Limitations of Learning via Embeddings in Euclidean Half-Spaces

The notion of embedding a class of dichotomies in a class of linear half spaces is central to the support vector machines paradigm. We examine the question of determining the minimal Euclidean dimension and the maximal margin that can be obtained when the embedded class has a finite VC dimension. We show that an overwhelming majority of the family of finite concept classes of any constant VC dimension cannot be embedded in low-dimensional half spaces. (In fact, we show that the Euclidean dimension must be almost as high as the size of the instance space.) We strengthen this result even further by showing that an overwhelming majority of the family of finite concept classes of any constant VC dimension cannot be embedded in half spaces (of arbitrarily high Euclidean dimension) with a large margin. (In fact, the margin cannot be substantially larger than the margin achieved by the trivial embedding.) Furthermore, these bounds are robust in the sense that allowing each image half space to err on a small fraction of the instances does not imply a significant weakening of these dimension and margin bounds. Our results indicate that any universal learning machine, which transforms data into the Euclidean space and then applies linear (or large margin) classification, cannot enjoy any meaningful generalization guarantees that are based on either VC dimension or margins considerations. This failure of generalization bounds applies even to classes for which ”straight forward” empirical risk minimization does enjoy meaningful generalization guarantees.