Recurrent neural networks of analog units are computers for realvalued functions. We study the time complexity of real computation in general recurrent neural networks. These have sigmoidal, linear, and product units of unlimited order as nodes and no restrictions on the weights. For networks operating in discrete time, we exhibit a family of functions with arbitrarily high complexity, and we derive almost tight bounds on the time required to compute these functions. Thus, evidence is given of the computational limitations that time-bounded analog recurrent neural networks are subject to. 1 Introduction Analog recurrent neural networks are known to have computational capabilities that exceed those of classical Turing machines (see, e.g., Siegelmann and Sontag, 1995; Kilian and Siegelmann, 1996; Siegelmann, 1999).

We study the dynamics of a Hebbian ICA algorithm extracting a single non-Gaussian component from a high-dimensional Gaussian background.

We give an unified convergence analysis of ensemble learning methods including e.g. AdaBoost, Logistic Regression and the Least-Square- Boost algorithm for regression. These methods have in common that they iteratively call a base learning algorithm which returns hypotheses that are then linearly combined. We show that these methods are related to the Gauss-Southwell method known from numerical optimization and state non-asymptotical convergence results for all these methods. Our analysis includes -norm regularized cost functions leading to a clean and general way to regularize ensemble learning.

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 study properties of popular near-uniform (Dirichlet) priors for learning undersampled probability distributions on discrete nonmetric spaces and show that they lead to disastrous results. However, an Occam-style phase space argument expands the priors into their infinite mixture and resolves most of the observed problems. This leads to a surprisingly good estimator of entropies of discrete distributions. Learning a probability distribution from examples is one of the basic problems in data analysis. Common practical approaches introduce a family of parametric models, leading to questions about model selection. In Bayesian inference, computing the total probability of the data arising from a model involves an integration over parameter space, and the resulting "phase space volume" automatically discriminates against models with larger numbers of parameters--hence the description of these volume terms as Occam factors [1, 2]. As we move from finite parameterizations to models that are described by smooth functions, the integrals over parameter space become functional integrals and methods from quantum field theory allow us to do these integrals asymptotically; again the volume in model space consistent with the data is larger for models that are smoother and hence less complex [3]. Further, at least under some conditions the relevant degree of smoothness can be determined self-consistently from the data, so that we approach something like a model independent method for learning a distribution [4]. The results emphasizing the importance of phase space factors in learning prompt us to look back at a seemingly much simpler problem, namely learning a distribution on a discrete, nonmetric space.

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.

It has been noted by several authors that Belief Propagation can can also give impressive results for graphs that are not trees [2]. The Cluster Variation Method (CVM), is a method that has been developed in the physics community for approximate inference in the Ising model [3]. The CVM approximates the joint probability distribution by a number of (overlapping) marginal distributions (clusters). The quality of the approximation is determined by the size and number of clusters. When the clusters consist of only two variables, the method is known as the Bethe approximation.

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.