Convergence Rate of Stochastic Approximation Algorithms in the Degenerate Case

Abstract

Let $f(\cdot)$ be an unknown function whose root $x^0$ is sought by stochastic approximation (SA). Convergence rate and asymptotic normality are usually established for the nondegenerate case $f'(x^0)\neq 0$. This paper demonstrates the convergence rate of SA algorithms for the degenerate case $f'(x^0)=0$. In comparison with the previous work, in this paper no growth rate restriction is imposed on $f(\cdot)$, no statistical property is required for the measurement noise, the general step size is considered, and the result is obtained for the multidimensional case, which is not a straightforward extension of the one-dimensional result. Although the observation noise may be either deterministic or random, the analysis is purely deterministic and elementary.