Gradient Estimation Using Lagrange Interpolation Polynomials

CentER Discussion Paper No. 2003-101

Posted: 16 Jun 2004

See all articles by R.C.M. Brekelmans

R.C.M. Brekelmans

Tilburg University - Tilburg University School of Economics and Management

Lonneke Driessen

Centre for Quantitative Methods

Herbert Hamers

Tilburg University - Center for Economic Research (CentER); Tilburg University - Department of Econometrics & Operations Research

Dick den Hertog

Tilburg University - Department of Econometrics & Operations Research

Date Written: 2003

Abstract

In this paper we use Lagrange interpolation polynomials to obtain good gradient estimations. This is e.g. important for nonlinear programming solvers. As an error criterion we take the mean squared error. This error can be split up into a deterministic and a stochastic error. We analyze these errors using (N times replicated) Lagrange interpolation polynomials. We show that the mean squared error is of order N-1+ 1 2d if we replicate the Lagrange estimation procedure N times and use 2d evaluations in each replicate. As a result the order of the mean squared error converges to N-1 if the number of evaluation points increases to infinity. Moreover, we show that our approach is also useful for deterministic functions in which numerical errors are involved. Finally, we consider the case of a fixed budget of evaluations. For this situation we provide an optimal division between the number of replicates and the number of evaluations in a replicate.

Keywords: Gradient estimation, noisy function, Lagrange interpolation

Suggested Citation

Brekelmans, R.C.M. and Driessen, Lonneke and Hamers, Herbert and den Hertog, Dick, Gradient Estimation Using Lagrange Interpolation Polynomials (2003). CentER Discussion Paper No. 2003-101. Available at SSRN: https://ssrn.com/abstract=556592

R.C.M. Brekelmans (Contact Author)

Tilburg University - Tilburg University School of Economics and Management ( email )

P.O. Box 90153
B 1029
5000 LE Tilburg
Netherlands
+31 13 466 8234 (Phone)

Lonneke Driessen

Centre for Quantitative Methods ( email )

P.O. Box 414
5600 AK Eindhoven
Netherlands

Herbert Hamers

Tilburg University - Center for Economic Research (CentER) ( email )

P.O. Box 90153
Tilburg, 5000 LE
Netherlands
+31 13 4666 2660 (Phone)

Tilburg University - Department of Econometrics & Operations Research

Tilburg, 5000 LE
Netherlands

Dick Den Hertog

Tilburg University - Department of Econometrics & Operations Research ( email )

Tilburg, 5000 LE
Netherlands

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