Active Learning for Derivative-Based Global Sensitivity Analysis with Gaussian Processes

TL;DR

This work proposes novel active learning acquisition functions that directly target key quantities of derivative-based global sensitivity measures (DGSMs) under Gaussian process surrogate models, and demonstrates how these active learning acquisition strategies substantially enhance the sample efficiency of DGSM estimation, particularly with limited evaluation budgets.

Abstract

We consider the problem of active learning for global sensitivity analysis of expensive black-box functions. Our aim is to efficiently learn the importance of different input variables, e.g., in vehicle safety experimentation, we study the impact of the thickness of various components on safety objectives. Since function evaluations are expensive, we use active learning to prioritize experimental resources where they yield the most value. We propose novel active learning acquisition functions that directly target key quantities of derivative-based global sensitivity measures (DGSMs) under Gaussian process surrogate models. We showcase the first application of active learning directly to DGSMs, and develop tractable uncertainty reduction and information gain acquisition functions for these measures. Through comprehensive evaluation on synthetic and real-world problems, our study demonstrates how these active learning acquisition strategies substantially enhance the sample efficiency of DGSM estimation, particularly with limited evaluation budgets. Our work paves the way for more efficient and accurate sensitivity analysis in various scientific and engineering applications.

Figures (16)

• Figure 1: (Left) Posteriors of $f$, $df/dx$, $|df/dx|$, and $(df/dx)^2$ are computed from a GP surrogate given six observations of $f$ (black dots). Posteriors are shown as posterior mean (line) and 95% credible interval (shaded). (Right) Acquisition functions are computed from these posteriors, targeting $f$ and derivative sensitivity measures. Dotted vertical lines show the maximizer. Acquisition functions that directly target DGSMs, not just $f$ generally, are required to learn the DGSMs efficiently.
• Figure 2: RMSE (mean over 50 runs, two standard errors shaded) for learning the DGSM, for 10 test problems. Results are shown for active learning methods targeting the raw derivative. Active learning targeting the derivative consistently outperformed space-filling designs and active learning targeting $f$. Derivative information gain was generally the best-performing acquisition function.
• Figure 3: RMSE results as in Fig. \ref{['fig:experiments_rmse_dv']}, here for the absolute derivative acquisition functions. These also outperformed the baselines, with absolute derivative information gain generally the best.
• Figure 4: RMSE results as in Fig. \ref{['fig:experiments_rmse_dv']}, here for the squared derivative acquisition functions. As with the other derivative active learning approaches, these outperformed the baselines, and squared derivative information gain generally performed best.
• Figure 5: An empirical evaluation of active subspace methods on the task of learning the square and absolute DGSM, with derivative information gain, derivative square information gain, and quasirandom as points of comparison.