Stochastic Approximation of Minima with Improved Asymptotic Speed

Abstract

It is shown that the Keifer-Wolfowitz procedure--for functions f sufficiently smooth at \&{\#}x3b8;, 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 \&{\#}x2202;if/{$[$}\&{\#}x2202; x(i){$]$}j, j = 3, 5 \&{\#}x22ef;, s - 1. Let \&{\#}x3b4;n be the distance from the approximating value to the approximated \&{\#}x3b8; after n observations have been made. Under similar conditions on f as those used by Dupa\&{\#}x10d; (1957), the results is E\&{\#}x3b4;n 2 = O(n-s/(s+1)). Under weaker conditions it is proved that \&{\#}x3b4;n 2n s/(s+1)-\&{\#}x3b5; \&{\#}x2192; 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 Yn in the scheme Xn+1 = Xn - anY n is described in Lemma 3.1. The proofs are similar to those used by Dupa\&{\#}x10d; (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 Dupa\&{\#}x10d;'s Theorem 3 and its proof, the consequences elaborated in Theorem 5.1.