A new learning algorithm is developed for the design of statistical classifiers minimizing the rate of misclassification. The method, which is based on ideas from information theory and analogies to statistical physics, assigns data to classes in probability. The distributions arechosen to minimize the expected classification error while simultaneously enforcing the classifier's structure and a level of "randomness" measured by Shannon's entropy. Achievement of the classifier structure is quantified by an associated cost. The constrained optimizationproblem is equivalent to the minimization of a Helmholtz free energy, and the resulting optimization method is a basic extension of the deterministic annealing algorithm that explicitly enforces structural constraints on assignments while reducing theentropy and expected cost with temperature. In the limit of low temperature, the error rate is minimized directly and a hard classifier with the requisite structure is obtained. This learning algorithmcan be used to design a variety of classifier structures. The approach is compared with standard methods for radial basis function design and is demonstrated to substantially outperform other design methods on several benchmark examples, while often retainingdesign complexity comparable to, or only moderately greater than that of strict descent-based methods.

However, learning simple behavior can be quite "Also with NEC Research Institute.

We propose a hierarchical scheme for rapid learning of context dependent "skills" that is based on the recently introduced "Parameterized Self Organizing Map" ("PSOM"). The underlying idea is to first invest some learning effort to specialize the system into a rapid learner for a more restricted range of contexts. The specialization is carried out by a prior "investment learning stage", during which the system acquires a set of basis mappings or "skills" for a set of prototypical contexts. Adaptation of a "skill" to a new context can then be achieved by interpolating in the space of the basis mappings and thus can be extremely rapid. We demonstrate the potential of this approach for the task of a 3D visuomotor mapfor a Puma robot and two cameras. This includes the forward and backward robot kinematics in 3D end effector coordinates, the 2D 2D retina coordinates and also the 6D joint angles. After the investment phasethe transformation can be learned for a new camera setup with a single observation.

Backpropagation learning algorithms typically collapse the network's structure into a single vector of weight parameters to be optimized. We suggest that their performance may be improved by utilizing the structural informationinstead of discarding it, and introduce a framework for ''tempering'' each weight accordingly. In the tempering model, activation and error signals are treated as approximately independentrandom variables. The characteristic scale of weight changes is then matched to that ofthe residuals, allowing structural properties suchas a node's fan-in and fan-out to affect the local learning rate and backpropagated error. The model also permits calculation of an upper bound on the global learning rate for batch updates, which in turn leads to different update rules for bias vs. non-bias weights. This approach yields hitherto unparalleled performance on the family relations benchmark,a deep multi-layer network: for both batch learning with momentum and the delta-bar-delta algorithm, convergence at the optimal learning rate is sped up by more than an order of magnitude.

Recently, several researchers have reported encouraging experimental results whenusing Gaussian or bump-like activation functions in multilayer perceptrons. Networks of this type usually require fewer hidden layers and units and often learn much faster than typical sigmoidal networks. To explain these results we consider a hyper-ridge network, which is a simple perceptron with no hidden units and a rid¥e activation function. If we are interested in partitioningp points in d dimensions into two classes then in the limit as d approaches infinity the capacity of a hyper-ridge and a perceptron is identical.

We compare two regularization methods which can be used to improve thegeneralization capabilities of Gaussian mixture density estimates. The first method uses a Bayesian prior on the parameter space.We derive EM (Expectation Maximization) update rules which maximize the a posterior parameter probability. In the second approachwe apply ensemble averaging to density estimation. This includes Breiman's "bagging", which recently has been found to produce impressive results for classification networks.

In particular, combining separately trained neural networks (commonly referred to as a neural-network ensemble) has been demonstrated to be particularly successful (Alpaydin, 1993; Drucker et al., 1994; Hansen and Salamon, 1990; Hashem et al., 1994; Krogh and Vedelsby, 1995; Maclin and Shavlik, 1995; Perrone, 1992). Both theoretical (Hansen and Salamon, 1990;Krogh and Vedelsby, 1995) and empirical (Hashem et al., 1994; 536 D.W. OPITZ, J. W. SHAVLIK Maclin and Shavlik, 1995) work has shown that a good ensemble is one where the individual networks are both accurate and make their errors on different parts of the input space; however, most previous work has either focussed on combining the output of multiple trained networks or only indirectly addressed how we should generate a good set of networks.

CART either split the data using axis-aligned hyperplanes or they perform a computationally expensivesearch in the continuous space of hyperplanes with unrestricted orientations. We show that the limitations of the former can be overcome without resorting to the latter. For every pair of training data-points, there is one hyperplane that is orthogonal tothe line joining the data-points and bisects this line. Such hyperplanes are plausible candidates for splits. In a comparison on a suite of 12 datasets we found that this method of generating candidate splits outperformed the standard methods, particularly when the training sets were small. 1 Introduction Binary decision trees come in many flavours, but they all rely on splitting the set of k-dimensional data-points at each internal node into two disjoint sets.

We study Bayesian networks for continuous variables using nonlinear conditionaldensity estimators. We demonstrate that useful structures can be extracted from a data set in a self-organized way and we present sampling techniques for belief update based on Markov blanket conditional density models. 1 Introduction One of the strongest types of information that can be learned about an unknown process is the discovery of dependencies and -even more important-of independencies. Asuperior example is medical epidemiology where the goal is to find the causes of a disease and exclude factors which are irrelevant.

We develop a refined mean field approximation for inference and learning in probabilistic neural networks. Our mean field theory, unlike most, does not assume that the units behave as independent degrees of freedom; instead, it exploits in a principled way the existence of large substructures that are computationally tractable. To illustrate the advantages of this framework, we show how to incorporate weak higher order interactions into a first-order hidden Markov model, treating the corrections (but not the first order structure) within mean field theory. 1 INTRODUCTION Learning the parameters in a probabilistic neural network may be viewed as a problem in statistical estimation.