Organizations: Institute of Computer Science of the Czech Academy of Sciences Pod Vodárenskou věží, 2 - 18207 Prague, Czech Republic · DIBRIS University of Genova Via Opera Pia, 13 - 16145 Genova, Italy
Approximation and learning of classifiers of large data sets by neural networks in terms of high-dimensional geometry and statistical learning theory are investigated. The influence of the VC dimension of sets of input-output functions of networks on approximation capabilities is compared with its influence on consistency in learning from samples of data. It is shown that, whereas finite VC dimension is desirable for uniform convergence of empirical errors, it may not be desirable for approximation of functions drawn from a probability distribution modeling the likelihood that they occur in a given type of application. Based on the concentration-of-measure properties of high dimensional geometry, it is proven that both errors in approximation and empirical errors behave almost deterministically for networks implementing sets of input-output functions with finite VC dimensions in processing large data sets. Practical limitations of the universal approximation property, the trade-offs between the accuracy of approximation and consistency in learning from data, and the influence of depth of networks with ReLU units on their accuracy and consistency are discussed.
Figures & tables
Figure 1: Illustration of Theorem 3 (i). N=ΠH(m) caps scattered on Sm−1 . As m grows, each cap shrinks exponentionally while N grows only polynomially.
Figure 2: Classifiers within λ of h in the case of uniform probability.
Figure 3: The cases of uniform and non-uniform probability.