Topic

# Function approximation

About: Function approximation is a research topic. Over the lifetime, 4242 publications have been published within this topic receiving 133659 citations.

##### Papers published on a yearly basis

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TL;DR: A general gradient descent boosting paradigm is developed for additive expansions based on any fitting criterion, and specific algorithms are presented for least-squares, least absolute deviation, and Huber-M loss functions for regression, and multiclass logistic likelihood for classification.

Abstract: Function estimation/approximation is viewed from the perspective of numerical optimization in function space, rather than parameter space. A connection is made between stagewise additive expansions and steepest-descent minimization. A general gradient descent “boosting” paradigm is developed for additive expansions based on any fitting criterion.Specific algorithms are presented for least-squares, least absolute deviation, and Huber-M loss functions for regression, and multiclass logistic likelihood for classification. Special enhancements are derived for the particular case where the individual additive components are regression trees, and tools for interpreting such “TreeBoost” models are presented. Gradient boosting of regression trees produces competitive, highly robust, interpretable procedures for both regression and classification, especially appropriate for mining less than clean data. Connections between this approach and the boosting methods of Freund and Shapire and Friedman, Hastie and Tibshirani are discussed.

12,602 citations

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AT&T Labs

^{1}TL;DR: This paper proves for the first time that a version of policy iteration with arbitrary differentiable function approximation is convergent to a locally optimal policy.

Abstract: Function approximation is essential to reinforcement learning, but the standard approach of approximating a value function and determining a policy from it has so far proven theoretically intractable. In this paper we explore an alternative approach in which the policy is explicitly represented by its own function approximator, independent of the value function, and is updated according to the gradient of expected reward with respect to the policy parameters. Williams's REINFORCE method and actor-critic methods are examples of this approach. Our main new result is to show that the gradient can be written in a form suitable for estimation from experience aided by an approximate action-value or advantage function. Using this result, we prove for the first time that a version of policy iteration with arbitrary differentiable function approximation is convergent to a locally optimal policy.

4,394 citations

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TL;DR: The paper describes a general methodology for the fitting of measured or calculated frequency domain responses with rational function approximations by replacing a set of starting poles with an improved set of poles via a scaling procedure.

Abstract: The paper describes a general methodology for the fitting of measured or calculated frequency domain responses with rational function approximations. This is achieved by replacing a set of starting poles with an improved set of poles via a scaling procedure. A previous paper (Gustavsen et al., 1997) described the application of the method to smooth functions using real starting poles. This paper extends the method to functions with a high number of resonance peaks by allowing complex starting poles. Fundamental properties of the method are discussed and details of its practical implementation are described. The method is demonstrated to be very suitable for fitting network equivalents and transformer responses. The computer code is in the public domain, available from the first author.

2,592 citations

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TL;DR: The approximation rate and the parsimony of the parameterization of the networks are shown to be advantageous in high-dimensional settings and the integrated squared approximation error cannot be made smaller than order 1/n/sup 2/d/ uniformly for functions satisfying the same smoothness assumption.

Abstract: Approximation properties of a class of artificial neural networks are established. It is shown that feedforward networks with one layer of sigmoidal nonlinearities achieve integrated squared error of order O(1/n), where n is the number of nodes. The approximated function is assumed to have a bound on the first moment of the magnitude distribution of the Fourier transform. The nonlinear parameters associated with the sigmoidal nodes, as well as the parameters of linear combination, are adjusted in the approximation. In contrast, it is shown that for series expansions with n terms, in which only the parameters of linear combination are adjusted, the integrated squared approximation error cannot be made smaller than order 1/n/sup 2/d/ uniformly for functions satisfying the same smoothness assumption, where d is the dimension of the input to the function. For the class of functions examined, the approximation rate and the parsimony of the parameterization of the networks are shown to be advantageous in high-dimensional settings. >

2,519 citations

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TL;DR: This presentation reports results of applying the Support Vector method to problems of estimating regressions, constructing multidimensional splines, and solving linear operator equations.

Abstract: The Support Vector (SV) method was recently proposed for estimating regressions, constructing multidimensional splines, and solving linear operator equations [Vapnik, 1995]. In this presentation we report results of applying the SV method to these problems.

2,447 citations