Concentration inequalities in product spaces and applications to statistical learning theory

In our dissertation we pursue two closely related directions of research. In the first part we study concentration inequalities for some functionals defined on product measurable spaces. This problem is related to and often phenomenon resembling classical isoperimetry---which describe how fast the measure is concentrating around the set of "large" measure of Talagrand, and as an application we prove new inequalities for a class of functionals called empirical processes, and show the advantages of this approach over other popular methods. Using this approach we also prove new Vapnik-Chervonenkis type inequalities that provide a tighter "individual" control for inhomogeneous classes of functions. Finally, we suggest a new symmetrization approach to proving concentration inequalities for empirical processes, and give several examples of application.; In the second part we study a number of machine learning algorithms from the point of view of explaining their generalization ability in terms of some important parameters. We show that this problem is closely related to concentration inequalities for empirical processes, and also to the theory of Gaussian and Rademacher processes. We develope several approaches to this problem and in many cases obtain the best known, and sometimes optimal, bounds on generalization error of these algorithms. Among algorithms that will be studied are voting methods, such as boosting and bagging, neural networks, and support vector machines.