Golden Section Search With Noisy Measurements

Question

I'm using a modified golden section search (brents) to find the maximum / minimum of a function. The function is a real time measurement from a laser that is measuring the height of a single peak on a piece of machined metal. The metal is machined into a roughly sinusoidal shape.

The task is to move the laser to the peak of the machined metal. In an ideal situation the laser measurement would be noise free. However in reality it has randomness to about 0.001 in height measurement.

Does anybody have any advice on modifying the algorithm to handle noisy measurements? I've noticed in my simulations that the algorithm converges to an incorrect point.

Answer 1

There is a version in the book Regret Analysis of Stochastic and Nonstochastic Multi-armed Bandit Problems by Bubeck. There are some convergence guarantees but frankly it involves a crazy amount of sampling.

https://www.nowpublishers.com/article/Details/MAL-024

For a more complete solution see Unimodal Bandits by Yu and Mannor (ICML 2011). They describe the method of sampling more thoroughly, so you can have some insight on how to choose the number of samples.

The algorithm they describe is called the Line Search Elimination Algorithm, which is the same as the Stochastic Golden Search here.