Dynamics and Generalization Ability of LVQ Algorithms

Learning vector quantization (LVQ) schemes constitute intuitive, powerful classification heuristics with numerous successful applications but, so far, limited theoretical background. We study LVQ rigorously within a simplifying model situation: two competing prototypes are trained from a sequence of examples drawn from a mixture of Gaussians. Concepts from statistical physics and the theory of on-line learning allow for an exact description of the training dynamics in highdimensional feature space. The analysis yields typical learning curves, convergence properties, and achievable generalization abilities. This is also possible for heuristic training schemes which do not relate to a cost function. We compare the performance of several algorithms, including Kohonen’s LVQ1 and LVQ+/-, a limiting case of LVQ2.1. The former shows close to optimal performance, while LVQ+/displays divergent behavior. We investigate how early stopping can overcome this difficulty. Furthermore, we study a crisp version of robust soft LVQ, which was recently derived from a statistical formulation. Surprisingly, it exhibits relatively poor generalization. Performance improves if a window for the selection of data is introduced; the resulting algorithm corresponds to cost function based LVQ2. The dependence of these results on the model parameters, for example, prior class probabilities, is investigated systematically, simulations confirm our analytical findings.