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Privacy-Resolution Tradeoff for Adaptive Noisy Twenty Questions Estimation

Chunsong Sun, Lin Zhou

TL;DR

This work addresses privacy concerns in adaptive noisy twenty questions estimation by introducing a two-stage private query procedure that combines a non-adaptive first stage with a parallel adaptive second stage. It provides non-asymptotic achievability bounds and second-order asymptotics for the minimal achievable resolution δ^*(N,L,ε) under a query-dependent channel, revealing how the privacy level L modulates performance. The analysis shows the proposed method bridges adaptive and non-adaptive querying as a function of L, and improves upon existing noiseless benchmarks in the appropriate limit. The results have practical implications for secure interactive sensing tasks and active beam alignment, with future directions including converse bounds and broader applications such as multi-armed identification problems.

Abstract

We revisit noisy twenty questions estimation and study the privacy-resolution tradeoff for adaptive query procedures. Specifically, in twenty questions estimation, there are two players: an oracle and a questioner. The questioner aims to estimate target variables by posing queries to the oracle that knows the variables and using noisy responses to form reliable estimates. Typically, there are adaptive and non-adaptive query procedures. In adaptive querying, one designs the current query using previous queries and their noisy responses while in non-adaptive querying, all queries are posed simultaneously. Generally speaking, adaptive query procedures yield better performance. However, adaptive querying leads to privacy concerns, which were first studied by Tsitsiklis, Xu and Xu (COLT 2018) and by Xu, Xu and Yang (AISTATS 2021) for the noiseless case, where the oracle always provides correct answers to queries. In this paper, we generalize the above results to the more practical noisy case, by proposing a two-stage private query procedure, analyzing its non-asymptotic and second-order asymptotic achievable performance and discussing the impact of privacy concerns. Furthermore, when specialized to the noiseless case, our private query procedure achieves better performance than above-mentioned query procedures (COLT 2018, AISTATS 2021).

Privacy-Resolution Tradeoff for Adaptive Noisy Twenty Questions Estimation

TL;DR

This work addresses privacy concerns in adaptive noisy twenty questions estimation by introducing a two-stage private query procedure that combines a non-adaptive first stage with a parallel adaptive second stage. It provides non-asymptotic achievability bounds and second-order asymptotics for the minimal achievable resolution δ^*(N,L,ε) under a query-dependent channel, revealing how the privacy level L modulates performance. The analysis shows the proposed method bridges adaptive and non-adaptive querying as a function of L, and improves upon existing noiseless benchmarks in the appropriate limit. The results have practical implications for secure interactive sensing tasks and active beam alignment, with future directions including converse bounds and broader applications such as multi-armed identification problems.

Abstract

We revisit noisy twenty questions estimation and study the privacy-resolution tradeoff for adaptive query procedures. Specifically, in twenty questions estimation, there are two players: an oracle and a questioner. The questioner aims to estimate target variables by posing queries to the oracle that knows the variables and using noisy responses to form reliable estimates. Typically, there are adaptive and non-adaptive query procedures. In adaptive querying, one designs the current query using previous queries and their noisy responses while in non-adaptive querying, all queries are posed simultaneously. Generally speaking, adaptive query procedures yield better performance. However, adaptive querying leads to privacy concerns, which were first studied by Tsitsiklis, Xu and Xu (COLT 2018) and by Xu, Xu and Yang (AISTATS 2021) for the noiseless case, where the oracle always provides correct answers to queries. In this paper, we generalize the above results to the more practical noisy case, by proposing a two-stage private query procedure, analyzing its non-asymptotic and second-order asymptotic achievable performance and discussing the impact of privacy concerns. Furthermore, when specialized to the noiseless case, our private query procedure achieves better performance than above-mentioned query procedures (COLT 2018, AISTATS 2021).
Paper Structure (15 sections, 2 theorems, 52 equations, 5 figures)

This paper contains 15 sections, 2 theorems, 52 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Adaptive Noisy Twenty Questions Estimation with Eavesdropper under Query-Dependent Channel
  4. Definition of Private Query Procedure
  5. Main Results
  6. Preliminaries
  7. Non-adaptive query procedure
  8. Adaptive query procedure
  9. Our Two-Stage Private Query Procedure
  10. Results, Discussions and Numerical Illustrations
  11. Conclusion
  12. Proof of the Non-Asymptotic Bound (Theorem \ref{['nonasymptotic']})
  13. Error Probability
  14. Average Stopping Time
  15. Proof of the Asymptotic Bound (Theorem \ref{['2order-asymptotic']})

Key Result

Theorem 1

Given any $M \in \mathbb{N}$ and any $L \in [2,M-1]$, for any $(p,\varepsilon_0,\varepsilon^{\prime},\lambda_1,\lambda_2) \in [0,1]^3 \times \mathbb{R}_+^2$, there exists an $(N,L,1/M,\varepsilon)$-private query procedure such that where

Figures (5)

  • Figure 1: System model for adaptive noisy twenty questions estimation with an eavesdropper.
  • Figure 2: Illustration of our two-stage private query procedure.
  • Figure 3: Plot of achievability bounds on the resolution decay rate $-\frac{\log \delta^*(N,L,\varepsilon)}{N}$ as a function of the number of queries $N$ for various privacy levels $L$ with the excess-resolution probability $\varepsilon=0.1$, under a query-dependent binary symmetric channel with $h(p)=0.1+0.3p$.
  • Figure 4: Plot of the achievable resolutions $-\log \delta^*(N,L,\varepsilon)$ of the proposed two-stage private query procedure and the benchmark procedure xu2021optimal, as a function of the privacy level $L$ for the fixed number of queries $N=100$. For the noisy case, the excess-resolution probability is set to $\varepsilon=0.1$, and the channel is a query-dependent binary symmetric channel with $h(p)=0.1+0.3p$.
  • Figure 5: Plot of the achievable resolutions $-\log \delta^*(N,L,\varepsilon)$ of the proposed two-stage private query procedure and the benchmark procedure xu2021optimal, as a function of the privacy level $L$ for the fixed number of queries $N=100$, when specialized to the noiseless case.

Theorems & Definitions (3)

  • Definition 1
  • Theorem 1
  • Theorem 2