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**A theory of learning and generalization. With applications to neural networks and control systems.**
*(English)*
Zbl 0928.68061

Berlin: Springer. xviii, 383 p. $ 79.95; DM 118.00; öS 862.00; sFr 107.50; £47.50 (1997).

The book presents recent developments in statistical learning theory and their applications to neural networks and control theory. The emphasis is on the so-called “probably approximately correct (PAC)” learning theory. A connection between PAC theory and some of the fundamental results in the theory of empirical processes is shown. The book consists of 12 chapters, the first two of which introduce the background for the rest of the book. Chapter 3 presents three learning problem formulations defining the universe of discourse of the book. Chapter 4 introduces two combinatorial parameters, the Vapnik-Chervonenkis dimension and the Pollard dimension. The definitions of these notions are given, and some useful inequalities are proved in respect of the so-called growth functions. Chapters 5 discusses the first learning problem formulated earlier in Chapter 3, namely the uniform convergence of empirical means to their true values as the number of samples approaches infinity. Chapter 6 presents the problems of concept and functional learning where samples are generated with known fixed probability, while in Chapter 7 it is assumed that the probability measure generating the samples is unknown. An intermediate solution assuming that learning samples are generated by a probability measure belonging to a family that is neither a singleton set, not the set of all probability measures, is presented in Chapter 8. Chapter 9 introduces the so-called “efficient learning”, and presents examples of concept classes that are efficiently learnable and such that are not efficiently learnable. In Chapters 10 and 11 several applications of presented in the book learning models to neural networks and control systems are discussed. Some open problems in statistical learning theory are stated in Chapter 12.

Reviewer: N.Zlatareva (New Britain)