Approximation Error Bounds via Rademacher's Complexity

Approximation properties of some connectionistic models, commonly used to construct approximation schemes for optimization problems with multivariable functions as admissible solutions, are investigated. Such models are made up of linear combinations of computational units with adjustable parameters. The relationship between model complexity (number of computational units) and approximation error is investigated using tools from Statistical Learning Theory, such as Talagrand’s inequality, fat-shattering dimension, and Rademacher’s complexity. For some families of multivariable functions, estimates of the approximation accuracy of models with certain computational units are derived in dependence of the Rademacher’s complexities of the families. The estimates improve previously-available ones, which were expressed in terms of V C dimension and derived by exploiting union-bound techniques. The results are applied to approximation schemes with certain radial-basis-functions as computational units, for which it is shown that the estimates do not exhibit the curse of dimensionality with respect to the number of variables.