Resolution Limits of Non-Adaptive Querying for Noisy 20 Questions - - PowerPoint PPT Presentation

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Resolution Limits of Non-Adaptive Querying for Noisy 20 Questions Estimation Lin Zhou

EECS Department, University of Michigan Joint work with Prof. Alfred Hero Paper 1043, ISIT 2020

Lin Zhou (University of Michigan) ISIT 2020 Paper 1043: Noisy 20 Questions 1 / 15

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Origin: 20 Questions Game

Lin Zhou (University of Michigan) ISIT 2020 Paper 1043: Noisy 20 Questions 2 / 15

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Origin: 20 Questions Game

Lin Zhou (University of Michigan) ISIT 2020 Paper 1043: Noisy 20 Questions 2 / 15

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Origin: 20 Questions Game

Lin Zhou (University of Michigan) ISIT 2020 Paper 1043: Noisy 20 Questions 2 / 15

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Origin: 20 Questions Game

Lin Zhou (University of Michigan) ISIT 2020 Paper 1043: Noisy 20 Questions 2 / 15

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Origin: 20 Questions Game

Lin Zhou (University of Michigan) ISIT 2020 Paper 1043: Noisy 20 Questions 2 / 15

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System Model: Searching with Noise

S ∈ [0, 1]

✲ Responder ❄

Queries

✲

Answers Noisy Channel

✲ Questioner ✲ ˆ

S

Target: estimate a continuous random variable S with pdf fS

Lin Zhou (University of Michigan) ISIT 2020 Paper 1043: Noisy 20 Questions 3 / 15

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System Model: Searching with Noise

S ∈ [0, 1]

✲ Responder ❄

Queries

✲

Answers Noisy Channel

✲ Questioner ✲ ˆ

S

Target: estimate a continuous random variable S with pdf fS Task: design queries and decoder (scheme/strategy/policy)

Lin Zhou (University of Michigan) ISIT 2020 Paper 1043: Noisy 20 Questions 3 / 15

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System Model: Searching with Noise

S ∈ [0, 1]

✲ Responder ❄

Queries

✲

Answers Noisy Channel

✲ Questioner ✲ ˆ

S

Target: estimate a continuous random variable S with pdf fS Task: design queries and decoder (scheme/strategy/policy) Motivation: applications such as localization with sensor networks, human brain interface

Lin Zhou (University of Michigan) ISIT 2020 Paper 1043: Noisy 20 Questions 3 / 15

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Adaptive and Non-Adaptive Query Schemes

A query asks whether S lies in a certain set Q ⊂ [0, 1]

Lin Zhou (University of Michigan) ISIT 2020 Paper 1043: Noisy 20 Questions 4 / 15

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Adaptive and Non-Adaptive Query Schemes

A query asks whether S lies in a certain set Q ⊂ [0, 1] Query schemes can be classified as adaptive and non-adaptive

Lin Zhou (University of Michigan) ISIT 2020 Paper 1043: Noisy 20 Questions 4 / 15

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Adaptive and Non-Adaptive Query Schemes

A query asks whether S lies in a certain set Q ⊂ [0, 1] Query schemes can be classified as adaptive and non-adaptive Adaptive

Qi

❄

S

✲

Oracle

❄

Xi Noisy Channel

❄

Yi

✛

Y1, . . . , Yi−1 Q1, . . . , Qi−1 Decoder

❄

Stop ˆ S

Lin Zhou (University of Michigan) ISIT 2020 Paper 1043: Noisy 20 Questions 4 / 15

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Adaptive and Non-Adaptive Query Schemes

A query asks whether S lies in a certain set Q ⊂ [0, 1] Query schemes can be classified as adaptive and non-adaptive Adaptive

Qi

❄

S

✲

Oracle

❄

Xi Noisy Channel

❄

Yi

✛

Y1, . . . , Yi−1 Q1, . . . , Qi−1 Decoder

❄

Stop ˆ S

Non-adaptive

Q1 Q2

r r r

Qn

❄ ❄ ❄

S

✲

Oracle

❄ ❄ ❄

X1 X2

r r r

Xn Noisy Channel

❄ ❄ ❄

Y1 Y2

r r r

Yn Decoder

❄

ˆ S

Lin Zhou (University of Michigan) ISIT 2020 Paper 1043: Noisy 20 Questions 4 / 15

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Measurement Independent and Dependent12 Noise

Measurement independent noise

S

✲ Oracle ❄

Qi

✲

Xi PY|X

✲

Yi

• 1Y. Kaspi, O. Shayevitz, and T. Javidi, “Searching with measurement dependent noise,” IEEE Trans. Inf.

Theory, vol. 64, no. 4, pp. 2690-2705, 2018.

• 2A. Lalitha, N. Ronquillo, and T. Javidi, “Improved target acquisition rates with feedback codes,” IEEE

Journal of Selected Topics in Signal Processing, vol. 12, no. 5, pp. 871-885, 2018.

Lin Zhou (University of Michigan) ISIT 2020 Paper 1043: Noisy 20 Questions 5 / 15

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Measurement Independent and Dependent12 Noise

Measurement independent noise

S

✲ Oracle ❄

Qi

✲

Xi PY|X

✲

Yi

Measurement dependent noise

S

✲ Oracle ❄

Qi

✲

Xi P|Qi|

Y|X

✲

Yi

❄

• 1Y. Kaspi, O. Shayevitz, and T. Javidi, “Searching with measurement dependent noise,” IEEE Trans. Inf.

Theory, vol. 64, no. 4, pp. 2690-2705, 2018.

• 2A. Lalitha, N. Ronquillo, and T. Javidi, “Improved target acquisition rates with feedback codes,” IEEE

Journal of Selected Topics in Signal Processing, vol. 12, no. 5, pp. 871-885, 2018.

Lin Zhou (University of Michigan) ISIT 2020 Paper 1043: Noisy 20 Questions 5 / 15

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Problem Formulation

S ∈ [0, 1]

✲ Oracle ❄

Queries

✲

Answers Noisy Channel

✲ Decoder ✲ ˆ

S

❄

Searching with measurement dependent noise

Lin Zhou (University of Michigan) ISIT 2020 Paper 1043: Noisy 20 Questions 6 / 15

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Problem Formulation

S ∈ [0, 1]

✲ Oracle ❄

Queries

✲

Answers Noisy Channel

✲ Decoder ✲ ˆ

S

❄

Searching with measurement dependent noise Excess-resolution probability with respect to a tolerable resolution level δ ∈ R+ sup

fS

Pr{|ˆ S − S| > δ}

Lin Zhou (University of Michigan) ISIT 2020 Paper 1043: Noisy 20 Questions 6 / 15

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Problem Formulation

S ∈ [0, 1]

✲ Oracle ❄

Queries

✲

Answers Noisy Channel

✲ Decoder ✲ ˆ

S

❄

Searching with measurement dependent noise Excess-resolution probability with respect to a tolerable resolution level δ ∈ R+ sup

fS

Pr{|ˆ S − S| > δ}

Lin Zhou (University of Michigan) ISIT 2020 Paper 1043: Noisy 20 Questions 6 / 15

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Problem Formulation

S ∈ [0, 1]

✲ Oracle ❄

Queries

✲

Answers Noisy Channel

✲ Decoder ✲ ˆ

S

❄

Searching with measurement dependent noise Excess-resolution probability with respect to a tolerable resolution level δ ∈ R+ sup

fS

Pr{|ˆ S − S| > δ}

Analogous to excess-distortion probability in rate-distortion

Lin Zhou (University of Michigan) ISIT 2020 Paper 1043: Noisy 20 Questions 6 / 15

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Problem Formulation

S ∈ [0, 1]

✲ Oracle ❄

Queries

✲

Answers Noisy Channel

✲ Decoder ✲ ˆ

S

❄

Searching with measurement dependent noise Excess-resolution probability with respect to a tolerable resolution level δ ∈ R+ sup

fS

Pr{|ˆ S − S| > δ}

Analogous to excess-distortion probability in rate-distortion Probably approximately correct learning

Lin Zhou (University of Michigan) ISIT 2020 Paper 1043: Noisy 20 Questions 6 / 15

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Problem Formulation

S ∈ [0, 1]

✲ Oracle ❄

Queries

✲

Answers Noisy Channel

✲ Decoder ✲ ˆ

S

❄

Searching with measurement dependent noise Excess-resolution probability with respect to a tolerable resolution level δ ∈ R+ sup

fS

Pr{|ˆ S − S| > δ}

Analogous to excess-distortion probability in rate-distortion Probably approximately correct learning Similar to the criterion considered in Kaspi et al., TIT 2018

Lin Zhou (University of Michigan) ISIT 2020 Paper 1043: Noisy 20 Questions 6 / 15

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Formal Definition of Non-Adaptive Query Scheme

Given any n ∈ N, δ ∈ R+ and ε ∈ [0, 1), an (n, δ, ε)-non-adaptive query scheme for the noisy 20 question problem consists of n queries {Qi}i∈[n] ⊆ [0, 1]n and a decoder g : Yn → S, such that sup

fS∈F(S)

Pr{|g(Y n) − S| > δ} ≤ ε.

Lin Zhou (University of Michigan) ISIT 2020 Paper 1043: Noisy 20 Questions 7 / 15

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Formal Definition of Non-Adaptive Query Scheme

Given any n ∈ N, δ ∈ R+ and ε ∈ [0, 1), an (n, δ, ε)-non-adaptive query scheme for the noisy 20 question problem consists of n queries {Qi}i∈[n] ⊆ [0, 1]n and a decoder g : Yn → S, such that sup

fS∈F(S)

Pr{|g(Y n) − S| > δ} ≤ ε. An (n, δ, ε)-non-adaptive query scheme is universal with respect to the distribution of the target variable S.

Lin Zhou (University of Michigan) ISIT 2020 Paper 1043: Noisy 20 Questions 7 / 15

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Fundamental Limit of Non-Adaptive Query Schemes

Given any n ∈ N and ε ∈ [0, 1), δ∗(n, ε) := inf{δ : ∃ an (n, δ, ε)−non−adaptive−scheme}

Lin Zhou (University of Michigan) ISIT 2020 Paper 1043: Noisy 20 Questions 8 / 15

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Fundamental Limit of Non-Adaptive Query Schemes

Given any n ∈ N and ε ∈ [0, 1), δ∗(n, ε) := inf{δ : ∃ an (n, δ, ε)−non−adaptive−scheme}

achievable resolution of an optimal non-adaptive query procedure with n queries and excess-resolution probability ε

Lin Zhou (University of Michigan) ISIT 2020 Paper 1043: Noisy 20 Questions 8 / 15

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Fundamental Limit of Non-Adaptive Query Schemes

Given any n ∈ N and ε ∈ [0, 1), δ∗(n, ε) := inf{δ : ∃ an (n, δ, ε)−non−adaptive−scheme}

achievable resolution of an optimal non-adaptive query procedure with n queries and excess-resolution probability ε ⌊− log2 δ∗(n, ε)⌋ denotes the number of most significant bits extracted from the binary expansion of the target variable S.

Lin Zhou (University of Michigan) ISIT 2020 Paper 1043: Noisy 20 Questions 8 / 15

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Fundamental Limit of Non-Adaptive Query Schemes

Given any n ∈ N and ε ∈ [0, 1), δ∗(n, ε) := inf{δ : ∃ an (n, δ, ε)−non−adaptive−scheme}

achievable resolution of an optimal non-adaptive query procedure with n queries and excess-resolution probability ε ⌊− log2 δ∗(n, ε)⌋ denotes the number of most significant bits extracted from the binary expansion of the target variable S.

Dual quantity: n∗(δ, ε) := inf{n : ∃ an (n, δ, ε)−non−adaptive−scheme}

Lin Zhou (University of Michigan) ISIT 2020 Paper 1043: Noisy 20 Questions 8 / 15

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Fundamental Limit of Non-Adaptive Query Schemes

Given any n ∈ N and ε ∈ [0, 1), δ∗(n, ε) := inf{δ : ∃ an (n, δ, ε)−non−adaptive−scheme}

achievable resolution of an optimal non-adaptive query procedure with n queries and excess-resolution probability ε ⌊− log2 δ∗(n, ε)⌋ denotes the number of most significant bits extracted from the binary expansion of the target variable S.

Dual quantity: n∗(δ, ε) := inf{n : ∃ an (n, δ, ε)−non−adaptive−scheme} = inf{n : δ∗(n, ε) ≤ δ}.

Lin Zhou (University of Michigan) ISIT 2020 Paper 1043: Noisy 20 Questions 8 / 15

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Preliminaries

Let Pq

Y|X be a measurement dependent channel with parameter q ∈ [0, 1] where

X ∈ X = {0, 1} and Y ∈ Y.

Lin Zhou (University of Michigan) ISIT 2020 Paper 1043: Noisy 20 Questions 9 / 15

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Preliminaries

Let Pq

Y|X be a measurement dependent channel with parameter q ∈ [0, 1] where

X ∈ X = {0, 1} and Y ∈ Y. Let Pp,q

Y

be the marginal distribution on Y induced by PX = Bern(p) and Pq

Y|X.

Lin Zhou (University of Michigan) ISIT 2020 Paper 1043: Noisy 20 Questions 9 / 15

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Preliminaries

Let Pq

Y|X be a measurement dependent channel with parameter q ∈ [0, 1] where

X ∈ X = {0, 1} and Y ∈ Y. Let Pp,q

Y

be the marginal distribution on Y induced by PX = Bern(p) and Pq

Y|X.

For any (x, y) ∈ X × Y, define the mutual information density ıp,q(x; y) := log Pq

Y|X(y|x)

Pp,q

Y (y) .

For any (xn, yn) ∈ X n × Yn, let ıp(xn; yn) :=

n

∑

i=1

ıp,p(xi; yi).

Lin Zhou (University of Michigan) ISIT 2020 Paper 1043: Noisy 20 Questions 9 / 15

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Approximations to δ∗(n, ε): Second-Order Asymptotics

“Capacity” of measurement dependent channels {Pq

Y|X}q∈[0,1]

C := max

q∈[0,1] E[ıq,q(X; Y)] , (X, Y) ∼ Bern(q) × Pq Y|X

Lin Zhou (University of Michigan) ISIT 2020 Paper 1043: Noisy 20 Questions 10 / 15

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Approximations to δ∗(n, ε): Second-Order Asymptotics

“Capacity” of measurement dependent channels {Pq

Y|X}q∈[0,1]

C := max

q∈[0,1] E[ıq,q(X; Y)] , (X, Y) ∼ Bern(q) × Pq Y|X

“Dispersion” of measurement dependent channels Vε := { minq∈Pca Var[ıq,q(X; Y)] if ε ≤ 0.5, maxq∈Pca Var[ıq,q(X; Y)] if ε > 0.5.

Lin Zhou (University of Michigan) ISIT 2020 Paper 1043: Noisy 20 Questions 10 / 15

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Approximations to δ∗(n, ε): Second-Order Asymptotics

“Capacity” of measurement dependent channels {Pq

Y|X}q∈[0,1]

C := max

q∈[0,1] E[ıq,q(X; Y)] , (X, Y) ∼ Bern(q) × Pq Y|X

“Dispersion” of measurement dependent channels Vε := { minq∈Pca Var[ıq,q(X; Y)] if ε ≤ 0.5, maxq∈Pca Var[ıq,q(X; Y)] if ε > 0.5. Under mild conditions, for any ε ∈ (0, 1), we have − log δ∗(n, ε) = nC + √ nVεΦ−1(ε) + O(log n).

Lin Zhou (University of Michigan) ISIT 2020 Paper 1043: Noisy 20 Questions 10 / 15

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Optimal Query Procedure

Input: The number of queries n and two parameters (M, p) Output: An estimate ˆ s of a target variable s Partition the unit interval [0, 1] into M disjoint subintervals (S1, . . . , SM) with equal size. Generate M binary vectors (xn(1), . . . , xn(M)) where each binary vector is generated i.i.d. from a Bernoulli distribution with parameter p. i ← 1. for i ≤ n do Form the i-th query as Ai := ∪

t∈[M]:xi(t)=1

St. Obtain a noisy responses yi from the oracle to the query Ai. i ← i + 1. end for Generate an estimate ˆ s as ˆ s = 2 ˆ w − 1 2M , where ˆ w is obtained via the maximum mutual information density estimator, i.e, ˆ w = max

w∈[M] ıp(xn(w); yn).

Lin Zhou (University of Michigan) ISIT 2020 Paper 1043: Noisy 20 Questions 11 / 15

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Remarks for Second-Order Asymptotics

− log δ∗(n, ε) = nC + √ nVεΦ−1(ε) + O(log n). Proof sketch

Achievability: change of measure technique and modification of random coding union bound Converse: i) relate the excess-resolution probability to error probability in channel coding and ii) apply non-asymptotic converse result in channel coding.

Lin Zhou (University of Michigan) ISIT 2020 Paper 1043: Noisy 20 Questions 12 / 15

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Remarks for Second-Order Asymptotics

− log δ∗(n, ε) = nC + √ nVεΦ−1(ε) + O(log n). Proof sketch

Achievability: change of measure technique and modification of random coding union bound Converse: i) relate the excess-resolution probability to error probability in channel coding and ii) apply non-asymptotic converse result in channel coding.

Refined the results by Kaspi et al., TIT 2018:

Non-asymptotic, non-vanishing versus asymptotic, vanishing (i.e., n → ∞ and ε → 0) Any measurement dependent channel versus a special case of measurement dependent binary symmetric channel Strengthened with strong converse: for any ε ∈ (0, 1), lim

n→∞ −1

n log δ∗(n, ε) = C.

Lin Zhou (University of Michigan) ISIT 2020 Paper 1043: Noisy 20 Questions 12 / 15

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Phase Transition

Resolution decay rate − log δ∗(n,ε)

n

0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5

Excess-resolution probability ε

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

n = 200 n = 400 n = 1000 n = 10000 capacity C

Lin Zhou (University of Michigan) ISIT 2020 Paper 1043: Noisy 20 Questions 13 / 15

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Numerical Example

Consider a uniformly distributed target variable S over the unit interval [0, 1], Binary symmetric channel with cross over probability 0.4|Q| Excess-resolution probability 0.1

n

10 20 30 40 50 60 70 80 90

− log2 δ∗(n, ε)

5 10 15 20 25 30

Simulated Theoretical Lin Zhou (University of Michigan) ISIT 2020 Paper 1043: Noisy 20 Questions 14 / 15

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Reflections and Future Research Directions

Resolution limits of 20 questions estimation with measurement-dependent noise

Lin Zhou (University of Michigan) ISIT 2020 Paper 1043: Noisy 20 Questions 15 / 15

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Reflections and Future Research Directions

Resolution limits of 20 questions estimation with measurement-dependent noise

Searching for a one dimensional target over the unit interval

Lin Zhou (University of Michigan) ISIT 2020 Paper 1043: Noisy 20 Questions 15 / 15

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Reflections and Future Research Directions

Resolution limits of 20 questions estimation with measurement-dependent noise

Searching for a one dimensional target over the unit interval Non-adaptive query schemes

Lin Zhou (University of Michigan) ISIT 2020 Paper 1043: Noisy 20 Questions 15 / 15

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Reflections and Future Research Directions

Resolution limits of 20 questions estimation with measurement-dependent noise

Searching for a one dimensional target over the unit interval Non-adaptive query schemes Second-order asymptotics, approximation to the finite blocklength performance

Lin Zhou (University of Michigan) ISIT 2020 Paper 1043: Noisy 20 Questions 15 / 15

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Reflections and Future Research Directions

Resolution limits of 20 questions estimation with measurement-dependent noise

Searching for a one dimensional target over the unit interval Non-adaptive query schemes Second-order asymptotics, approximation to the finite blocklength performance Phase transition holds with a critical resolution decay rate

Lin Zhou (University of Michigan) ISIT 2020 Paper 1043: Noisy 20 Questions 15 / 15

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Reflections and Future Research Directions

Resolution limits of 20 questions estimation with measurement-dependent noise

Searching for a one dimensional target over the unit interval Non-adaptive query schemes Second-order asymptotics, approximation to the finite blocklength performance Phase transition holds with a critical resolution decay rate

Extended version available at arXiv 1909.12954

Lin Zhou (University of Michigan) ISIT 2020 Paper 1043: Noisy 20 Questions 15 / 15

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Reflections and Future Research Directions

Resolution limits of 20 questions estimation with measurement-dependent noise

Searching for a one dimensional target over the unit interval Non-adaptive query schemes Second-order asymptotics, approximation to the finite blocklength performance Phase transition holds with a critical resolution decay rate

Extended version available at arXiv 1909.12954

Searching for a Multidimensional target over the unit cube

Lin Zhou (University of Michigan) ISIT 2020 Paper 1043: Noisy 20 Questions 15 / 15

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Reflections and Future Research Directions

Resolution limits of 20 questions estimation with measurement-dependent noise

Searching for a one dimensional target over the unit interval Non-adaptive query schemes Second-order asymptotics, approximation to the finite blocklength performance Phase transition holds with a critical resolution decay rate

Extended version available at arXiv 1909.12954

Searching for a Multidimensional target over the unit cube Simultaneous searching for multiple targets over the unit cube

Lin Zhou (University of Michigan) ISIT 2020 Paper 1043: Noisy 20 Questions 15 / 15

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Reflections and Future Research Directions

Resolution limits of 20 questions estimation with measurement-dependent noise

Searching for a one dimensional target over the unit interval Non-adaptive query schemes Second-order asymptotics, approximation to the finite blocklength performance Phase transition holds with a critical resolution decay rate

Extended version available at arXiv 1909.12954

Searching for a Multidimensional target over the unit cube Simultaneous searching for multiple targets over the unit cube Lower bound on the benefit of adaptivity

Lin Zhou (University of Michigan) ISIT 2020 Paper 1043: Noisy 20 Questions 15 / 15

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Reflections and Future Research Directions

Resolution limits of 20 questions estimation with measurement-dependent noise

Searching for a one dimensional target over the unit interval Non-adaptive query schemes Second-order asymptotics, approximation to the finite blocklength performance Phase transition holds with a critical resolution decay rate

Extended version available at arXiv 1909.12954

Searching for a Multidimensional target over the unit cube Simultaneous searching for multiple targets over the unit cube Lower bound on the benefit of adaptivity

Future research directions

Lin Zhou (University of Michigan) ISIT 2020 Paper 1043: Noisy 20 Questions 15 / 15

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Reflections and Future Research Directions

Resolution limits of 20 questions estimation with measurement-dependent noise

Searching for a one dimensional target over the unit interval Non-adaptive query schemes Second-order asymptotics, approximation to the finite blocklength performance Phase transition holds with a critical resolution decay rate

Extended version available at arXiv 1909.12954

Searching for a Multidimensional target over the unit cube Simultaneous searching for multiple targets over the unit cube Lower bound on the benefit of adaptivity

Future research directions

Practical query schemes with low complexity

Lin Zhou (University of Michigan) ISIT 2020 Paper 1043: Noisy 20 Questions 15 / 15

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Reflections and Future Research Directions

Resolution limits of 20 questions estimation with measurement-dependent noise

Searching for a one dimensional target over the unit interval Non-adaptive query schemes Second-order asymptotics, approximation to the finite blocklength performance Phase transition holds with a critical resolution decay rate

Extended version available at arXiv 1909.12954

Searching for a Multidimensional target over the unit cube Simultaneous searching for multiple targets over the unit cube Lower bound on the benefit of adaptivity

Future research directions

Practical query schemes with low complexity Universal query schemes ignorant of the noise or the channel

Lin Zhou (University of Michigan) ISIT 2020 Paper 1043: Noisy 20 Questions 15 / 15

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Reflections and Future Research Directions

Resolution limits of 20 questions estimation with measurement-dependent noise

Searching for a one dimensional target over the unit interval Non-adaptive query schemes Second-order asymptotics, approximation to the finite blocklength performance Phase transition holds with a critical resolution decay rate

Extended version available at arXiv 1909.12954

Searching for a Multidimensional target over the unit cube Simultaneous searching for multiple targets over the unit cube Lower bound on the benefit of adaptivity

Future research directions

Practical query schemes with low complexity Universal query schemes ignorant of the noise or the channel Searching for a moving target with unknown speed (Kaspi et al., ITW 2014)

Lin Zhou (University of Michigan) ISIT 2020 Paper 1043: Noisy 20 Questions 15 / 15