Optimal estimators of cross-partial derivatives and surrogates of functions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916325449793536 |
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| author | Lamboni, Matieyendou |
| author_facet | Lamboni, Matieyendou |
| contents | Computing cross-partial derivatives using fewer model runs is relevant in modeling, such as stochastic approximation, derivative-based ANOVA, exploring complex models, and active subspaces. This paper introduces surrogates of all the cross-partial derivatives of functions by evaluating such functions at $N$ randomized points and using a set of $L$ constraints. Randomized points rely on independent, central, and symmetric variables. The associated estimators, based on $NL$ model runs, reach the optimal rates of convergence (i.e., $\mathcal{O}(N^{-1})$), and the biases of our approximations do not suffer from the curse of dimensionality for a wide class of functions. Such results are used for i) computing the main and upper-bounds of sensitivity indices, and ii) deriving emulators of simulators or surrogates of functions thanks to the derivative-based ANOVA. Simulations are presented to show the accuracy of our emulators and estimators of sensitivity indices. The plug-in estimates of indices using the U-statistics of one sample are numerically much stable. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_11035 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Optimal estimators of cross-partial derivatives and surrogates of functions Lamboni, Matieyendou Methodology Optimization and Control Probability Statistics Theory Computation Machine Learning Computing cross-partial derivatives using fewer model runs is relevant in modeling, such as stochastic approximation, derivative-based ANOVA, exploring complex models, and active subspaces. This paper introduces surrogates of all the cross-partial derivatives of functions by evaluating such functions at $N$ randomized points and using a set of $L$ constraints. Randomized points rely on independent, central, and symmetric variables. The associated estimators, based on $NL$ model runs, reach the optimal rates of convergence (i.e., $\mathcal{O}(N^{-1})$), and the biases of our approximations do not suffer from the curse of dimensionality for a wide class of functions. Such results are used for i) computing the main and upper-bounds of sensitivity indices, and ii) deriving emulators of simulators or surrogates of functions thanks to the derivative-based ANOVA. Simulations are presented to show the accuracy of our emulators and estimators of sensitivity indices. The plug-in estimates of indices using the U-statistics of one sample are numerically much stable. |
| title | Optimal estimators of cross-partial derivatives and surrogates of functions |
| topic | Methodology Optimization and Control Probability Statistics Theory Computation Machine Learning |
| url | https://arxiv.org/abs/2407.11035 |