Stochastic Approximation of Minima with Improved Asymptotic Speed

Abstract

It is shown that the Keifer-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 Robins-Monro method. The modification consists in making more observations at every step and in utilizing these so as to eliminate the effect of all derivatives $\partial^if/\lbrack\partial x^{(i)}\rbrack^j, j = 3, 5 \cdots, s - 1$. Let $\delta_n$ be the distance from the approximating value to the approximated $\theta$ after $n$ observations have been made. Under similar conditions on $f$ as those used by Dupac (1957), the results is $E\delta_n^2 = O(n^{-s/(s+1)})$. Under weaker conditions it is proved that $\delta_n^2n^{s/(s+1)-\epsilon} \rightarrow 0$ with probability one for every $\epsilon > 0$. Both results are given for the multidimensional case in Theorems 5.1 and 5.3. The modified choice of $Y_n$ in the scheme $X_{n+1} = X_n - a_nY_n$ is described in Lemma 3.1. The proofs are similar to those used by Dupac (1957) and are based on Chung's (1954) lemmas and, in Theorem 5.3, on a modification of one of these lemmas. The result of Theorem 5.3 is new also for the usual Kiefer-Wolfowitz procedure. The main and very simple idea, however, is in Lemma 3.1; it will suggest, to a reader acquainted with Dupac's Theorem 3 and its proof, the consequences elaborated in Theorem 5.1.