A companion for the Kiefer–Wolfowitz–Blum stochastic approximation algorithm

Abstract

A stochastic algorithm for the recursive approximation of the location θ of a maximum of a regression function was introduced by Kiefer and Wolfowitz [Ann. Math. Statist. 23 (1952) 462-466] in the univariate framework, and by Blum [Ann. Math. Statist. 25 (1954) 737-744] in the multivariate case. The aim of this paper is to provide a companion algorithm to the Kiefer-Wolfowitz-Blum algorithm, which allows one to simultaneously recursively approximate the size µ of the maximum of the regression function. A precise study of the joint weak convergence rate of both algorithms is given; it turns out that, unlike the location of the maximum, the size of the maximum can be approximated by an algorithm which converges at the parametric rate. Moreover, averaging leads to an asymptotically efficient algorithm for the approximation of the couple (θ, µ).

The stochastic approximation method for the estimation of a multivariate probability density

“…Condition (4) was introduced by Galambos and Seneta (1973) to define regularly varying sequences (see also Bojanic and Seneta, 1973), and by Mokkadem and Pelletier (2007a) in the context of stochastic approximation algorithms. Typical sequences in GS( ) are, for b ∈ R, n (log n) b , n (log log n) b , and so on.…”

“…Their well-known algorithm was widely discussed and extended in many directions (see, among many others, Blum, 1954;Fabian, 1967;Kushner and Clark, 1978;Hall and Heyde, 1980;Ruppert, 1982;Chen, 1988;Spall, 1988Spall, , 1997Polyak and Tsybakov, 1990;Duflo, 1996;Dippon and Renz, 1997;Chen et al, 1999;Dippon, 2003, andPelletier, 2007a). Stochastic approximation algorithms were also introduced by Révész (1973Révész ( , 1977 to estimate a regression function, and by Tsybakov (1990) to approximate the mode of a probability density.…”

Bandwidth Selection in Deconvolution Kernel Distribution Estimators Defined by Stochastic Approximation Method with Laplace Errors

“…Condition (5) was introduced by Galambos & Seneta (1973) to define regularly varying sequences (see also Bojanic & Seneta (1973), and by Mokkadem & Pelletier (2007) in the context of stochastic approximation algorithms. Noting that the acronym GS stand for (Galambos & Seneta).…”

Bayesian mode and maximum estimation and accelerated rates of contraction

“…However if one is allowed to choose samples based on information gathered from past samples, the structure of the problem changes and we are in the sequential design setting. In this case, the minimax sequential rates of estimating µ and M are respectively n −(α−1)/(2α) and n −1/2 (see Chen [5], Polyak and Tsybakov [23], Mokkadem and Pelletier [20]). When compared with the fixed design case, it is clear that sequential rates are uniformly better and in fact M has successfully achieved the parametric rate.…”

“…Moreover, it also shows that judicious use of past information to guide future actions removes the effect of dimension d on the rates. On the more practical side, Kiefer and Wolfowitz [16] and Blum [2] used Robbins-Monro type procedures that is consistent; while Fabian [9], Dippon [8] and Mokkadem and Pelletier [20] each constructed sequential procedures that actually attain the minimax rates.…”