Convergence Analysis of the Random Bisection Method

TL;DR

This paper analyzes a generalized random bisection method where the cut point is chosen randomly, deriving convergence rates based on the distribution of the cut, and extends the analysis to multiple cuts with validation through simulations.

Contribution

It introduces a generalized random bisection method with arbitrary distributions for the cut point and provides convergence analysis for both single and multiple cuts.

Findings

Expected convergence rate depends only on the expectation of c(1-c).

The method generalizes to K random cuts with similar convergence properties.

Numerical simulations validate the theoretical convergence results.

Abstract

We propose a generalized version of the bisection method where the cutting point between the two subintervals is chosen at random following an arbitrary distribution. We compute expected convergence rates with respect to any arbitrary a priori distribution for the position of the root in the initial interval and proved that it depends only on the the expectation $\mathbb{E}[c(1-c)]$ of the cut $c$. We also provide a generalization of the method for $K$ random cuts and study its convergence properties. Most probabilistic derivations are kept fairly simple for the ease of understanding of a larger audience. Our theoretical results are then validated numerically using statistical simulation.