The Annals of Statistics

Adaptive optimization and $D$-optimum experimental design

Luc Pronzato

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Abstract

We consider the situation where one has to maximize a function $\eta(\theta, \mathbf{x})$ with respect to $\mathbf{x} \epsilon \mathbb{R}^q$, when $\theta$ is unknown and estimated by least squares through observations $y_k = \mathbf{f}^{\top(\mathbf{x}_k)\theta + \varepsilon_k$, with $\varepsilon_k$ some random error. Classical applications are regulation and extremum control problems. The approach we adopt corresponds to maximizing the sum of the current estimated objective and a penalization for poor estimation: $\mathbf{x}_{k + 1}$ maximizes $\eta(\hat{\theta}^k, \mathbf{x}) + (\alpha_k/k), d_k(\mathbf{x})$, with $\hat{\theta}^k$ the estimated value of $\theta$ at step $k$ and $d_k$ the penalization function. Sufficient conditions for strong consistency of $\hat{\theta}^k$ and for almost sure convergence of $(1/k) \Sigma_{i=1}^k \eta(\theta, \mathbf{x}_i)$ to the maximum value of $\eta(\theta, \mathbf{x})$ are derived in the case where $d_k(\cdot)$ is the variance function used in the sequential construction of $D$-optimum designs. A classical sequential scheme from adaptive control is shown not to satisfy these conditions, and numerical simulations confirm that it indeed has convergence problems.

Article information

Source
Ann. Statist. Volume 28, Number 6 (2000), 1743-1761.

Dates
First available in Project Euclid: 12 March 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1015957479

Digital Object Identifier
doi:10.1214/aos/1015957479

Mathematical Reviews number (MathSciNet)
MR1835050

Zentralblatt MATH identifier
1103.93320

Subjects
Primary: 62F12: Asymptotic properties of estimators 62L05: Sequential design
Secondary: 93C40: Adaptive control

Keywords
Adaptive control least-squares estimation $D$-optimum design sequential design strong consistency

Citation

Pronzato, Luc. Adaptive optimization and $D$-optimum experimental design. Ann. Statist. 28 (2000), no. 6, 1743--1761. doi:10.1214/aos/1015957479. https://projecteuclid.org/euclid.aos/1015957479


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