Mini-Minimax Uncertainty Quantification for Emulators

TL;DR

This paper introduces a method to quantify the worst-case error in emulating a black box function using minimal assumptions, providing bounds on uncertainty based on limited data, with applications to climate modeling.

Contribution

It develops the concept of mini-minimax uncertainty to assess emulator accuracy under minimal regularity assumptions, especially for high-dimensional models.

Findings

Lower bounds on observation requirements for desired accuracy.

Application to Community Atmosphere Model shows significant uncertainty bounds.

Provides confidence bounds for uncertainty quantiles and averages.

Abstract

Consider approximating a "black box" function $f$ by an emulator $\hat{f}$ based on $n$ noiseless observations of $f$. Let $w$ be a point in the domain of $f$. How big might the error $|\hat{f}(w) - f(w)|$ be? If $f$ could be arbitrarily rough, this error could be arbitrarily large: we need some constraint on $f$ besides the data. Suppose $f$ is Lipschitz with known constant. We find a lower bound on the number of observations required to ensure that for the best emulator $\hat{f}$ based on the $n$ data, $|\hat{f}(w) - f(w)| \le \epsilon$. But in general, we will not know whether $f$ is Lipschitz, much less know its Lipschitz constant. Assume optimistically that $f$ is Lipschitz-continuous with the smallest constant consistent with the $n$ data. We find the maximum (over such regular $f$) of $|\hat{f}(w) - f(w)|$ for the best possible emulator $\hat{f}$; we call this the "mini-minimax uncertainty" at $w$. In reality, $f$ might not be Lipschitz or---if it is---it might not attain its Lipschitz constant on the data. Hence, the mini-minimax uncertainty at $w$ could be much smaller than $|\hat{f}(w) - f(w)|$. But if the mini-minimax uncertainty is large, then---even if $f$ satisfies the optimistic regularity assumption---$|\hat{f}(w) - f(w)|$ could be large, no matter how cleverly we choose $\hat{f}$. For the Community Atmosphere Model, the maximum (over $w$) of the mini-minimax uncertainty based on a set of 1154~observations of $f$ is no smaller than it would be for a single observation of $f$ at the centroid of the 21-dimensional parameter space. We also find lower confidence bounds for quantiles of the mini-minimax uncertainty and its mean over the domain of $f$. For the Community Atmosphere Model, these lower confidence bounds are an appreciable fraction of the maximum.