, pointing to deallocated memory space The sayHello message is sent to the first object in myArray the problem is that the object no longer exists Although in this simple example

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t ~ i n t If we were to substitute H for C in the lower bound, this would result in a tighter bound on : in the case H > C

5:

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