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Robust Super-Level Set Estimation Using Gaussian Processes

  • Andrea Zanette
  • Junzi Zhang
  • Mykel J. KochenderferEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11052)

Abstract

This paper focuses on the problem of determining as large a region as possible where a function exceeds a given threshold with high probability. We assume that we only have access to a noise-corrupted version of the function and that function evaluations are costly. To select the next query point, we propose maximizing the expected volume of the domain identified as above the threshold as predicted by a Gaussian process, robustified by a variance term. We also give asymptotic guarantees on the exploration effect of the algorithm, regardless of the prior misspecification. We show by various numerical examples that our approach also outperforms existing techniques in the literature in practice.

Keywords

Active learning Gaussian processes Level set estimation 

Notes

Acknowledgments

Blake Wulfe provided the simulator for the simulations experiments. The authors are grateful to the reviewers for their comments.

Supplementary material

478890_1_En_17_MOESM1_ESM.pdf (829 kb)
Supplementary material 1 (pdf 829 KB)

References

  1. 1.
    Gotovos, A., Casati, N., Hitz, G., Krause, A.: Active learning for level set estimation. In: International Joint Conference on Artificial Intelligence (IJCAI), pp. 1344–1350 (2013)Google Scholar
  2. 2.
    Rasmussen, C.E., Williams, C.K.I.: Gaussian Processes for Machine Learning. MIT Press, Cambridge (2006)zbMATHGoogle Scholar
  3. 3.
    Shahriari, B., Swersky, K., Wang, Z., Adams, R.P., de Freitas, N.: Taking the human out of the loop: a review of Bayesian optimization. Proc. IEEE 104(1), 148–175 (2016)CrossRefGoogle Scholar
  4. 4.
    Kushner, H.J.: A new method of locating the maximum point of an arbitrary multipeak curve in the presence of noise. J. Basic Eng. 86(1), 97–106 (1964)CrossRefGoogle Scholar
  5. 5.
    Mockus, J., Tiesis, V., Zilinskas, A.: The application of Bayesian methods for seeking the extremum. In: Dixon, L.C.W., Szego, G.P. (eds.) Towards Global Optimization, vol. 2. North-Holland Publishing Company, Amsterdam (1978)Google Scholar
  6. 6.
    Srinivas, N., Krause, A., Kakade, S.M., Seeger, M.: Gaussian process optimization in the bandit setting: no regret and experimental design. In: International Conference on Machine Learning (ICML), pp. 1015–1022 (2010)Google Scholar
  7. 7.
    Shahriari, B., Wang, Z., Hoffman, M.W., Bouchard-Cote, A., de Freitas, N.: An entropy search portfolio for Bayesian optimization. In: Advances on Neural Information Processing Systems (NIPS) (2014)Google Scholar
  8. 8.
    Willet, R.M., Nowak, R.D.: Minimax optimal level-set estimation. IEEE Trans. Image Process. 16(12), 2965–2979 (2007)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Soni, A., Haupt, J.: Level set estimation from compressive measurements using box constrained total variation regularization. In: IEEE International Conference on Image Processing (ICIP), pp. 2573–2576 (2012)Google Scholar
  10. 10.
    Krishnamurthy, K., Bajwa, W.U., Willett, R.: Level set estimation from projection measurements: performance guarantees and fast computation. SIAM J. Imaging Sci. 6(4), 2047–2074 (2013)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Krause, A., Singh, A., Guestrin, C.: Near-optimal sensor placements in Gaussian processes: theory, efficient algorithms and empirical studies. J. Mach. Learn. Res. 9, 235–284 (2008)zbMATHGoogle Scholar
  12. 12.
    Martino, L., Vicent, J., Camps-Valls, G.: Automatic emulator and optimized look-up table generation for radiative transfer models. In: IEEE International Geoscience and Remote Sensing Symposium, pp. 1457–1460 (2017)Google Scholar
  13. 13.
    Busby, D.: Hierarchical adaptive experimental design for Gaussian process emulators. Reliab. Eng. Syst. Saf. 94(7), 1183–1193 (2009)CrossRefGoogle Scholar
  14. 14.
    Bryan, B., Nichol, R.C., Genovese, C.R., Schneider, J., Miller, C.J., Wasserman, L.: Active learning for identifying function threshold boundaries. In: Advances in Neural Information Processing Systems (NIPS), pp. 163–170 (2005)Google Scholar
  15. 15.
    Bogunovic, I., Scarlett, J., Krause, A., Cevher, V.: Truncated variance reduction: a unified approach to Bayesian optimization and level-set estimation. In: Advances in Neural Information Processing Systems (NIPS), pp. 1507–1515 (2016)Google Scholar
  16. 16.
    Yang, J., Wang, Z., Wu, Z.: Level set estimation with dynamic sparse sensing. In: IEEE Global Conference on Signal and Information Processing (GlobalSIP), pp. 487–491 (2014)Google Scholar
  17. 17.
    Ma, Y., Garnett, R., Schneider, J.: Active area search via Bayesian quadrature. In: Artificial Intelligence and Statistics (AISTATS), pp. 595–603 (2014)Google Scholar
  18. 18.
    Ma, Y., Sutherland, D., Garnett, R., Schneider, J.: Active pointillistic pattern search. In: Artificial Intelligence and Statistics (AISTATS), pp. 672–680 (2015)Google Scholar
  19. 19.
    Bect, J., Ginsbourger, D., Li, L., Picheny, V., Vazquez, E.: Sequential design of computer experiments for the estimation of a probability of failure. Stat. Comput. 22(3), 773–793 (2012)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Chevalier, C., Bect, J., Ginsbourger, D., Vazquez, E., Picheny, V., Richet, Y.: Fast parallel kriging-based stepwise uncertainty reduction with application to the identification of an excursion set. Technometrics 56(4), 455–465 (2014)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Treiber, M., Hennecke, A., Helbing, D.: Congested traffic states in empirical observations and microscopic simulations. Phys. Rev. E 62(2), 1805 (2000)CrossRefGoogle Scholar
  22. 22.
    Kesting, A., Treiber, M., Helbing, D.: General lane-changing model MOBIL for car-following models. Transp. Res. Rec. 1999(1), 86–94 (2007)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Andrea Zanette
    • 1
  • Junzi Zhang
    • 1
  • Mykel J. Kochenderfer
    • 1
    Email author
  1. 1.Stanford UniversityStanfordUSA

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