Guessing with a bit of help

Nir Weinberger*, Ofer Shayevitz

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

What is the value of just a few bits to a guesser? We study this problem in a setup where Alice wishes to guess an independent and identically distributed (i.i.d.) random vector and can procure a fixed number of k information bits from Bob, who has observed this vector through a memoryless channel. We are interested in the guessing ratio, which we define as the ratio of Alice's guessing-moments with and without observing Bob's bits. For the case of a uniform binary vector observed through a binary symmetric channel, we provide two upper bounds on the guessing ratio by analyzing the performance of the dictator (for general k ≥ 1) and majority functions (for k = 1). We further provide a lower bound via maximum entropy (for general k ≥ 1) and a lower bound based on Fourier-analytic/hypercontractivity arguments (for k = 1). We then extend our maximum entropy argument to give a lower bound on the guessing ratio for a general channel with a binary uniform input that is expressed using the strong data-processing inequality constant of the reverse channel. We compute this bound for the binary erasure channel and conjecture that greedy dictator functions achieve the optimal guessing ratio.

Original languageEnglish
Pages (from-to)39
Number of pages1
JournalEntropy
Volume22
Issue number1
DOIs
StatePublished - 1 Jan 2020

Funding

FundersFunder number
MIT-Technion
Horizon 2020 Framework Programme639573
Engineering Research Centers
European Research Council

    Keywords

    • Boolean functions
    • Fourier analysis
    • Guessing moments
    • Guessing with a helper
    • Hypercontractivity
    • Maximum entropy
    • Strong data-processing inequalities

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