Convergence of the Kiefer-Wolfowitz algorithm in the presence of discontinuities

TL;DR

This paper analyzes the convergence rate of the Kiefer-Wolfowitz stochastic approximation algorithm when dealing with discontinuous stochastic representations, achieving an $O(n^{-1/5})$ error rate under certain conditions.

Contribution

It extends previous convergence results by allowing the stochastic representation to be discontinuous and dependent, providing an error estimate for this more general setting.

Findings

Achieves an $O(n^{-1/5})$ error rate for the $n$th iteration.

Allows for discontinuous and dependent stochastic representations.

Provides theoretical convergence guarantees under mixing conditions.

Abstract

In this paper we estimate the expected error of a stochastic approximation algorithm where the maximum of a function is found using finite differences of a stochastic representation of that function. An error estimate of $O(n^{-1/5})$ for the $n$th iteration is achieved using suitable parameters. The novelty with respect to previous studies is that we allow the stochastic representation to be discontinuous and to consist of possibly dependent random variables (satisfying a mixing condition).