Discretized-Vapnik-Chervonenkis Dimension for Analyzing Complexity of Real Function Classes


In this paper, we have a tendency to introduce the discretized-Vapnik-Chervonenkis (VC) dimension for finding out the complexity of a true perform class, and then analyze properties of real operate classes and neural networks. We tend to 1st prove that a countable traversal set is enough to achieve the VC dimension for a real operate category, whereas its classical definition states that the traversal set is that the output range of the perform category. Based on this result, we propose the discretized-VC dimension outlined by employing a countable traversal set consisting of rational numbers in the range of a true function class. By using the discretized-VC dimension, we show that if a true function class contains a finite VC dimension, solely a finite traversal set is needed to attain the VC dimension. We have a tendency to then point out that the real operate classes, that have the infinite VC dimension, will be grouped into two classes: KIND-A and KIND-B. Subsequently, based mostly on the obtained results, we tend to discuss the relationship between the VC dimension of an indicator-output network and that of the important-output network, when each networks have the same structure except for the output activation functions. Finally, we present the chance bound based mostly on the discretized-VC dimension for a true operate class that has infinite VC dimension and is of SORT-A. We have a tendency to prove that, with such a function class, the empirical risk minimization (ERM) principle for the perform category remains in step with overwhelming chance. This is a development of the prevailing data that the ERM learning is consistent if and solely if the operate class includes a finite VC dimension.

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