Cautious optimization via data informativity

TL;DR

This paper introduces a robust, data-driven approach for determining guaranteed bounds and cautious optimization of unknown, noisy cost functions, applicable in control scenarios like system analysis and regulation.

Contribution

It develops a set-valued regression framework with closed-form bounds and regularity properties, enabling verification of suboptimality and cautious optimization in one-shot and online settings.

Findings

Provides data-based conditions for bounds of unknown functions.

Establishes convexity and Lipschitzness of compatible functions.

Demonstrates effectiveness in control problems with simulations.

Abstract

This paper deals with the problem of accurately determining guaranteed suboptimal values of an unknown cost function on the basis of noisy measurements. We consider a set-valued variant to regression where, instead of finding a best estimate of the cost function, we reason over all functions compatible with the measurements and apply robust methods explicitly in terms of the data. Our treatment provides data-based conditions under which closed-forms expressions of upper bounds of the unknown function can be obtained, and regularity properties like convexity and Lipschitzness can be established. These results allow us to provide tests for point- and set-wise verification of suboptimality, and tackle the cautious optimization of the unknown function in both one-shot and online scenarios. We showcase the versatility of the proposed methods in two control-relevant problems: data-driven contraction analysis of unknown nonlinear systems and suboptimal regulation with unknown dynamics and cost. Simulations illustrate our results.