We study a mistake-driven variant of an online Bayesian learning algorithm(similar to one studied by Cesa-Bianchi, Helmbold, and Panizza [CHP96]). This variant only updates its state (learns) on trials in which it makes a mistake. The algorithm makes binary classifications using a linear-threshold classifier and runs in time linear inthe number of attributes seen by the learner. We have been able to show, theoretically and in simulations, that this algorithm performs well under assumptions quite different from those embodied inthe prior of the original Bayesian algorithm. It can handle situations that we do not know how to handle in linear time with Bayesian algorithms. We expect our techniques to be useful in deriving and analyzing other apobayesian algorithms. 1 Introduction We consider two styles of online learning.

A Bayesian formulation of the problem leads to a clear concept of a solution whose computation, however, appears to entail an examination of an intractably-large number of hyperstates. This paper hassuggested extending the Gittins index approach (which applies with great power and elegance to the special class of multi-armed bandit processes) to general adaptive MDP's. The hope has been that if certain salient features of the value of information could be captured, even approximately, then one could be led to a reasonable method for avoiding certain defects of certainty-equivalence approaches (problems with identifiability, "metastability"). Obviously, positive evidence, in the form of empirical results from simulation experiments, would lend support to these ideas-work along these lines is underway. Local bandit approximation is but one approximate computational approach for problems of optimal learning and dual control. Most prominent in the literature of control theory is the "wide-sense" approach of [Bar-Shalom & Tse, 1976], which utilizes localquadratic approximations about nominal state/control trajectories. For certain problems, this method has demonstrated superior performance compared to a certainty-equivalence approach, but it is computationally very intensive and unwieldy, particularly for problems with controller dimension greater than one. One could revert to the view of the bandit problem, or general adaptive MDP, as simply a very large MDP defined over hyperstates, and then consider a some- Local Bandit Approximationfor Optimal Learning Problems 1025 what direct approach in which one performs approximate dynamic programming with function approximation over this domain-details of function-approximation, feature-selection, and "training" all become important design issues.

Department of Computer Science, Carnegie Mellon University 5000 Forbes Ave, Pittsburgh, PA 15213 Abstract Dynamic Programming, Q-Iearning and other discrete Markov Decision Process solvers can be -applied to continuous d-dimensional state-spaces by quantizing the state space into an array of boxes. This is often problematic above two dimensions: a coarse quantization can lead to poor policies, and fine quantization is too expensive. Possible solutions are variable-resolution discretization, or function approximation by neural nets. A third option, which has been little studied in the reinforcement learning literature, is interpolation on a coarse grid. In this paper we study interpolation techniques thatcan result in vast improvements in the online behavior of the resulting control systems: multilinear interpolation, and an interpolation algorithm based on an interesting regular triangulation of d-dimensional space.

The learning properties of a universal approximator, a normalized committee machine with adjustable biases, are studied for online back-propagation learning. Within a statistical mechanics framework, numericalstudies show that this model has features which do not exist in previously studied two-layer network models without adjustablebiases, e.g., attractive suboptimal symmetric phases even for realizable cases and noiseless data. 1 INTRODUCTION Recently there has been much interest in the theoretical breakthrough in the understanding ofthe online learning dynamics of multi-layer feedforward perceptrons (MLPs) using a statistical mechanics framework. In the seminal paper (Saad & Solla, 1995), a two-layer network with an arbitrary number of hidden units was studied, allowing insight into the learning behaviour of neural network models whose complexity is of the same order as those used in real world applications. The model studied, a soft committee machine (Biehl & Schwarze, 1995), consists of a single hidden layer with adjustable input-hidden, but fixed hidden-output weights. The average learning dynamics of these networks are studied in the thermodynamic limit of infinite input dimensions in a student-teacher scenario, where a stu.dent network is presented serially with training examples (elS, (IS) labelled by a teacher network of the same architecture but possibly different number of hidden units.

We study the effect of noise and regularization in an online gradient-descent learning scenario for a general two-layer student network with an arbitrary number of hidden units. Training examples arerandomly drawn input vectors labeled by a two-layer teacher network with an arbitrary number of hidden units; the examples arecorrupted by Gaussian noise affecting either the output or the model itself. We examine the effect of both types of noise and that of weight-decay regularization on the dynamical evolution ofthe order parameters and the generalization error in various phases of the learning process. 1 Introduction One of the most powerful and commonly used methods for training large layered neural networks is that of online learning, whereby the internal network parameters {J} are modified after the presentation of each training example so as to minimize the corresponding error.

Stochastic (online) learning can be faster than batch learning. However, at late times, the learning rate must be annealed to remove thenoise present in the stochastic weight updates. In this annealing phase, the convergence rate (in mean square) is at best proportional to l/T where T is the number of input presentations. An alternative is to increase the batch size to remove the noise. In this paper we explore convergence for LMS using 1) small but fixed batch sizes and 2) an adaptive batch size. We show that the best adaptive batch schedule is exponential and has a rate of convergence whichis the same as for annealing, Le., at best proportional to l/T. 1 Introduction Stochastic (online) learning can speed learning over its batch training particularly,,,,hen data sets are large and contain redundant information [M0l93J. However, at late times in learning, noise present in the weight updates prevents complete convergence fromtaking place. To reduce the noise, the learning rate is slowly decreased (annealed{ at late times. The optimal annealing schedule is asymptotically proportional toT where t is the iteration [GoI87, L093, Orr95J.

Shun-ichi Amari RIKEN Frontier Research Program, RIKEN, Hirosawa 2-1, Wako-shi 351-01, Japan amari@zoo.riken.go.jp Abstract The parameter space of neural networks has a Riemannian metric structure.The natural Riemannian gradient should be used instead of the conventional gradient, since the former denotes the true steepest descent direction of a loss function in the Riemannian space. The behavior of the stochastic gradient learning algorithm is much more effective if the natural gradient is used. The present paper studies the information-geometrical structure of perceptrons and other networks, and prove that the online learning method based on the natural gradient is asymptotically as efficient as the optimal batch algorithm. Adaptive modification of the learning constant is proposed and analyzed in terms of the Riemannian measure andis shown to be efficient. The natural gradient is finally applied to blind separation of mixtured independent signal sources. 1 Introd uction Neural learning takes place in the parameter space of modifiable synaptic weights of a neural network.

New additions in 1998 (UAI-98), July 24-26, 1998 February 1, 1998, use of the new 650 will be the Integrated Systems Track of --papers due February 23, 1998 code will be mandatory; so, please update the technical program and the Intelligent --www.uai98.cbmi.upmc.edu The Computing Research Association All regular members in good February 27: Intelligent Systems is planning a Career Development standing are encouraged to consider Demonstration proposals due. Details can be found at www. (at least one from a current is available by writing to ncai@ cra.org. AAAI fellow) must accompany nominations. Nomination materials are also available on AAAI is delighted to announce the AAAI-98 Workshop Program.

This article posits the idea of robot learning as a new subfield. The results of the Robolearn-96 Workshop provide evidence that learning in modern robotics is distinct from traditional machine learning. The article examines the role of robotics in the social and natural sciences and the potential impact of learning on robotics, generating both a continuum of research issues and a description of the divergent terminology, target domains, and standards of proof associated with robot learning. The article argues that although robot learning is a new subfield, there is significant potential for synergy with traditional machine learning if the differences in research cultures can be overcome.

Machine-learning research has been making great progress in many directions. This article summarizes four of these directions and discusses some current open problems. The four directions are (1) the improvement of classification accuracy by learning ensembles of classifiers, (2) methods for scaling up supervised learning algorithms, (3) reinforcement learning, and (4) the learning of complex stochastic models.