Subsequently, SVMs have been modified to handle regression [12] and GPs have been adapted to the problem of classification [8]. Both schemes essentially work in the same function space that is characterised by kernels (SVM) and covariance functions (GP), respectively. While the formal similarity of the two methods is striking the underlying paradigms of inference are very different. The SVM was inspired by results from statistical/PAC learning theory while GPs are usually considered in a Bayesian framework. This ideological clash can be viewed as a continuation in machine learning of the by now classical disagreement between Bayesian and frequentistic statistics.

We present an algorithm that samples the hypothesis space of kernel classifiers. Given a uniform prior over normalised weight vectors and a likelihood based on a model of label noise leads to a piecewise constant posterior that can be sampled by the kernel Gibbs sampler (KGS). The KGS is a Markov Chain Monte Carlo method that chooses a random direction in parameter space and samples from the resulting piecewise constant density along the line chosen. The KGS can be used as an analytical tool for the exploration of Bayesian transduction, Bayes point machines, active learning, and evidence-based model selection on small data sets that are contaminated with label noise. For a simple toy example we demonstrate experimentally how a Bayes point machine based on the KGS outperforms an SVM that is incapable of taking into account label noise. 1 Introduction Two great ideas have dominated recent developments in machine learning: the application of kernel methods and the popularisation of Bayesian inference.

In this paper we give necessary and sufficient conditions under which kernels of dot product type k(x, y) k(x.

Vapnik's result that the expectation of the generalisation error ofthe optimal hyperplane is bounded by the expectation of the ratio of the number of support vectors to the number of training examples is extended to a broad class of kernel machines. The class includes Support Vector Machines for soft margin classification and regression, and Regularization Networks with a variety of kernels and cost functions. We show that key inequalities in Vapnik's result become equalities once "the classification error" is replaced by "the margin error", with the latter defined as an instance with positive cost. In particular we show that expectations of the true margin error and the empirical margin error are equal, and that the sparse solutions for kernel machines are possible only if the cost function is "partially" insensitive. 1 Introduction Minimization of regularized risk is a backbone of several recent advances in machine learning, including Support Vector Machines (SVM) [13], Regularization Networks (RN) [5] or Gaussian Processes [15]. Such a machine is typically implemented as a weighted sum of a kernel function evaluated for pairs composed of a data vector in question and a number of selected training vectors, so called support vectors.

We present an improvement of Novikoff's perceptron convergence theorem. Reinterpreting this mistake bound as a margin dependent sparsity guarantee allows us to give a PACstyle generalisation error bound for the classifier learned by the perceptron learning algorithm. The bound value crucially depends on the margin a support vector machine would achieve on the same data set using the same kernel. Ironically, the bound yields better guarantees than are currently available for the support vector solution itself.

We present a simple sparse greedy technique to approximate the maximum a posteriori estimate of Gaussian Processes with much improved scaling behaviour in the sample size m.

Subsequently, SVMs have been modified to handle regression [12] and GPs have been adapted to the problem of classification [8]. Both schemes essentially work in the same function space that is characterised by kernels (SVM) and covariance functions (GP), respectively. While the formal similarity of the two methods is striking the underlying paradigms of inference are very different. The SVM was inspired by results from statistical/PAC learning theory while GPs are usually considered in a Bayesian framework. This ideological clash can be viewed as a continuation in machine learning of the by now classical disagreement between Bayesian and frequentistic statistics.

We present an algorithm that samples the hypothesis space of kernel classifiers. Given a uniform prior over normalised weight vectors and a likelihood based on a model of label noise leads to a piecewise constant posterior that can be sampled by the kernel Gibbs sampler (KGS). The KGS is a Markov Chain Monte Carlo method that chooses a random direction in parameter space and samples from the resulting piecewise constant density along the line chosen. The KGS can be used as an analytical tool for the exploration of Bayesian transduction, Bayes point machines, active learning, and evidence-based model selection on small data sets that are contaminated with label noise. For a simple toy example we demonstrate experimentally how a Bayes point machine based on the KGS outperforms an SVM that is incapable of taking into account label noise. 1 Introduction Two great ideas have dominated recent developments in machine learning: the application of kernel methods and the popularisation of Bayesian inference.

In this paper we give necessary and sufficient conditions under which kernels of dot product type k(x, y) k(x.

Vapnik's result that the expectation of the generalisation error ofthe optimal hyperplane is bounded by the expectation of the ratio of the number of support vectors to the number of training examples is extended to a broad class of kernel machines. The class includes Support Vector Machines for soft margin classification and regression, and Regularization Networks with a variety of kernels and cost functions. We show that key inequalities in Vapnik's result become equalities once "the classification error" is replaced by "the margin error", with the latter defined as an instance with positive cost. In particular we show that expectations of the true margin error and the empirical margin error are equal, and that the sparse solutions for kernel machines are possible only if the cost function is "partially" insensitive. 1 Introduction Minimization of regularized risk is a backbone of several recent advances in machine learning, including Support Vector Machines (SVM) [13], Regularization Networks (RN) [5] or Gaussian Processes [15]. Such a machine is typically implemented as a weighted sum of a kernel function evaluated for pairs composed of a data vector in question and a number of selected training vectors, so called support vectors.