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Optimally Adaptive Integration of Univariate Lipschitz Functions

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Abstract

We consider the problem of approximately integrating a Lipschitz function f (with a known Lipschitz constant) over an interval. The goal is to achieve an additive error of at most ε using as few samples of f as possible. We use the adaptive framework: on all problem instances an adaptive algorithm should perform almost as well as the best possible algorithm tuned for the particular problem instance. We distinguish between \({\rm DOPT}\) and \({\rm ROPT}\) , the performances of the best possible deterministic and randomized algorithms, respectively. We give a deterministic algorithm that uses \(O({\rm DOPT}(f,\epsilon)\cdot\log(\epsilon^{-1}/{\rm DOPT}(f,\epsilon)))\) samples and show that an asymptotically better algorithm is impossible. However, any deterministic algorithm requires \(\Omega({\rm ROPT}(f,\epsilon)^{2})\) samples on some problem instance. By combining a deterministic adaptive algorithm and Monte Carlo sampling with variance reduction, we give an algorithm that uses at most \(O({\rm ROPT}(f,\epsilon)^{4/3}+{\rm ROPT}(f,\epsilon)\cdot\log(1/\epsilon))\) samples. We also show that any algorithm requires \(\Omega({\rm ROPT}(f,\epsilon)^{4/3}+{\rm ROPT}(f,\epsilon)\cdot\log(1/\epsilon))\) samples in expectation on some problem instance (f,ε), which proves that our algorithm is optimal.

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Correspondence to Ilya Baran.

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Baran, I., Demaine, E.D. & Katz, D.A. Optimally Adaptive Integration of Univariate Lipschitz Functions. Algorithmica 50, 255–278 (2008). https://doi.org/10.1007/s00453-007-9093-7

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Keywords

  • Adaptive algorithms
  • Lipschitz functions