Adaptive optimization and D-optimum experimental design

• 成果类型: Article
• 署名作者: Pronzato, L
• 署名单位: Centre National de la Recherche Scientifique (CNRS); Universite Cote d'Azur
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/aos/1015957479
• 发表日期: 2000
• 页码: 1743-1761
• 关键词: algorithms

摘要

We consider the situation where one has to maximize a function eta(theta, x) with respect to x is an element of R-q, when a is unknown and estimated by least squares through observations y(k) = f(inverted perpendicular)(x(k))theta + epsilon (k), with epsilon (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: x(k+1) maximizes eta(<()over cap>(k), x) + (alpha (k)/k), d(k)(x), with <()over cap>(k) the estimated value of theta at step k and dk the penalization function. Sufficient conditions for strong consistency of <()over cap>(k) and for almost sure convergence of(1/k) Sigma (k)(i=1) eta(theta, x(i)) to the maximum value of eta(theta, x) are derived in the case where d(k)(.) 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.