Machine Learning with Data Dependent Hypothesis Classes

We extend the VC theory of statistical learning to data dependent spaces of classifiers. This theory can be viewed as a decomposition of classifier design into two components; the first component is a restriction to a data dependent hypothesis class and the second is empirical risk minimization within that class. We define a measure of complexity for data dependent hypothesis classes and provide data dependent versions of bounds on error deviance and estimation error. We also provide a structural risk minimization procedure over data dependent hierarchies and prove consistency. We use this theory to provide a framework for studying the trade-offs between performance and computational complexity in classifier design. As a consequence we obtain a new family of classifiers with dimension independent performance bounds and efficient learning procedures.