16 September 2008 

Geometrical Approaches to Active Learning

Prof. Dr. rer. nat. Klaus Obermayer 
TU Berlin and Bernstein Center for Computational Neuroscience

Learning from examples is a key property of automous agents, and - consequently - many learning algorithms for inductive learning have been developed over the years in the machine learning field. In my contribution I will focus on a particular class of algorithms referred to as "optimal experimental design" or "active learning".Learning machines, which employ these algorithms, request examples which are maximal "informative" for learning a predictor rather than "passively" scanning their environment. There is a large body of empirical evidence, that active learning algorithms - though computationally more expensive - are more efficient in terms of the required number of training examples. Hence active learning should be preferred if training examples are costly to obtain. In my contribution I will present an approach to active learning for binary classification, which is based on the geometrical concept of a "version space". Using an adapted Riemannian metric for the data space, I construct a space of classifiers, where the distance between classifiers is given by their probability of disagreement. Using isometries induced by charts of the sphere, I then relate general binary classification problems to binary classification problems on the hypersphere. Active learning then proceeds by optimally subdividing the "version space", defined as a submanifold of the hypersphere whose classifiers are consistent with the training data. I provide hard upper bounds on the generalization error and provide evidence for an exponential (rather than polynomial) decrease of the generalization error with the number of training examples. Finally, I will discuss the performance of practical algorithms for subdividing version space using kernel methods and Monte-Carlo techniques. The presentation covers joint work with F. Henrich and A. Paus

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