An adaptive online algorithm extending the learning of learning idea is proposed and theoretically motivated. Relying only on gradient flow information it can be applied to learning continuous functions or distributions, even when no explicit loss function is given and the Hessian is not available. Its efficiency is demonstrated for a non-stationary blind separation task of acoustic signals.

To obtain classification systems with both good generalization performance and efficiency in space and time, we propose a learning method based on combinations of weak classifiers, where weak classifiers are linear classifiers (perceptrons) which can do a little better than making random guesses. A randomized algorithm is proposed to find the weak classifiers. They· are then combined through a majority vote. As demonstrated through systematic experiments, the method developed is able to obtain combinations of weak classifiers with good generalization performance and a fast training time on a variety of test problems and real applications.

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, numerical studies show that this model has features which do not exist in previously studied two-layer network models without adjustable biases, 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 of the 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 (e lS, (IS) labelled by a teacher network of the same architecture but possibly different number of hidden units.

By an extension of statistical mechanics methods, we obtain exact results for the time-dependent generalization error of a linear network with a large number of weights N. We find, for example, that for small training sets of size p N, larger learning rates can be used without compromising asymptotic generalization performance or convergence speed. Encouragingly, for optimal settings of TJ (and, less importantly, weight decay,\) at given final learning time, the generalization performance of online learning is essentially as good as that of offline learning.

Stochastic (online) learning can be faster than batch learning. However, at late times, the learning rate must be annealed to remove the noise 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 which is the same as for annealing, Le., at best proportional to l/T.

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 in the 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 in the 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.

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 and is shown to be efficient.

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, numerical studies show that this model has features which do not exist in previously studied two-layer network models without adjustable biases, 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 of the 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 (e lS, (IS) labelled by a teacher network of the same architecture but possibly different number of hidden units.

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 has suggested 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 local quadratic 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 that can 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.