On optimal quantization rules for some sequential decision problems

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On optimal quantization rules for some sequential
decision problems

• by X. Nguyen, M. Wainwright M. Jordan

Discussion led by Qi An ECE, Duke University
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Outline

• Introduction
• Background
• Approximate design
• Suboptimality of stationary designs
• Discussion

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Introduction

• Whats decentralized detection problem?
• A hypothesis problem in which the decision maker
  have no direct access to the raw data, but
  instead must infer based only on a set of local
  quantization functions and a sequence of summary
  statistics.
• Whats the goal?
• The goal is to design both the local quantization
  functions and to specify a global decision rules
  so as to predict the underlying hypothesis H in a
  manner that optimally trades off accuracy and
  delay

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Introduction

• For general framework of sequential decentralized
  problems, Veeravalli defined five problems (case
  A to E) distinguished by amount of information
  available to the local sensors.
• Veeravalli use likelihood ratio test to solve the
  case E, where each local sensor has its current
  data, On, as well as the summary statistics form
  all of the other local sensor.

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In Veeravallis paper, he conjectured that
stationary local decision functions may actually
be optimal for case A, where neither local memory
nor feedback are assumed to be available.
In this paper, the author shows that stationary
decision functions are, in fact, not optimal for
decentralized problem of case A.

O1n
Otn
Usually, O?1,,M and U?1,,K, where MgtgtK
S1
St
U1nfn(O1n)
Utnfn(Otn)
quantization functions
S0
fusion center
H?n(U11,,Ut1,U12,,Ut2,,U1n,,Utn)
decision rule
Let XnO1n,,Otn and ZnU1n,,Utn .
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Sequential detection

• Consider a general centralized sequential
  detection problem. Let P0 and P1 represent the
  class-conditional distributions of Z, when
  conditioned on H0 and H1, and and
  are corresponding density functions.
• Focusing the Bayesian formulation, we let
  and denote the
  prior probabilities of the two hypothesis
• The cost function we want to minimize is a
  weighted sum

stopping time
cost per step
We want to choose the pair so as to
minimized the expected loss
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Dynamic programming

• The previous centralized sequential detection can
  be solved by DP approach by iteratively updating
  the cost function over horizon.
• But it is not straightforward to apply DP
  approach to decentralized versions of sequential
  detection.

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Walds approximation
The optimal stopping rule for the cost function
takes the form
The optimal decision rule has the form
Lets define two types of errors
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Walds approximation
The cost function of the decision
rule based on envelop a and b can be written as
where
and
It is hard to calculate this cost function but it
can be easily approximated.
The errors a,ß are related to a and b by the
classical inequalities
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Walds approximation
If we ignore the overshoot and replace the
inequalities with the equalities, i.e.
we can therefore get the approximate cost
function
where
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Approximate design

• Now consider the decentralized setting. Given a
  fixed stationary quantizer f, Walds
  approximation suggests the following strategy
• For a given set of admissible errors a and ß,
  first assign the values of thresholds
  and
• Then use the quantity G(a,ß) as an approximation
  to the true cost J(a,b).

By assuming
, the author proves
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Suboptimallity of the stationary design

• It was shown by Tsitsiklis that optimal
  quantizers fn take the form of threshold rules
  based on the likelihood ratio
• Veeravalli asked whether these rules can be taken
  to be stationary, a problem that has remained
  open.

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Suboptimallity of the stationary design

• A simple counterexample to show the optimal
  quantizer is not stationary

Consider a problem in which and
the conditional distributions are
Suppose the prior are and
, and the cost for each step is
If we restrict to binary quantizers, then there
are only three possible quantizers
1. Design A .
As a result, the pdf for Zn is
2. Design B .
As a result, the pdf is
3. Design C
. As a result, the pdf is
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Suboptimallity of the stationary design
Where is corresponding to the nonstationary
design by applying design A for only the first
step and applying the design B for the remaining
steps.
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Asymptotic suboptimality of stationary designs
As we can see in the approximate cost function,
it is composed of two KL divergences. If we want
to achieve a small cost we need to choose a
quantizer f which make both divergences, µ1 and
µ0, as large as possible.
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Discussion

• The problem of decentralized sequential detection
  encompasses a wide range of problems involving
  different assumptions about the memory and
  feedback.
• They have provided an asymptotic characterization
  of the cost of the optimal sequential test in the
  setting of case A.
• They provided an explicit counterexample to the
  stationary conjecture and showed that under some
  conditions there is a guaranteed range of prior
  probabilities for which stationary strategies are
  suboptimal.