On the optimal boolean function for prediction under quadratic loss

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

Suppose Y n is obtained by observing a uniform Bernoulli random vector Xn through a binary symmetric channel. Courtade and Kumar asked how large the mutual information between Y n and a Boolean function b(Xn) could be, and conjectured that the maximum is attained by the dictator function. An equivalent formulation of this conjecture is that dictator minimizes the prediction cost in sequentially predicting Y n under logarithmic loss, given b(Xn). In this paper, we study the question of minimizing the sequential prediction cost under a different (proper) loss function - the quadratic loss. In the noiseless case, we show that majority asymptotically minimizes this prediction cost among all Boolean functions. We further show that for weak noise, majority is better than dictator, and that for strong noise dictator outperforms majority. We conjecture that for quadratic loss, there is no single Boolean function that is simultaneously optimal at all noise levels.

Original languageEnglish
Title of host publicationProceedings - ISIT 2016; 2016 IEEE International Symposium on Information Theory
Pages495-499
Number of pages5
ISBN (Electronic)9781509018062
DOIs
StatePublished - 10 Aug 2016
Event2016 IEEE International Symposium on Information Theory, ISIT 2016 - Barcelona, Spain
Duration: 10 Jul 201615 Jul 2016

Publication series

NameIEEE International Symposium on Information Theory - Proceedings
Volume2016-August

Conference

Conference2016 IEEE International Symposium on Information Theory, ISIT 2016
Country/TerritorySpain
CityBarcelona
Period10/07/1615/07/16

Keywords

  • Boolean functions
  • Pinsker's inequality
  • logarithmic loss function
  • quadratic loss function
  • sequential prediction

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Information Systems
  • Modelling and Simulation
  • Applied Mathematics

Fingerprint

Dive into the research topics of 'On the optimal boolean function for prediction under quadratic loss'. Together they form a unique fingerprint.

Cite this