H. Schwarze Dept. of Theoretical Physics Lund University Solvegatan 14A 223 62 Lund Sweden J.Hertz Nordita Blegdamsvej 17 2100 Copenhagen 0 Denmark Abstract The problem of learning from examples in multilayer networks is studied within the framework of statistical mechanics. Using the replica formalism we calculate the average generalization error of a fully connected committee machine in the limit of a large number of hidden units. If the number of training examples is proportional to the number of inputs in the network, the generalization error as a function of the training set size approaches a finite value. If the number of training examples is proportional to the number of weights in the network we find first-order phase transitions with a discontinuous drop in the generalization error for both binary and continuous weights. 1 INTRODUCTION Feedforward neural networks are widely used as nonlinear, parametric models for the solution of classification tasks and function approximation. Trained from examples of a given task, they are able to generalize, i.e. to compute the correct output for new, unknown inputs.

Based on precise anatomical data of the bee's olfactory system, we propose an investigation of the possible mechanisms of modulation and control between the two levels of olfactory information processing: the antennallobe glomeruli and the mushroom bodies. We use simplified neurons, but realistic architecture. As a first conclusion, we postulate that the feature extraction performed by the antennallobe (glomeruli and interneurons) necessitates central input from the mushroom bodies for fine tuning.

A.C.C. Coolen R.W. Penney D. Sherrington Dept. of Physics - Theoretical Physics University of Oxford 1 Keble Road, Oxford OXI 3NP, U.K. Abstract A simple model of coupled dynamics of fast neurons and slow interactions, modellingself-organization in recurrent neural networks, leads naturally to an effective statistical mechanics characterized by a partition function which is an average over a replicated system. This is reminiscent of the replica trick used to study spin-glasses, but with the difference that the number of replicas has a physical meaningas the ratio of two temperatures and can be varied throughout the whole range of real values. The model has interesting phaseconsequences as a function of varying this ratio and external stimuli, and can be extended to a range of other models. 1 A SIMPLE MODEL WITH FAST DYNAMIC NEURONS AND SLOW DYNAMIC INTERACTIONS As the basic archetypal model we consider a system of Ising spin neurons (J'i E {-I, I}, i E {I, ..., N}, interacting via continuous-valued symmetric interactions, Iij, which themselves evolve in response to the states of the neurons. The neurons are taken to have a stochastic field-alignment dynamics which is fast compared with the evolution rate of the interactions hj, such that on the timescale of Iii-dynamics the neurons are effectively in equilibrium according to a Boltzmann distribution, (1) 447 448 Cooien, Penney, and Sherrington where HVoj}({O"d) JijO"iO"j (2) i j and the subscript {Jij} indicates that the {Jij} are to be considered as quenched variables. In practice, several specific types of dynamics which obey detailed balance lead to the equilibrium distribution (1), such as a Markov process with single-spin flip Glauber dynamics [1]. The second term acts to limit the magnitude of hj; f3 is the characteristic inverse temperature of the interaction system. VNTJij(t) (4) where the effective Hamiltonian 11. ({ hj}) is given by 1 1 We now recognise (4) as having the form of a Langevin equation, so that the equilibrium distributionof the interaction system is given by a Boltzmann form. Z{3 (6) Coupled Dynamics of Fast Neurons and Slow Interactions 449 where n _ /j3. We may use Z as a generating functional to produce thermodynamic averagesof state variables I ( {O"d; {Jij}) in the combined system by adding suitable infinitesimal source terms to the neuron Hamiltonian (2): HP.j}({O"d) In fact, any real n is possible by tuning the ratio between the two {3's. In the formulation presented in this paper n is always nonnegative, but negative values are possible if the Hebbian rule of (3) is replaced by an anti-Hebbian form with (UiO"j) replaced by - (O"iO"j) (the case of negative n is being studied by Mezard and coworkers [7]).

EE., Linkoping University S-58183 Linkoping Sweden Abstract We describe a number of learning rules that can be used to train unsupervised parallelfeature extraction systems. The learning rules are derived using gradient ascent of a quality function. We consider anumber of quality functions that are rational functions of higher order moments of the extracted feature values. We show that one system learns the principle components of the correlation matrix.Principal component analysis systems are usually not optimal feature extractors for classification. Therefore we design quality functions which produce feature vectors that support unsupervised classification.The properties of the different systems are compared with the help of different artificially designed datasets and a database consisting of all Munsell color spectra. 1 Introduction There are a number of unsupervised Hebbian learning algorithms (see Oja, 1992 and references therein) that perform some version of the Karhunen-Loeve expansion.

Center positions are continuously updated through soft competitive learning. The width of the radial basis functions is derived from the distance to topological neighbors. During the training the observed error is accumulated locally and used to determine where to insert the next unit. This leads (in case of classification problems) to the placement of units near class borders rather than near frequency peaks as is done by most existing methods. The resulting networks need few training epochs and seem to generalize very well. This is demonstrated by examples.

Bumptrees are geometric data structures introduced by Omohundro (1991) to provide efficient access to a collection of functions on a Euclidean space of interest. We describe a modified bumptree structure that has been employed as a neural network classifier, and compare its performance on several classification tasks against that of radial basis function networks and the standard mutIi-Iayer perceptron. 1 INTRODUCTION A number of neural network studies have demonstrated the utility of the multi-layer perceptron (MLP) and shown it to be a highly effective paradigm. Studies have also shown, however, that the MLP is not without its problems, in particular it requires an extensive training time, is susceptible to local minima problems and its perfonnance is dependent upon its internal network architecture. In an attempt to improve upon the generalisation performance and computational efficiency a number of studies have been undertaken principally concerned with investigating the parametrisation of the MLP. It is well known, for example, that the generalisation performance of the MLP is affected by the number of hidden units in the network, which have to be determined empirically since theory provides no guidance.

Reinforcement Learning methods based on approximating dynamic programming (DP) are receiving increased attention due to their utility in forming reactive control policies for systems embedded in dynamic environments. Environments are usually modeled as controlled Markov processes, but when the environment model is not known a priori, adaptive methods are necessary. Adaptive control methodsare often classified as being direct or indirect. Direct methods directly adapt the control policy from experience, whereas indirect methods adapt a model of the controlled process and compute controlpolicies based on the latest model. Our focus is on indirect adaptive DPbased methods in this paper. We present a convergence result for indirect adaptive asynchronous value iteration algorithmsfor the case in which a lookup table is used to store the value function. Our result implies convergence of several existing reinforcementlearning algorithms such as adaptive real-time dynamic programming (ARTDP) (Barto, Bradtke, & Singh, 1993) and prioritized sweeping (Moore & Atkeson, 1993). Although the emphasis of researchers studying DPbased reinforcement learning has been on direct adaptive methods such as Q-Learning (Watkins, 1989) and methods using TD algorithms (Sutton, 1988), it is not clear that these direct methods are preferable in practice to indirect methods such as those analyzed in this paper.

Many events such as flight delays or the absence of a member require the crew pool rescheduling team to change the initial schedule (rescheduling). In this paper, we show that the neural network comparison paradigm applied to the backgammon game by Tesauro (Tesauro and Sejnowski, 1989)can also be applied to the rescheduling problem of an aircrew pool. Indeed both problems correspond to choosing the best solut.ion

Transition point dynamic programming (TPDP) is a memorybased, reinforcementlearning, direct dynamic programming approach toadaptive optimal control that can reduce the learning time and memory usage required for the control of continuous stochastic dynamic systems. TPDP does so by determining an ideal set of transition points (TPs) which specify only the control action changes necessary for optimal control. TPDP converges to an ideal TP set by using a variation of Q-Iearning to assess the merits ofadding, swapping and removing TPs from states throughout the state space. When applied to a race track problem, TPDP learned the optimal control policy much sooner than conventional Q-Iearning, and was able to do so using less memory. 1 INTRODUCTION Dynamic programming (DP) approaches can be utilized to determine optimal control policiesfor continuous stochastic dynamic systems when the state spaces of those systems have been quantized with a resolution suitable for control (Barto et al., 1991). DP controllers, in lheir simplest form, are memory-based controllers that operate by repeatedly updating cost values associated with every state in the discretized state space (Barto et al., 1991).

We prove that the so called "loading problem" for (recurrent) neural networks isunsolvable. This extends several results which already demonstrated thattraining and related design problems for neural networks are (at least) NPcomplete. Our result also implies that it is impossible to find or to formulate a universal training algorithm, which for any neural networkarchitecture could determine a correct set of weights. For the simple proof of this, we will just show that the loading problem is equivalent to "Hilbert's tenth problem" which is known to be unsolvable.