A Newton-Raphson Version of the Multivariate Robbins-Monro Procedure

Abstract

Suppose that ff is a function from Rk\mathbb{R}^k to Rk\mathbb{R}^k and for some θ,f(θ)=0\theta, f(\theta) = 0. Initially ff is unknown, but for any xx in Rk\mathbb{R}^k we can observe a random vector Y(x)Y(x) with expectation f(x)f(x). The unknown θ\theta can be estimated recursively by Blum's (1954) multivariate version of the Robbins-Monro procedure. Blum's procedure requires the rather restrictive assumption that infimum of the inner product (x−θ)tf(x)(x - \theta)^tf(x) over any compact set not containing θ\theta be positive. Thus at each x,f(x)x, f(x) gives information about the direction towards θ\theta. Blum's recursion is Xn+1=Xn−anYnX_{n+1} = X_n - a_n Y_n where the conditional expectation of YnY_n given X1,⋯,XnX_1, \cdots, X_n is f(Xn)f(X_n) and an>0a_n > 0. Unlike Blum's method, the procedure introduced in this paper does not necessarily attempt to move in a direction that decreases ∥Xn−θ∥\|X_n - \theta\|, at least not during the initial stage of the procedure. Rather, except for random fluctuations it moves in a direction which decreases ∥f∥2\|f\|^2, and it may follow a circuitous route to θ\theta. Consequently, it does not require that (x−θ)tf(x)(x - \theta)^tf(x) have a constant signum. This new procedure is somewhat similar to the multivariate Kiefer-Wolfowitz procedure applied to ∥f∥2\|f\|^2, but unlike the latter it converges to θ\theta at rate n−1/2n^{-1/2}. Deterministic root finding methods are briefly discussed. The method of this paper is a stochastic analog of the Newton-Raphson and Gauss-Newton techniques.