In many discrimination problems a large amount of data is available but only a few of them are labeled. This provides a strong motivation to improve or develop methods for semi-supervised learning. In this paper, boosting is generalized to this task within the optimization framework of MarginBoost . We extend the margin definition to unlabeled data and develop the gradient descent algorithm that corresponds to the resulting margin cost function. This meta-learning scheme can be applied to any base classifier able to benefit from unlabeled data. We propose here to apply it to mixture models trained with an Expectation-Maximization algorithm. Promising results are presented on benchmarks with different rates of labeled data.

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].

In this paper we introduce new algorithms for unsupervised learning basedon the use of a kernel matrix. All the information required bysuch algorithms is contained in the eigenvectors of the matrix or of closely related matrices. We use two different but related costfunctions, the Alignment and the'cut cost'. The first one is discussed in a companion paper [3], the second one is based on graph theoretic concepts. Both functions measure the level of clustering of a labeled dataset, or the correlation between data clusters andlabels.

We study online learning in Boolean domains using kernels which capture featureexpansions equivalent to using conjunctions over basic features. Wedemonstrate a tradeoff between the computational efficiency with which these kernels can be computed and the generalization ability ofthe resulting classifier. We first describe several kernel functions which capture either limited forms of conjunctions or all conjunctions. We show that these kernels can be used to efficiently run the Perceptron algorithmover an exponential number of conjunctions; however we also prove that using such kernels the Perceptron algorithm can make an exponential number of mistakes even when learning simple functions. Wealso consider an analogous use of kernel functions to run the multiplicative-update Winnow algorithm over an expanded feature space of exponentially many conjunctions. While known upper bounds imply that Winnow can learn DNF formulae with a polynomial mistake bound in this setting, we prove that it is computationally hard to simulate Winnow's behaviorfor learning DNF over such a feature set, and thus that such kernel functions for Winnow are not efficiently computable.

We develop an intuitive geometric framework for support vector regression (SVR). By examining when ɛ-tubes exist, we show that SVR can be regarded as a classification problem in the dual space. Hard and soft ɛ-tubes are constructed by separating the convex or reduced convex hulls respectively of the training data with the response variable shifted up and down by ɛ. A novel SVR model is proposed based on choosing the max-margin plane between the two shifted datasets.

Locally Linear Embedding (LLE) is an elegant nonlinear dimensionality-reduction technique recently introduced by Roweis and Saul [2]. It fails when the data is divided into separate groups. We study a variant of LLE that can simultaneously group the data and calculate local embedding of each group. An estimate for the upper bound on the intrinsic dimension of the data set is obtained automatically. 1 Introduction

In this paper we show that online algorithms for classification and regression canbe naturally used to obtain hypotheses with good datadependent tailbounds on their risk. Our results are proven without requiring complicated concentration-of-measure arguments and they hold for arbitrary online learning algorithms. Furthermore, when applied to concrete online algorithms, our results yield tail bounds that in many cases are comparable or better than the best known bounds.

We propose randomized techniques for speeding up Kernel Principal Component Analysis on three levels: sampling and quantization of the Gram matrix in training, randomized rounding in evaluating the kernel expansions, and random projections in evaluating the kernel itself. In all three cases, we give sharp bounds on the accuracy of the obtained approximations. Ratherintriguingly, all three techniques can be viewed as instantiations of the following idea: replace the kernel function by a "randomized kernel" which behaves like in expectation.

We present an extension to the Mixture of Experts (ME) model, where the individual experts are Gaussian Process (GP) regression models. Using aninput-dependent adaptation of the Dirichlet Process, we implement agating network for an infinite number of Experts. Inference in this model may be done efficiently using a Markov Chain relying on Gibbs sampling. The model allows the effective covariance function to vary with the inputs, and may handle large datasets - thus potentially overcoming twoof the biggest hurdles with GP models.

Discriminative classifiers model the posterior p(ylx)directly, or learn a direct map from inputs x to the class labels. There are several compelling reasons for using discriminative rather than generative classifiers, oneof which, succinctly articulated by Vapnik [6], is that "one should solve the [classification] problem directly and never solve a more general problem as an intermediate step [such as modeling p(xly)]." Indeed, leaving aside computational issues and matters such as handling missing data, the prevailing consensus seems to be that discriminative classifiers are almost always to be preferred to generative ones. Anotherpiece of prevailing folk wisdom is that the number of examples needed to fit a model is often roughly linear in the number of free parameters of a model. This has its theoretical basis in the observation that for "many" models, the VC dimension is roughly linear or at most some low-order polynomial in the number of parameters (see, e.g., [1, 3]), and it is known that sample complexity in the discriminative setting is linear in the VC dimension [6]. In this paper, we study empirically and theoretically the extent to which these beliefs are true. A parametric family of probabilistic models p(x, y) can be fit either to optimize the joint likelihood of the inputs and the labels, or fit to optimize the conditional likelihood p(ylx), or even fit to minimize the 0-1 training error obtained by thresholding p(ylx) to make predictions.