We develop a principled strategy to sample a function optimally for function approximation tasks within a Bayesian framework. Using ideas from optimal experiment design, we introduce an objective function (incorporating both bias and variance) to measure the degree of approximation, and the potential utility of the data points towards optimizing this objective. We show how the general strategy can be used to derive precise algorithms to select data for two cases: learning unit step functions and polynomial functions. In particular, we investigate whether such active algorithms can learn the target with fewer examples. We obtain theoretical and empirical results to suggest that this is the case. 1 INTRODUCTION AND MOTIVATION Learning from examples is a common supervised learning paradigm that hypothesizes a target concept given a stream of training examples that describes the concept. In function approximation, example-based learning can be formulated as synthesizing an approximation function for data sampled from an unknown target function (Poggio and Girosi, 1990). Active learning describes a class of example-based learning paradigms that seeks out new training examples from specific regions of the input space, instead of passively accepting examples from some data generating source.

We develop a principled strategy to sample a function optimally for function approximation tasks within a Bayesian framework. Using ideas from optimal experiment design, we introduce an objective function (incorporating both bias and variance) to measure the degree ofapproximation, and the potential utility of the data points towards optimizing this objective. We show how the general strategy canbe used to derive precise algorithms to select data for two cases: learning unit step functions and polynomial functions. In particular, we investigate whether such active algorithms can learn the target with fewer examples. We obtain theoretical and empirical resultsto suggest that this is the case. 1 INTRODUCTION AND MOTIVATION Learning from examples is a common supervised learning paradigm that hypothesizes atarget concept given a stream of training examples that describes the concept. In function approximation, example-based learning can be formulated as synthesizing anapproximation function for data sampled from an unknown target function (Poggio and Girosi, 1990). Active learning describes a class of example-based learning paradigms that seeks out new training examples from specific regions of the input space, instead of passively accepting examples from some data generating source. By judiciously selecting ex- 594 KahKay Sung, Parlha Niyogi amples instead of allowing for possible random sampling, active learning techniques can conceivably have faster learning rates and better approximation results than passive learning methods. This paper presents a Bayesian formulation for active learning within the function approximation framework.

Such decisions include which function approximator to use, how to trade smoothness for goodness of fit and which features are relevant.

Such decisions include which function approximator to use, how to trade smoothness for goodness of fit and which features are relevant.

Such decisions include which function approximator to use, how to trade smoothness for goodness of fit and which features arerelevant.

This article summarizes the findings of a 1992 study of knowledge-based systems research and applications in Japan. Representatives of universities and businesses were chosen by the Japan Technology Evaluation Center to investigate the state of the technology in Japan relative to the United States. The panel's report focused on applications, tools, and research and development in universities and industry and on major national projects.

A three-step method for function approximation with a fuzzy system is proposed. First, the membership functions and an initial rule representation are learned; second, the rules are compressed as much as possible using information theory; and finally, a computational network is constructed to compute the function value. This system is applied to two control examples: learning the truck and trailer backer-upper control system, and learning a cruise control system for a radio-controlled model car. 1 Introduction Function approximation is the problem of estimating a function from a set of examples of its independent variables and function value. If there is prior knowledge of the type of function being learned, a mathematical model of the function can be constructed and the parameters perturbed until the best match is achieved. However, if there is no prior knowledge of the function, a model-free system such as a neural network or a fuzzy system may be employed to approximate an arbitrary nonlinear function. A neural network's inherent parallel computation is efficient for speed; however, the information learned is expressed only in the weights of the network. The advantage of fuzzy systems over neural networks is that the information learned is expressed in terms of linguistic rules. In this paper, we propose a method for learning a complete fuzzy system to approximate example data.

A three-step method for function approximation with a fuzzy system is proposed. First, the membership functions and an initial rule representation are learned; second, the rules are compressed as much as possible using information theory; and finally, a computational network is constructed to compute the function value. This system is applied to two control examples: learning the truck and trailer backer-upper control system, and learning a cruise control system for a radio-controlled model car. 1 Introduction Function approximation is the problem of estimating a function from a set of examples of its independent variables and function value. If there is prior knowledge of the type of function being learned, a mathematical model of the function can be constructed and the parameters perturbed until the best match is achieved. However, if there is no prior knowledge of the function, a model-free system such as a neural network or a fuzzy system may be employed to approximate an arbitrary nonlinear function. A neural network's inherent parallel computation is efficient for speed; however, the information learned is expressed only in the weights of the network. The advantage of fuzzy systems over neural networks is that the information learned is expressed in terms of linguistic rules. In this paper, we propose a method for learning a complete fuzzy system to approximate example data.

First, the membership functions and an initial rule representation are learned; second, the rules are compressed as much as possible using information theory; and finally, a computational networkis constructed to compute the function value. This system is applied to two control examples: learning the truck and trailer backer-upper control system, and learning a cruise control systemfor a radio-controlled model car. 1 Introduction Function approximation is the problem of estimating a function from a set of examples ofits independent variables and function value. If there is prior knowledge of the type of function being learned, a mathematical model of the function can be constructed and the parameters perturbed until the best match is achieved. However, ifthere is no prior knowledge of the function, a model-free system such as a neural network or a fuzzy system may be employed to approximate an arbitrary nonlinear function. A neural network's inherent parallel computation is efficient for speed; however, the information learned is expressed only in the weights of the network. The advantage of fuzzy systems over neural networks is that the information learnedis expressed in terms of linguistic rules. In this paper, we propose a method for learning a complete fuzzy system to approximate example data.

The First International Workshop on Rough Sets: State of the Art and Perspectives was held on 2-4 September 1992 in Kiekrz, Poland. To stimulate the discussion, the participation was limited to 40 researchers who are involved in fundamental research in rough set theory and its extensions, logic for approximate reasoning, machine learning, knowledge representation and transfer, and applications of rough set methodology. The workshop focused primarily on applications of the basic idea of the approximate definition of a set and its consequences in other areas of science and engineering. Applications discussed at the workshop included machine learning, medical diagnosis, fault detection, medical image processing, neural net training, database organization, drug research, and digital circuit design.