The partition function for a Boltzmann machine can be bounded from above and below. We can use this to bound the means and the correlations. For networks with small weights, the values of these statistics can be restricted to nontrivial regions (i.e. a subset of [-1, 1]). Experimental results show that reasonable bounding occurs for weight sizes where mean field expansions generally give good results. 1 Introduction Over the last decade, bounding techniques have become a popular tool to deal with graphical models that are too complex for exact computation. A nice property of bounds is that they give at least some information you can rely on.

We derive an equivalence between AdaBoost and the dual of a convex optimization problem, showing that the only difference between minimizing the exponential loss used by AdaBoost and maximum likelihood for exponential models is that the latter requires the model to be normalized to form a conditional probability distribution over labels. In addition to establishing a simple and easily understood connection between the two methods, this framework enables us to derive new regularization procedures for boosting that directly correspond to penalized maximum likelihood. Experiments on UCI datasets support our theoretical analysis and give additional insight into the relationship between boosting and logistic regression.

We give results about the learnability and required complexity of logical formulae to solve classification problems. These results are obtained by linking propositional logic with kernel machines. In particular we show that decision trees and disjunctive normal forms (DNF) can be represented by the help of a special kernel, linking regularized risk to separation margin. Subsequently we derive a number of lower bounds on the required complexity of logic formulae using properties of algorithms for generation of linear estimators, such as perceptron and maximal perceptron learning.

The problem of searching for information in networks like the World Wide Web can be approached in a variety of ways, ranging from centralized indexing schemes to decentralized mechanisms that navigate the underlying network without knowledge of its global structure. The decentralized approach appears in a variety of settings: in the behavior of users browsing the Web by following hyperlinks; in the design of focused crawlers [4, 5, 8] and other agents that explore the Web's links to gather information; and in the search protocols underlying decentralized peer-to-peer systems such as Gnutella [10], Freenet [7], and recent research prototypes [21, 22, 23], through which users can share resources without a central server. In recent work, we have been investigating the problem of decentralized search in large information networks [14, 15]. Our initial motivation was an experiment that dealt directly with the search problem in a decidedly pre-Internet context: Stanley Milgram's famous study of the small-world phenomenon [16, 17]. Milgram was seeking to determine whether most pairs of people in society were linked by short chains of acquaintances, and for this purpose he recruited individuals to try forwarding a letter to a designated "target" through people they knew on a firstname basis.

We study online learning in Boolean domains using kernels which capture feature expansions equivalent to using conjunctions over basic features. We demonstrate a tradeoff between the computational efficiency with which these kernels can be computed and the generalization ability of the 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 algorithm over 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. We also 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 behavior for learning DNF over such a feature set, and thus that such kernel functions for Winnow are not efficiently computable.

The mutual information of two random variables z and J with joint probabilities {7rij} is commonly used in learning Bayesian nets as well as in many other fields. The chances 7rij are usually estimated by the empirical sampling frequency nij In leading to a point estimate J(nij In) for the mutual information. To answer questions like "is J (nij In) consistent with zero?" or "what is the probability that the true mutual information is much larger than the point estimate?"

The recent introduction of the 'relevance vector machine' has effectively demonstrated how sparsity may be obtained in generalised linear models within a Bayesian framework. Using a particular form of Gaussian parameter prior, 'learning' is the maximisation, with respect to hyperparameters, of the marginal likelihood of the data.

We introduce the notion of kernel-alignment, a measure of similarity between two kernel functions or between a kernel and a target function. This quantity captures the degree of agreement between a kernel and a given learning task, and has very natural interpretations in machine learning, leading also to simple algorithms for model selection and learning. We analyse its theoretical properties, proving that it is sharply concentrated around its expected value, and we discuss its relation with other standard measures of performance. Finally we describe some of the algorithms that can be obtained within this framework, giving experimental results showing that adapting the kernel to improve alignment on the labelled data significantly increases the alignment on the test set, giving improved classification accuracy. Hence, the approach provides a principled method of performing transduction.

In this paper we show that online algorithms for classification and regression can be naturally used to obtain hypotheses with good datadependent tail bounds 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 consider noisy Euclidean traveling salesman problems in the plane, which are random combinatorial problems with underlying structure. Gibbs sampling is used to compute average trajectories, which estimate the underlying structure common to all instances. This procedure requires identifying the exact relationship between permutations and tours. In a learning setting, the average trajectory is used as a model to construct solutions to new instances sampled from the same source. Experimental results show that the average trajectory can in fact estimate the underlying structure and that overfitting effects occur if the trajectory adapts too closely to a single instance.