The Annals of Mathematical Statistics

On the Choice of Design in Stochastic Approximation Methods

Vaclav Fabian

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Abstract

In a previous paper [Fabian (1967)], we have shown that the Kiefer-Wolfowitz procedure--for functions $f$ sufficiently smooth at $\theta$, the point of minimum--can be modified in such a way as to be almost as speedy as the original Robbins-Monro method. The modification consists of taking more observations at every step and utilizing these (according to a design $d$) so as to eliminate the effect of all derivatives $\partial^jf/\lbrack\partial x^{(i)}\rbrack^j, j = 3, 5, \cdots, s - 1$. Let $\delta_n$ be the distance of the approximating value to the approximated $\theta$ after $n$ observations taken. Under some regularity conditions it was shown that $E\delta^2_n = O(n^{-s/(s + 1)}).$ There are many designs $d$ achieving this speed. For selection of the best one, i.e. the one which minimizes $\lim n^{s/(s + 1)}E\delta^2_n$ we have to derive the dependence of this limit on the design $d$, which is done in Section 4. The best choice of the design $d = \lbrack u, \xi\rbrack$ is that which minimizes the right-hand side of (2.7) below; here $u = \lbrack u_1, u_2, \cdots, u_m\rbrack, \xi = \lbrack\xi_1, \cdots, \xi_m\rbrack$ with $0 < u_1 < u_2 < \cdots < u_m \leqq 1, \xi_i \geqq 0, \sum^m_{i = 1}\xi_i = 1; \xi_i$ indicates how many observations should be taken (roughly speaking) at $u_i$. The vector $v = \lbrack v_1, \cdots, v_m\rbrack$ is determined by $v = \frac{1}{2}U^{-1}e_1 (e_1 = \lbrack 1, 0, \cdots, 0\rbrack, \lbrack\cdots\rbrack$ denotes column vectors), $U^{(ij)} = u^{2i - 1}_j, i, j = 1, \cdots, m$. It seems difficult to minimize (2.7) given $K_0, K_1$. Moreover we usually do not know these constants. So in this paper we solve the question of minimizing the first term $\sum^m_{i = 1}(v^2_i/\xi_i)$ only. The result is formulated in Theorem 5.1.

Article information

Source
Ann. Math. Statist., Volume 39, Number 2 (1968), 457-465.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177698409

Digital Object Identifier
doi:10.1214/aoms/1177698409

Mathematical Reviews number (MathSciNet)
MR225445

Zentralblatt MATH identifier
0162.21601

JSTOR
links.jstor.org

Citation

Fabian, Vaclav. On the Choice of Design in Stochastic Approximation Methods. Ann. Math. Statist. 39 (1968), no. 2, 457--465. doi:10.1214/aoms/1177698409. https://projecteuclid.org/euclid.aoms/1177698409


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