A method for transforming performance evaluation signals distal both in space and time into proximal signals usable by supervised learning algorithms, presentedin [Jordan & Jacobs 90], is examined. A simple observation concerning differentiation through models trained with redundant inputs (as one of their networks is) explains a weakness in the original architecture and suggests a modification: an internal world model that encodes action-space exploration and, crucially, cancels input redundancy to the forward model is added. Learning time on an example task, cartpole balancing,is thereby reduced about 50 to 100 times. 1 INTRODUCTION In many learning control problems, the evaluation used to modify (and thus improve) controlmay not be available in terms of the controller's output: instead, it may be in terms of a spatial transformation of the controller's output variables (in which case we shall term it as being "distal in space"), or it may be available only several time steps into the future (termed as being "distal in time"). For example, control of a robot arm may be exerted in terms of joint angles, while evaluation may be in terms of the endpoint cartesian coordinates; furthermore, we may only wish to evaluate the endpoint coordinates reached after a certain period of time: the co- ·Current address: Computation and Neural Systems Program, California Institute of Technology, Pasadena CA. 563 564 Brody ordinatesreached at the end of some motion, for instance. In such cases, supervised learning methods are not directly applicable, and other techniques must be used. Here we study one such technique (proposed for cases where the evaluation is distal in both space and time by [Jordan & Jacobs 90)), analyse a source of its problems, and propose a simple solution for them which leads to fast, efficient learning. We first describe two methods, and then combine them into the "predictive forward modeling" technique with which we are concerned.

An alternative to the typical technique of selecting training examples independently from a fixed distribution is fonnulated and analyzed, in which the current example is presented repeatedly until the error for that item is reduced to some criterion value,; then, another item is randomly selected.The convergence time can be dramatically increased or decreased by this heuristic, depending on the task, and is very sensitive to the value of .

Learning an input-output mapping from a set of examples can be regarded as synthesizing an approximation of a multidimensional function.

Learning an input-output mapping from a set of examples can be regarded as synthesizing an approximation of a multidimensional function.

Bruno Caprile I.R.S.T. Povo, Italy, 38050 Learning an input-output mapping from a set of examples can be regarded as synthesizing an approximation of a multidimensional function.

The issues of scaling and generalization have emerged as key issues in current studies of supervised learning from examples in neural networks. Questions such as how many training patterns and training cycles are needed for a problem of a given size and difficulty, how to represent the inllUh and how to choose useful training exemplars, are of considerable theoretical and practical importance. Several intuitive rules of thumb have been obtained from empirical studies, but as yet there are few rigorous results. In this paper we summarize a study Qf generalization in the simplest possible case-perceptron networks learning linearly separable functions. The task chosen was the majority function (i.e. return a 1 if a majority of the input units are on), a predicate with a number of useful properties. We find that many aspects of.generalization in multilayer networks learning large, difficult tasks are reproduced in this simple domain, in which concrete numerical results and even some analytic understanding can be achieved.

The issues of scaling and generalization have emerged as key issues in current studies of supervised learning from examples in neural networks. Questions such as how many training patterns and training cycles are needed for a problem of a given size and difficulty, how to represent the inllUh and how to choose useful training exemplars, are of considerable theoretical and practical importance. Several intuitive rules of thumb have been obtained from empirical studies, but as yet there are few rigorous results. In this paper we summarize a study Qf generalization in the simplest possible case-perceptron networks learning linearly separable functions. The task chosen was the majority function (i.e. return a 1 if a majority of the input units are on), a predicate with a number of useful properties. We find that many aspects of.generalization in multilayer networks learning large, difficult tasks are reproduced in this simple domain, in which concrete numerical results and even some analytic understanding can be achieved.

The issues of scaling and generalization have emerged as key issues in current studies of supervised learning from examples in neural networks. Questions such as how many training patterns and training cycles are needed for a problem of a given size and difficulty, how to represent the inllUh and how to choose useful training exemplars, are of considerable theoretical and practical importance. Several intuitive rules of thumb have been obtained from empirical studies, but as yet there are few rigorous results.In this paper we summarize a study Qf generalization in the simplest possible case-perceptron networks learning linearly separable functions.The task chosen was the majority function (i.e. return a 1 if a majority of the input units are on), a predicate with a number ofuseful properties. We find that many aspects of.generalization in multilayer networks learning large, difficult tasks are reproduced in this simple domain, in which concrete numerical results and even some analytic understanding can be achieved.

Over the last eight years, four workshops on machine learning have been held. Participation in these workshops was by invitation only. In response to the rapid growth in the number of researchers active in machine learning, it was decided that the fifth meeting should be a conference with open attendance and full review for presented papers. Thus, the first open conference on machine learning took place 12 to 14 June 1988 at The University of Michigan at Ann Arbor.

We address the question of when a network can be expected to generalize from m random training examples chosen from some arbitrary probability distribution, assuming that future test examples are drawn from the same distribution. Among our results are the following bounds on appropriate sample vs. network size. We show that if m O(W/ log N/) random examples can be loaded on a feedforward network of linear threshold functions with N nodes and W weights, so that at least a fraction 1 /2 of the examples are correctly classified, then one has confidence approaching certainty that the network will correctly classify a fraction 1 of future test examples drawn from the same distribution. Conversely, for fully-connected feedforward nets with one hidden layer, any learning algorithm using fewer than Ω(W/) random training examples will, for some distributions of examples consistent with an appropriate weight choice, fail at least some fixed fraction of the time to find a weight choice that will correctly classify more than a 1 fraction of the future test examples.