Mini-Minimax Uncertainty Quantification for Emulators

作者：Regier Jeffrey C^*; Stark Philip B

来源：SIAM-ASA Journal on Uncertainty Quantification, 2015, 3(1): 686-708.

DOI：10.1137/130917909

• emulator
• surrogate function
• metamodel
• minimax
• Lipschitz
• information-based complexity

Consider approximating, a "black box" function f by an emulator (f) over cap based on 77 noiseless observations of f. Let w be a point in the domain of f How big might the error vertical bar(f) over cap (w)-f(w)vertical bar 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 a known constant. We find a lower bound on the number of observations required to ensure that for the best emulator (f ) over cap based on the n data, vertical bar(f) over cap (w) - f (w)vertical bar <= is an element of. 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 vertical bar(f) over cap (w)-f(w)vertical bar for the best possible emulator f; we call this the "mini-mininiax uncertainty" nt w. To 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 vertical bar(f) over cap (w)-f(w)vertical bar. But if the mini-miniinax uncertainty is large, then even it f satisfies the optimistic regularity assumption vertical bar(f) over cap (w)-f(w)vertical bar could he large, no matter how cleverly we choose. 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. To know that the emulator estimates f accurately would require evidence that f is typically more regular than it is across the n sample values.

• 出版日期2015