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f F p using .net framework toget 3 of 9 barcode in web,windows application QR Code Overview (10.13). Note the impo rtant improvement over the VC theorem (10.9), with a factor rather than 2 in the exponent of the right hand side of the inequality. Considering again the thermodynamic limit p and dVC with a xed value of the ratio = p/dVC we conclude that (for > 1) the accuracy threshold is given by th ( ) = 1 + ln(2 ) .

ln 2 (10.14). By a more re Code 39 for .NET ned argument and replacing the upper bound by the exact asymptotic expression for the combinatorial sum appearing in (10.7), the accuracy threshold may be further improved to [161] th ( ) = 2 ln(2 ) (2 1) ln(2 1) =: VC ( ).

ln 2 (10.15). We conclude t hat all the error probabilities f of the compatible students are smaller than VC ( ) with probability 1 in the thermodynamic limit. Since the VC dimension characterizes the classi cation diversity of a hypothesis class F, we expect that it must also be related to the storage capacity of the classi ers under consideration. In fact, it is not dif cult to prove from the Sauer.

10 Making Contact with Statistics lemma (10.8) that the storage capacity c = pc /dVC , where pc is the maximum number of random examples that can be correctly stored, must be smaller than 2 (cf. problem 10.

5). The appearance of the VC dimension in the problems of both storage and generalization points to an interesting relation between them. If a class of classi ers is confronted with an example set of size much smaller than its VC dimension it can reproduce the set without exploiting possible correlations coded in the classi cations.

In this region the system works as a pure memory and the generalization ability is rather poor. If, on the other hand, the size of the training set is much larger than the VC dimension the system is well above its storage capacity and has in general no chance to implement all the classi cations. The only way to nevertheless reproduce the whole training set is to exploit all the possible correlations coming from the fact that the training set classi cations were produced by a teacher function.

In this way an alignment between teacher and student builds up which results in an appreciable generalization ability. In this sense generalization sets in where memorization ends. A system that can implement all possible classi cations is a perfect storage device and (for this very reason!) fails completely to generalize.

On the other hand a system with very small VC dimension can only store very few items but generalizes rather quickly (of course only to one of the few classi cations it is able to realize). This highlights the trade-off always present in the modelling of data: a complicated model will be able to reproduce all the features but needs large data sets for suf cient generalization. A simple model will quickly generalize but may be unable to reproduce the data, so that the problem becomes unrealizable.

The best model is hence the simplest one that can reproduce all the data, a principle sometimes referred to as Occam s razor.. 10.4 Comparis Code 39 Full ASCII for .NET on with statistical mechanics The VC bounds derived above are very general and consequently it is not known how tight they are in special situations.

It is hence tempting to test them against explicit results obtained by statistical mechanics for the speci c scenario of a student perceptron learning from a teacher perceptron. As we have argued before, statistical mechanics is designed to study typical properties rather than worst case situations. However, as we will see, it is also possible to analyse partial worst case scenarios and to extract in this way information on some typical worst case .

As an example we will analyse the generalization error of the worst student from the version space. This corresponds to a partial worst case analysis since we do not include the worst possible teacher and the worst possible distribution of the examples. Instead we will use our standard distribution (1.

5) for the examples and.
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