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- W2061033294 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 $partial^jf/lbrackpartial x^{(i)}rbrack^j, j = 3, 5, cdots, s - 1$. 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 $Edelta^2_n = O(n^{-s/(s + 1)}).$ There are many designs $d$ achieving this speed. For selection of the best one, i.e. the one which minimizes $lim n^{s/(s + 1)}Edelta^2_n$ 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 $d = lbrack u, xirbrack$ is that which minimizes the right-hand side of (2.7) below; here $u = lbrack u_1, u_2, cdots, u_mrbrack, xi = lbrackxi_1, cdots, xi_mrbrack$ 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 $v = lbrack v_1, cdots, v_mrbrack$ is determined by $v = frac{1}{2}U^{-1}e_1 (e_1 = lbrack 1, 0, cdots, 0rbrack, lbrackcdotsrbrack$ 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." @default.
- W2061033294 created "2016-06-24" @default.
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- W2061033294 date "1968-04-01" @default.
- W2061033294 modified "2023-10-16" @default.
- W2061033294 title "On the Choice of Design in Stochastic Approximation Methods" @default.
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- W2061033294 doi "https://doi.org/10.1214/aoms/1177698409" @default.
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