The Kernel Gibbs Sampler. Graepel, Thore; Herbrich, Ralf (2000). 514–520.

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.

Generalisation Error Bounds for Sparse Linear Classifiers. Graepel, Thore; Herbrich, Ralf; Shawe-Taylor, John (2000). 298–303.

We provide small sample size bounds on the generalisation error of linear classi�ers that are sparse in their dual representation given by the expansion coe�cients of the weight vector in terms of the training data. These results theoretically justify algorithms like the Support Vector Machine, the Relevance Vector Machine and K-nearest-neighbour. The bounds are a-posteriori bounds to be evaluated after learning when the attained level of sparsity is known. In a PAC-Bayesian style prior knowledge about the expected sparsity is incorporated into the bounds. The proofs avoid the use of double sample arguments by taking into account the sparsity that leaves unused training points for the evaluation of classi�ers. We furthermore give a PAC-Bayesian bound on the average generalisation error over subsets of parameter space that may pave the way combining sparsity in the expansion coefficients and margin in a single bound. Finally, reinterpreting a mistake bound for the classical perceptron algorithm due to Noviko� we demonstrate that our new results put classifiers found by this algorithm on a �firm theoretical basis.

From Margin to Sparsity. Graepel, Thore; Herbrich, Ralf; Williamson, Robert C (2000). 210–216.

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 PAC-style generalisation error bound for the classifier learned by the dual 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.

*Learning Linear Classifiers - Theory and Algorithms*. Technical Report (PhD dissertation), Herbrich, Ralf (2000).

Large Scale Bayes Point Machines. Herbrich, Ralf; Graepel, Thore (2000). 528–534.

The concept of averaging over classifiers is fundamental to the Bayesian analysis of learning. Based on this viewpoint, it has recently been demonstrated for linear classifiers that the centre of mass of version space (the set of all classifiers consistent with the training set) - also known as the Bayes point - exhibits excellent generalisation abilities. However, the billiard algorithm as presented in [Herbrich et al., 2000] is restricted to small sample size because it requires O(m*m) of memory and O(N*m*m) computational steps where m is the number of training patterns and N is the number of random draws from the posterior distribution. In this paper we present a method based on the simple perceptron learning algorithm which allows to overcome this algorithmic drawback. The method is algorithmically simple and is easily extended to the multi-class case. We present experimental results on the MNIST data set of handwritten digits which show that Bayes Point Machines are competitive with the current world champion, the support vector machine. In addition, the computational complexity of BPMs can be tuned by varying the number of samples from the posterior. Finally, rejecting test points on the basis of their (approximative) posterior probability leads to a rapid decrease in generalisation error, e.g. 0.1% generalisation error for a given rejection rate of 10%.

A PAC-Bayesian Margin Bound for Linear Classifiers: Why SVMs work. Herbrich, Ralf; Graepel, Thore (2000). 224–230.

We present a bound on the generalisation error of linear classifiers in terms of a refined margin quantity on the training set. The result is obtained in a PAC-Bayesian framework and is based on geometrical arguments in the space of linear classifiers. The new bound constitutes an exponential improvement of the so far tightest margin bound by Shawe-Taylor et al. and scales logarithmically in the inverse margin. Even in the case of less training examples than input dimensions sufficiently large margins lead to non-trivial bound values and - for maximum margins - to a vanishing complexity term. Furthermore, the classical margin is too coarse a measure for the essential quantity that controls the generalisation error: the volume ratio between the whole hypothesis space and the subset of consistent hypotheses. The practical relevance of the result lies in the fact that the well-known support vector machine is optimal w.r.t. the new bound only if the feature vectors are all of the same length. As a consequence we recommend to use SVMs on normalised feature vectors only - a recommendation that is well supported by our numerical experiments on two benchmark data sets.

Robust Bayes Point Machines. Herbrich, Ralf; Graepel, Thore; Campbell, Colin (2000). 49–54.

Support Vector Machines choose the hypothesis corresponding to the centre of the largest hypersphere that can be inscribed in version space. If version space is elongated or irregularly shaped a potentially superior approach is take into account the whole of version space. We propose to construct the Bayes point which is approximated by the centre of mass. Our implementation of a Bayes Point Machine (BPM) uses an ergodic billiard to estimate this point in the kernel space. We show that BPMs outperform hard margin Support Vector Machines (SVMs) on real world datasets. We introduce a technique that allows the BPM to construct hypotheses with non-zero training error similar to soft margin SVMs with quadratic penelisation of the margin slacks. An experimental study reveals that with decreasing penelisation of training error the improvement of BPMs over SVMs decays, a finding that is explained by geometrical considerations.

Sparsity vs. Large Margins for Linear Classifiers. Herbrich, Ralf; Graepel, Thore; Shawe-Taylor, John (2000). 304–308.

We provide small sample size bounds on the generalisation error of linear classifiers that take advantage of large observed margins on the training set and sparsity in the data dependent expansion coefficients. It is already known from results in the luckiness framework that both criteria independently have a large impact on the generalisation error. Our new results show that they can be combined which theoretically justifies learning algorithms like the Support Vector Machine or the Relevance Vector Machine. In contrast to previous studies we avoid using the classical technique of symmetrisation by a ghost sample but directly using the sparsity for the estimation of the generalisation error. We demonstrate that our result leads to practical useful results even in case of small sample size if the training set witnesses our prior belief in sparsity and large margins.