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Active Learning for Derivative-Based Global Sensitivity Analysis with Gaussian Processes
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Proposes Bayesian active learning acquisition functions (variance reduction, information gain) that directly target derivative-based global sensitivity measures (raw, absolute, squared DGSMs) under Gaussian process surrogates.
Problem
Estimating derivative-based global sensitivity measures (DGSMs) of expensive black-box functions with very limited evaluation budgets; existing designs (quasirandom, or active learning targeting the function itself) are sample-inefficient for DGSMs.
Approach
Derives closed-form derivative look-ahead distributions under a GP and builds acquisition functions (max variance, variance reduction, information gain) on the posterior of the gradient, its absolute value, and its square, with novel folded-normal and noncentral chi-squared entropy approximations, so selection targets the DGSM quantities rather than f.
Key results
- Derivative information gain acquisitions (DIG, DAbIG_1, DSqIG_1) performed best in the majority of experiments, outperforming quasirandom sequences and f-targeted active learning (Var, fIG) across the test problems.
- On high-dimensional problems (Gsobol15, Morris), baselines showed little RMSE reduction across iterations, while derivative max-variance acquisitions were competitive with or better than derivative information gain.
- Global look-ahead variants did not substantially improve over local acquisitions while adding large computational cost.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| target acquisition at DGSM quantities rather than using quasirandom sampling or acquisitions targeting f | Active learning for DGSM estimation · Synthetic and real-world problems | DGSM RMSE | Helped |
| use derivative information gain instead of other acquisition strategies | Active learning for DGSM estimation · Majority of experiments | DGSM RMSE | Helped |
| use derivative max variance instead of derivative information gain | Active learning for DGSM estimation · GSobol15 and Morris | DGSM RMSE | Mixed |
| use global integrated acquisitions instead of local acquisitions | Active learning for DGSM estimation · Synthetic and real-world problems | DGSM RMSE | Mixed |
| add small perturbations to points in a space-filling design | Quasirandom sampling baseline · DGSM estimation | DGSM RMSE | No clear effect |
| use folded-normal entropy approximation 1 rather than approximation 3 (ablation) | Absolute derivative information gain · DGSM estimation | DGSM RMSE | No clear effect |
| use squared-derivative entropy approximation 2 rather than approximation 1 (ablation) | Squared derivative information gain · Some problems | DGSM RMSE | Helped |
| replace derivative information gain with active subspace learning (Trace, Var1, or Var2) | Squared DGSM estimation · Three representative problems | DGSM RMSE | Hurt |
Surprising findings
- Information gain tends to select points near existing observations, which is counter to f-targeted active learning that selects far-away points; the paper argues adjacency is valuable for derivative estimation.
- A space-filling design plus small perturbations of its points did not significantly improve over quasirandom sequences.
Limitations
- One-step myopic look-ahead for information gain is insufficiently exploratory in high dimensions.
- Squared DGSM information gain is computationally expensive due to hypergeometric function evaluation (up to 214 s acquisition optimization).
- Restricted to Gaussian process surrogates and noiseless observations in experiments.
Experimental setup
Datasets: Ishigami1 Ishigami2 Gsobol6 a-function Gsobol10 Gsobol15 Morris Branin Hartmann3 Hartmann4 Car Side Impact Weight Vehicle Safety
Benchmarks: Ishigami1 Ishigami2 Gsobol6 a-function Gsobol10 Gsobol15 Morris Branin Hartmann3 Hartmann4 Car Side Impact Weight Vehicle Safety
Metrics: RMSE of DGSM estimate vs ground truth NDCG of variable importance ranking
Baselines: quasirandom sequences (scrambled Sobol) maximum variance of f information gain about f (fIG) active subspace methods (Trace, Var1, Var2)
Builds on: Bayesian active learning derivative-based global sensitivity measures Gaussian process derivative posterior BoTorch
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