Set-valued regression and cautious suboptimization: From noisy data to optimality

TL;DR

This paper introduces a method for finding suboptimal values of an unknown, noisy function using basis function parameterization, worst-case suboptimization, and online iterative measurements, ensuring safety and convergence to the true optimizer.

Contribution

It provides closed-form solutions and convexity analysis for worst-case suboptimization of noisy functions, along with an online safe measurement and optimization procedure.

Findings

Closed-form solutions for worst-case suboptimization

Convexity results enabling efficient optimization

An online iterative procedure converging to the true optimizer

Abstract

This paper deals with the problem of finding suboptimal values of an unknown function on the basis of measured data corrupted by bounded noise. As a prior, we assume that the unknown function is parameterized in terms of a number of basis functions. Inspired by the informativity approach, we view the problem as the suboptimization of the worst-case estimate of the function. The paper provides closed form solutions and convexity results for this function, which enables us to solve the problem. After this, an online implementation is investigated, where we iteratively measure the function and perform a suboptimization. This nets a procedure that is safe at each step, and which, under mild assumptions, converges to the true optimizer.