Abstract

Bisection search is the most efficient algorithm for locating a unique point $X^{\ast} \in [0,1]$ when we are able to query an oracle only about whether $X^{\ast}$ lies to the left or right of a point $x$ of our choosing. We study a noisy version of this classic problem, where the oracle's response is correct only with probability $p$. The probabilistic bisection algorithm (PBA) introduced by Horstein [IEEE Trans. Inform. Theory, 9 (1963), pp. 136--143] can be used to locate $X^{\ast}$ in this setting. While the method works extremely well in practice, very little is known about its theoretical properties. In this paper, we provide several key findings about the PBA, which lead to the main conclusion that the expected absolute residuals of successive search results, i.e., $\mathbb{E}[|X^{\ast} - X_n|]$, converge to 0 at a geometric rate.

Keywords

  1. probabilistic bisection
  2. noisy bisection
  3. geometric rate of convergence
  4. sequential analysis
  5. Bayesian performance analysis

MSC codes

  1. 93E20
  2. 68W40
  3. 62C10
  4. 60G35

Get full access to this article

View all available purchase options and get full access to this article.

References

1.
M. Ben-Or and A. Hassidim, Quantum Search in an Ordered List via Adaptive Learning, preprint, arXiv:quant-ph/0703231v2, 2007.
2.
M. Ben-Or and A. Hassidim, The Bayesian learner is optimal for noisy binary search, in Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, IEEE, Piscataway, NJ, 2008, pp. 221--230.
3.
M. Burnashev and K. Zigangirov, An interval estimation problem for controlled observations, Problemy Peredachi Informatsii, 10 (1974), pp. 51--61.
4.
R. Castro and R. Nowak, Active learning and sampling, in Foundations and Applications of Sensor Management, A. O. Hero III, D. A. Castan͂ón, D. Cochran, and K. Kastella, eds., Springer, New York, 2008, pp. 177--200.
5.
R. M. Castro and R. D. Nowak, Minimax bounds for active learning, IEEE Trans. Inform. Theory, 54 (2008), pp. 2339--2353.
6.
T. M. Cover and J. A. Thomas, Elements of Information Theory, John Wiley & Sons, New York, 1991.
7.
R. Durrett, Probability: Theory and Examples, 3rd ed., Duxbury Advanced Series, Duxbury Press, Belmont, CA, 2005.
8.
U. Feige, P. Raghavan, D. Peleg, and E. Upfal, Computing with noisy information, SIAM J. Comput., 23 (1994), pp. 1001--1018.
9.
P. I. Frazier, W. B. Powell, and S. Dayanik, A knowledge-gradient policy for sequential information collection, SIAM J. Control Optim., 47 (2008), pp. 2410--2439.
10.
W. Hoeffding, Probability inequalities for sums of bounded random variables, J. Amer. Statist. Assoc., (1963), pp. 13--30.
11.
M. Horstein, Sequential transmission using noiseless feedback, IEEE Trans. Inform. Theory, 9 (1963), pp. 136--143.
12.
B. Jedynak, P. I. Frazier, and R. Sznitman, Twenty questions with noise: Bayes optimal policies for entropy loss, J. Appl. Probab., 49 (2012), pp. 114--136.
13.
R. Karp and R. Kleinberg, Noisy binary search and its applications, in Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, ACM, New York, 2007, pp. 881--890.
14.
A. I. Levin, On an algorithm for the minimization of convex functions, Dokl. Akad. Nauk, 160 (1965), pp. 1244--1247 (in Russian). English translation in Dokl. Math., 6 (1965), pp. 286--290.
15.
A. S. Nemirovski and D. B. Yudin, Problem Complexity and Method Efficiency in Optimization, John Wiley & Sons, New York, 1983.
16.
D. J. Newman, Location of the maximum on unimodal surfaces, J. ACM, 12 (1965), pp. 395--398.
17.
R. Nowak, Generalized binary search, in Proceedings of the 46th Annual Allerton Conference on Communication, Control, and Computing, IEEE, Piscataway, NJ, 2008, pp. 568--574.
18.
R. Nowak, Noisy generalized binary search, in Advances in Neural Information Processing Systems 22, Y. Bengio, D. Schuurmans, J. Lafferty, C. K. I. Williams, and A. Culotta, eds., Neural Information Processing Systems Foundation, La Jolla, CA, 2009, pp. 1366--1374.
19.
A. Pelc, Searching with known error probability, Theoret. Comput. Sci., 63 (1989), pp. 185--202.
20.
R. Rivest, A. Meyer, D. Kleitman, K. Winklmann, and J. Spencer, Coping with errors in binary search procedures, J. Comput. System Sci., 20 (1980), pp. 396--404.
21.
R. Waeber, P. I. Frazier, and S. G. Henderson, A Bayesian approach to stochastic root finding, in Proceedings of the 2011 Winter Simulation Conference, S. Jain, R. R. Creasey, J. Himmelspach, K. P. White, and M. Fu, eds., IEEE, Piscataway, NJ, 2011, pp. 4038--4050.
22.
S. Zangenehpour, Method and Apparatus for Positioning Head of Disk Drive Using Zone-Bit-Recording, U.S. Patent 5257143, 1993.

Information & Authors

Information

Published In

cover image SIAM Journal on Control and Optimization
SIAM Journal on Control and Optimization
Pages: 2261 - 2279
ISSN (online): 1095-7138

History

Submitted: 10 January 2012
Accepted: 21 February 2013
Published online: 28 May 2013

Keywords

  1. probabilistic bisection
  2. noisy bisection
  3. geometric rate of convergence
  4. sequential analysis
  5. Bayesian performance analysis

MSC codes

  1. 93E20
  2. 68W40
  3. 62C10
  4. 60G35

Authors

Affiliations

Metrics & Citations

Metrics

Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By

View Options

View options

PDF

View PDF

Media

Figures

Other

Tables

Share

Share

Copy the content Link

Share with email

Email a colleague

Share on social media

The SIAM Publications Library now uses SIAM Single Sign-On for individuals. If you do not have existing SIAM credentials, create your SIAM account https://my.siam.org.