skip to main content
research-article

High-confidence predictions under adversarial uncertainty

Published:22 August 2013Publication History
Skip Abstract Section

Abstract

We study the setting in which the bits of an unknown infinite binary sequence x are revealed sequentially to an observer. We show that very limited assumptions about x allow one to make successful predictions about unseen bits of x. First, we study the problem of successfully predicting a single 0 from among the bits of x. In our model, we have only one chance to make a prediction, but may do so at a time of our choosing. This model is applicable to a variety of situations in which we want to perform an action of fixed duration, and need to predict a “safe” time-interval to perform it.

Letting Nt denote the number of 1s among the first t bits of x, we say that x is “ε-weakly sparse” if lim inf (Nt/t) ≤ ε. Our main result is a randomized algorithm that, given any ε-weakly sparse sequence x, predicts a 0 of x with success probability as close as desired to 1 - ε. Thus, we can perform this task with essentially the same success probability as under the much stronger assumption that each bit of x takes the value 1 independently with probability ε.

We apply this result to show how to successfully predict a bit (0 or 1) under a broad class of possible assumptions on the sequence x. The assumptions are stated in terms of the behavior of a finite automaton M reading the bits of x. We also propose and solve a variant of the well-studied “ignorant forecasting” problem. For every ε>0, we give a randomized forecasting algorithm Sε that, given sequential access to a binary sequence x, makes a prediction of the form: “A p fraction of the next N bits will be 1s.” (The algorithm gets to choose p, N, and the time of the prediction.) For any fixed sequence x, the forecast fraction p is accurate to within ±ε with probability 1 - ε.

References

  1. Athreya, K. B., Hitchcock, J. M., Lutz, J. H., and Mayordomo, E. 2007. Effective strong dimension in algorithmic information and computational complexity. SIAM J. Comput. 37, 3, 671--705. DOI:http://dx.doi.org/10.1137/S0097539703446912. Google ScholarGoogle ScholarDigital LibraryDigital Library
  2. Billingsley, P. 1965. Ergodic Theory and Information. John Wiley and Sons.Google ScholarGoogle Scholar
  3. Dawid, A. 1982. The well-calibrated Bayesian. J. Amer. Statist. Assoc. 77, 379, 605--610.Google ScholarGoogle Scholar
  4. Eggleston, H. 1949. The fractional dimension of a set defined by decimal properties. Quart. J. Math. 20, 31--36.Google ScholarGoogle ScholarCross RefCross Ref
  5. Fortnow, L. and Vohra, R. V. 2009. The complexity of forecast testing. Econometrica 77, 93--105.Google ScholarGoogle ScholarCross RefCross Ref
  6. Foster, D. P. and Vohra, R. V. 1998. Asymptotic calibration. Biometrika 85, 2, 379--390. http://www.jstor.org/stable/2337364.Google ScholarGoogle ScholarCross RefCross Ref
  7. Hemaspaandra, L. A. 2005. SIGACT news complexity theory column 48. SIGACT News 36, 3, 24--38. (Guest Column: The fractal geometry of complexity classes, by J. M. Hitchcock, J. H. Lutz, and E. Mayordomo.) Google ScholarGoogle ScholarDigital LibraryDigital Library
  8. Kapralov, M. and Panigrahy, R. 2011. Prediction strategies without loss. In Proceedings of the 25th Annual Conference on Neural Information Processing Systems. 828--836.Google ScholarGoogle Scholar
  9. Lutz, J. H. 2003a. Dimension in complexity classes. SIAM J. Comput. 32, 5, 1236--1259. DOI:http://dx.doi.org/10.1137/S0097539701417723. Google ScholarGoogle ScholarDigital LibraryDigital Library
  10. Lutz, J. H. 2003b. The dimensions of individual strings and sequences. Inf. Comput. 187, 1, 49--79. Google ScholarGoogle ScholarDigital LibraryDigital Library
  11. Merhav, N. and Feder, M. 1998. Universal prediction. IEEE Trans. Inf. Theory 44, 6, 2124--2147. Google ScholarGoogle ScholarDigital LibraryDigital Library
  12. Sandroni, A. 2003. The reproducible properties of correct forecasts. Int. J. Game Theory 32, 1, 151--159.Google ScholarGoogle ScholarDigital LibraryDigital Library

Index Terms

  1. High-confidence predictions under adversarial uncertainty

        Recommendations

        Comments

        Login options

        Check if you have access through your login credentials or your institution to get full access on this article.

        Sign in

        Full Access

        • Published in

          cover image ACM Transactions on Computation Theory
          ACM Transactions on Computation Theory  Volume 5, Issue 3
          Special issue on innovations in theoretical computer science 2012
          August 2013
          94 pages
          ISSN:1942-3454
          EISSN:1942-3462
          DOI:10.1145/2493252
          Issue’s Table of Contents

          Copyright © 2013 ACM

          Publisher

          Association for Computing Machinery

          New York, NY, United States

          Publication History

          • Published: 22 August 2013
          • Revised: 1 March 2013
          • Accepted: 1 March 2013
          • Received: 1 September 2012
          Published in toct Volume 5, Issue 3

          Permissions

          Request permissions about this article.

          Request Permissions

          Check for updates

          Qualifiers

          • research-article
          • Research
          • Refereed

        PDF Format

        View or Download as a PDF file.

        PDF

        eReader

        View online with eReader.

        eReader
        About Cookies On This Site

        We use cookies to ensure that we give you the best experience on our website.

        Learn more

        Got it!