Optimal uncertainty quantification for legacy data observations of Lipschitz functions

TL;DR

This paper develops an optimal uncertainty quantification framework for partially observed Lipschitz functions, capable of handling small or large datasets without assuming probability distributions, and identifies key data points influencing bounds.

Contribution

It introduces a novel UQ approach that computes optimal bounds from legacy data, even with non-monotonic dependencies, and proposes an efficient simplex-like algorithm for high-dimensional systems.

Findings

Optimal bounds often depend on only a few data points.

The proposed method efficiently handles high-dimensional, high-cardinality data.

Bounds can be non-monotonic and discontinuous with respect to data.

Abstract

We consider the problem of providing optimal uncertainty quantification (UQ) --- and hence rigorous certification --- for partially-observed functions. We present a UQ framework within which the observations may be small or large in number, and need not carry information about the probability distribution of the system in operation. The UQ objectives are posed as optimization problems, the solutions of which are optimal bounds on the quantities of interest; we consider two typical settings, namely parameter sensitivities (McDiarmid diameters) and output deviation (or failure) probabilities. The solutions of these optimization problems depend non-trivially (even non-monotonically and discontinuously) upon the specified legacy data. Furthermore, the extreme values are often determined by only a few members of the data set; in our principal physically-motivated example, the bounds are determined by just 2 out of 32 data points, and the remainder carry no information and could be neglected without changing the final answer. We propose an analogue of the simplex algorithm from linear programming that uses these observations to offer efficient and rigorous UQ for high-dimensional systems with high-cardinality legacy data. These findings suggest natural methods for selecting optimal (maximally informative) next experiments.