Sequential Learning of Active Subspaces
This work addresses the challenge of gradient estimation in sensitivity analysis for researchers using black-box simulators, though it is incremental as it builds on existing active subspace and Gaussian process methods.
The paper tackled the problem of performing sensitivity analysis with active subspace methods when gradients are unavailable due to noisy or expensive simulators, by deriving a closed-form estimator using Gaussian processes and developing sequential learning acquisition functions, resulting in model-based confidence intervals for sensitivity analysis.
In recent years, active subspace methods (ASMs) have become a popular means of performing subspace sensitivity analysis on black-box functions. Naively applied, however, ASMs require gradient evaluations of the target function. In the event of noisy, expensive, or stochastic simulators, evaluating gradients via finite differencing may be infeasible. In such cases, often a surrogate model is employed, on which finite differencing is performed. When the surrogate model is a Gaussian process, we show that the ASM estimator is available in closed form, rendering the finite-difference approximation unnecessary. We use our closed-form solution to develop acquisition functions focused on sequential learning tailored to sensitivity analysis on top of ASMs. We also show that the traditional ASM estimator may be viewed as a method of moments estimator for a certain class of Gaussian processes. We demonstrate how uncertainty on Gaussian process hyperparameters may be propagated to uncertainty on the sensitivity analysis, allowing model-based confidence intervals on the active subspace. Our methodological developments are illustrated on several examples.