Using methods of Statistical Physics, we investigate the rOle of model complexity in learning with support vector machines (SVMs). We show the advantages of using SVMs with kernels of infinite complexity on noisy target rules, which, in contrast to common theoretical beliefs, are found to achieve optimal generalization error although the training error does not converge to the generalization error. Moreover, we find a universal asymptotics of the learning curves which only depend on the target rule but not on the SVM kernel. 1 Introduction Powerful systems for data inference, like neural networks implement complex inputoutput relations by learning from example data. The price one has to pay for the flexibility of these models is the need to choose the proper model complexity for a given task, i.e. the system architecture which gives good generalization ability for novel data. This has become an important problem also for support vector machines [1].

We combine the replica approach from statistical physics with a variational approach to analyze learning curves analytically. We apply the method to Gaussian process regression. As a main result we derive approximative relations between empirical error measures, the generalization error and the posterior variance.

The adaptive TAP Gibbs free energy for a general densely connected probabilistic model with quadratic interactions and arbritary single site constraints is derived. We show how a specific sequential minimization of the free energy leads to a generalization of Minka's expectation propagation. Lastly, we derive a sparse representation version of the sequential algorithm. The usefulness of the approach is demonstrated on classification and density estimation with Gaussian processes and on an independent component analysis problem.

The adaptive TAP Gibbs free energy for a general densely connected probabilistic model with quadratic interactions and arbritary single site constraints is derived. We show how a specific sequential minimization of the free energy leads to a generalization of Minka's expectation propagation. Lastly,we derive a sparse representation version of the sequential algorithm. The usefulness of the approach is demonstrated on classification anddensity estimation with Gaussian processes and on an independent componentanalysis problem.

Using methods of Statistical Physics, we investigate the rOle of model complexity in learning with support vector machines (SVMs). We show the advantages of using SVMs with kernels of infinite complexity on noisy target rules, which, in contrast to common theoretical beliefs, are found to achieve optimal generalization erroralthough the training error does not converge to the generalization error. Moreover, we find a universal asymptotics of the learning curves which only depend on the target rule but not on the SVM kernel. 1 Introduction Powerful systems for data inference, like neural networks implement complex inputoutput relationsby learning from example data. The price one has to pay for the flexibility of these models is the need to choose the proper model complexity for a given task, i.e. the system architecture which gives good generalization ability for novel data. This has become an important problem also for support vector machines [1].

We combine the replica approach from statistical physics with a variational approachto analyze learning curves analytically. We apply the method to Gaussian process regression. As a main result we derive approximative relationsbetween empirical error measures, the generalization error and the posterior variance.

We develop an approach for a sparse representation for Gaussian Process (GP) models in order to overcome the limitations of GPs caused by large data sets.

We develop an approach for a sparse representation for Gaussian Process (GP) models in order to overcome the limitations of GPs caused by large data sets. The method is based on a combination of a Bayesian online algorithm together with a sequential construction of a relevant subsample of the data which fully specifies the prediction of the model. Experimental results on toy examples and large real-world data sets indicate the efficiency of the approach.

Based on a statistical mechanics approach, we develop a method for approximately computing average case learning curves for Gaussian processregression models. The approximation works well in the large sample size limit and for arbitrary dimensionality of the input space. We explain how the approximation can be systematically improvedand argue that similar techniques can be applied to general likelihood models. 1 Introduction Gaussian process (GP) models have gained considerable interest in the Neural Computation Community(see e.g.[I, 2, 3, 4]) in recent years. Being nonparametric models by construction their theoretical understanding seems to be less well developed comparedto simpler parametric models like neural networks. We are especially interested in developing theoretical approaches which will at least give good approximations togeneralization errors when the number of training data is sufficiently large.

Based on a statistical mechanics approach, we develop a method for approximately computing average case learning curves for Gaussian process regression models. The approximation works well in the large sample size limit and for arbitrary dimensionality of the input space. We explain how the approximation can be systematically improved and argue that similar techniques can be applied to general likelihood models. 1 Introduction Gaussian process (GP) models have gained considerable interest in the Neural Computation Community (see e.g.[I, 2, 3, 4]) in recent years. Being nonparametric models by construction their theoretical understanding seems to be less well developed compared to simpler parametric models like neural networks. We are especially interested in developing theoretical approaches which will at least give good approximations to generalization errors when the number of training data is sufficiently large. In this paper we present a step in this direction which is based on a statistical mechanics approach.