The Annals of Statistics

Adaptive optimization and $D$-optimum experimental design

Luc Pronzato

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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

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

First available in Project Euclid: 12 March 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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

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