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The simplex gradient, a popular numerical differentiation method due to itsflexibility, lacks a principled method by which to construct the sample set,specifically the location of function evaluations. Such evaluations, especiallyfrom real-world systems, are often noisy and expensive to obtain, making itessential that each evaluation is carefully chosen to reduce cost and increaseaccuracy. This paper introduces the curvature aligned simplex gradient (CASG),which provably selects the optimal sample set under a mean squared errorobjective. As CASG requires function-dependent information often not availablein practice, we additionally introduce a framework which exploits a history offunction evaluations often present in practical applications. Our numericalresults, focusing on applications in sensitivity analysis and derivative freeoptimization, show that our methodology significantly outperforms or matchesthe performance of the benchmark gradient estimator given by forwarddifferences (FD) which is given exact function-dependent information that isnot available in practice. Furthermore, our methodology is comparable to theperformance of central differences (CD) that requires twice the number offunction evaluations.

Curvature Aligned Simplex Gradient: Principled Sample Set Construction For Numerical Differentiation