We have calculated analytical expressions for how the bias and variance of the estimators provided by various temporal difference value estimation algorithms change with offline updates over trials in absorbing Markov chains using lookup table representations. We illustrate classes of learning curve behavior in various chains, and show the manner in which TD is sensitive to the choice of its stepsize andeligibility trace parameters. 1 INTRODUCTION A reassuring theory of asymptotic convergence is available for many reinforcement learning (RL) algorithms. What is not available, however, is a theory that explains the finite-term learning curve behavior of RL algorithms, e.g., what are the different kinds of learning curves, what are their key determinants, and how do different problem parameters effect rate of convergence. Answering these questions is crucial not only for making useful comparisons between algorithms, but also for developing hybrid and new RL methods. In this paper we provide preliminary answers to some of the above questions for the case of absorbing Markov chains, where mean square error between the estimated and true predictions is used as the quantity of interest in learning curves.

Images are ambiguous at each of many levels of a contextual hierarchy. Nevertheless,the high-level interpretation of most scenes is unambiguous, as evidenced by the superior performance of humans. Thisobservation argues for global vision models, such as deformable templates.Unfortunately, such models are computationally intractable for unconstrained problems. We propose a compositional modelin which primitives are recursively composed, subject to syntactic restrictions, to form tree-structured objects and object groupings. Ambiguity is propagated up the hierarchy in the form of multiple interpretations, which are later resolved by a Bayesian, equivalently minimum-description-Iength, cost functional.

Wolfgang Maass Institute for Theoretical Computer Science Technische Universitat Graz* PekkaOrponen Department of Mathematics University of Jyvaskylat Abstract We introduce a model for noise-robust analog computations with discrete time that is flexible enough to cover the most important concrete cases, such as computations in noisy analog neural nets and networks of noisy spiking neurons. We show that the presence of arbitrarily small amounts of analog noise reduces the power of analog computational models to that of finite automata, and we also prove a new type of upper bound for the VC-dimension of computational models with analog noise. 1 Introduction Analog noise is a serious issue in practical analog computation. However there exists no formal model for reliable computations by noisy analog systems which allows us to address this issue in an adequate manner. The investigation of noise-tolerant digital computations in the presence of stochastic failures of gates or wires had been initiated by [von Neumann, 1956]. We refer to [Cowan, 1966] and [Pippenger, 1989] for a small sample of the nllmerous results that have been achieved in this direction. The same framework (with stochastic failures of gates or wires) hac; been applied to analog neural nets in [Siegelmann, 1994].

By now it is widely accepted that learning a task from scratch, i.e., without any prior knowledge, is a daunting undertaking.

We propose a new method to compute prediction intervals. Especially forsmall data sets the width of a prediction interval does not only depend on the variance of the target distribution, but also on the accuracy of our estimator of the mean of the target, i.e., on the width of the confidence interval. The confidence interval follows from the variation in an ensemble of neural networks, each of them trained and stopped on bootstrap replicates of the original data set. A second improvement is the use of the residuals on validation patterns insteadof on training patterns for estimation of the variance of the target distribution. As illustrated on a synthetic example, our method is better than existing methods with regard to extrapolation andinterpolation in data regimes with a limited amount of data, and yields prediction intervals which actual confidence levels are closer to the desired confidence levels. 1 STATISTICAL INTERVALS In this paper we will consider feedforward neural networks for regression tasks: estimating an underlying mathematical function between input and output variables based on a finite number of data points possibly corrupted by noise.

Sometimes local features group into regions, as in texture segmentation; at other times they group into contours which may represent object boundaries. Although much is known about the processing steps that extract local features such as oriented input edges, it is still unclear how local features are grouped into global ones more meaningful for objects.

There is now considerable interest in using ensembles or committees of learning machines to improve the performance of the system over that of a single learning machine. In most neural network ensembles, the ensemble members are trained on either the same data (Hansen & Salamon 1990) or different subsets of the data (Perrone & Cooper 1993) . The ensemble members typically have different initial conditions and/ordifferent architectures. The subsets of the data may be chosen at random, with prior knowledge or by some principled approach e.g.

Time series prediction is one of the major applications of neural networks. Aftera short introduction into the basic theoretical foundations we argue that the iterated prediction of a dynamical system may be interpreted asa model of the system dynamics. By means of RBF neural networks we describe a modeling approach and extend it to be able to model instationary systems. As a practical test for the capabilities of the method we investigate the modeling of musical and speech signals and demonstrate that the model may be used for synthesis of musical and speech signals. 1 Introduction Since the formulation of the reconstruction theorem by Takens [10] it has been clear that a nonlinear predictor of a dynamical system may be directly derived from a systems time series. The method has been investigated extensively and with good success for the prediction oftime series of nonlinear systems. Especially the combination of reconstruction techniques and neural networks has shown good results [12].

Neural Computing Research Group Aston University Aston Triangle, Birmingham B4 7ET, UK. Abstract Given unlimited computational resources, it is best to use a criterion ofminimal expected generalisation error to select a model and determine its parameters. However, it may be worthwhile to sacrifice somegeneralisation performance for higher learning speed. A method for quantifying sub-optimality is set out here, so that this choice can be made intelligently. Furthermore, the method is applicable to a broad class of models, including the ultra-fast memory-based methods such as RAMnets. This brings the added benefit of providing, for the first time, the means to analyse the generalisation properties of such models in a Bayesian framework . 1 Introduction In order to quantitatively predict the performance of methods such as the ultra-fast RAMnet, which are not trained by minimising a cost function, we develop a Bayesian formalism for estimating the generalisation cost of a wide class of algorithms.

Many popular learning rules are formulated in terms of continuous, analoginputs and outputs. Biological systems, however, use action potentials, which are digital-amplitude events that encode analog information in the inter-event interval. Action-potential representations are now being used to advantage in neuromorphic VLSI systems as well. We report on a simple learning rule, based on the Riccati equation described by Kohonen [1], modified for action-potential neuronal outputs. We demonstrate this learning rule in an analog VLSI chip that uses volatile capacitive storage for synaptic weights. We show that our time-dependent learning rule is sufficient to achieve approximate weight normalization and can detect temporal correlations in spike trains.