Limitations of Learning via Embeddings in Euclidean Half-Spaces

Abstract

This paper considers the embeddability of general concept classes in Euclidean half spaces. By embedding in half spaces we refer to a mapping from some concept class to half spaces so that the labeling given to points in the instance space is retained. The existence of an embedding for some class may be used to learn it using an algorithm for the class it is embedded into. The Support Vector Machines paradigm employs this idea for the construction of a general learning system. We show that an overwhelming majority of the family of finite concept classes of constant VC dimension d cannot be embedded in lowdimensional 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 constant VC dimension d 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.