Partial Updates: Losing Information for Freshness Melih Ba stop cu - - PowerPoint PPT Presentation

Partial Updates: Losing Information for Freshness

Melih Ba¸ stop¸ cu and S ¸ennur Uluku¸ s

Department of Electrical and Computer Engineering, University of Maryland, College Park, MD

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Motivation

◮ In this work, we study the problem of generating partial updates [1] which

◮ have smaller information compared to the original updates ◮ also, have smaller update transmission (service) times

◮ Our aim is to generate the partial updates and find their corresponding

real-valued codeword lengths

◮ in order to minimize the average age ◮ while keeping the information content of partial updates at a desired level

◮ Codeword lengths represent the service times

◮ We design transmission times through source coding schemes ◮ This problem is different than the traditional source coding problem

[1] D. Ramirez, E. Erkip, and H. V. Poor. Age of information with finite horizon and partial updates. In IEEE ICASSP, pages 4965-4969, 2020.

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Information Update Model

source receiver Xi 101 : : : processing encoder transmitter ^ Xi

◮ The source generates updates as soon as requested by the transmitter ◮ The transmitter further processes it to generate a partial update ◮ The partial updates are encoded by using a binary alphabet ◮ The channel between the transmitter and the receiver is noiseless ◮ Our goal is to optimize the partial update generation process, and the

following codebook design

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Partial Updates

◮ The source generates i.i.d. status updates

◮ from a set X = {x1, x2, . . . , xn} with a pmf PX (xi) = {p1, p2, . . . , pn}

◮ The transmitter processes the update by using a function g(X) to

generate a partial update where

◮ g : X → ˆ

X and the cardinality of ˆ X is k, and 1 ≤ k ≤ n

◮ When k < n, some of the original updates from the set X is mapped to

• ne partial update from the set ˆ

X

◮ The pmf of the partial updates is equal to

P ˆ

X(ˆ

xi) = {ˆ pi|ˆ pi =

• i∈Si

pi, Si = {j|g(xj) = ˆ xi, j = 1, . . . , n}, i = 1, . . . , k}

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Partial Updates: Example

updates

• riginal

updates partial a b c d a b fc; dg pmf 0:5 0:25 0:125 0:125 pmf 0:5 0:25 0:25 g(a) g(b) g(c) g(d)

◮ When update a or b is realized at the source, the receiver fully knows the

realized update once the corresponding partial update is received

◮ When update c or d is realized at the source, the partial update {c, d} is

transmitted

◮ the receiver has the partial information about the update at the source ◮ it knows that c or d is realized but does not know which one specifically 5 / 16

Encoding Partial Updates

◮ The transmitter assigns codewords c(ˆ

xi) with lengths ℓ(ˆ xi) to each partial update by using a binary alphabet

◮ The first and second moments of the codeword lengths are

E[L] =

k

• i=1

P ˆ

X(ˆ

xi)ℓ(ˆ xi) E[L2] =

k

• i=1

P ˆ

X(ˆ

xi)ℓ(ˆ xi)2

◮ If update ˆ

xi is transmitted, it takes ℓ(ˆ xi) units of time to deliver this partial update to the receiver

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Average Age Analysis

a(t) t T s1 s2 s3 s4 r

◮ We define ∆T as the average AoI in the time interval [0, T], which is

∆T = 1 T T a(t)dt

◮ The long term average AoI ∆ is equal to

∆ = lim

T→∞ ∆T = E[S2]

2E[S] + E[S], where E[S] = E[L] and E[S2] = E[L2]

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Problem Formulation

◮ In order to quantify the information retained by the partial updates, we

use the mutual information between the original and partial updates I(X; ˆ X) = H( ˆ X) − H( ˆ X|X)

◮ We impose constraint on the mutual information between the original and

the partial updates,

◮ i.e., I(X; ˆ

X) = β where β is the desired level of mutual information

◮ We write the optimization problem as

min

{ˆ pi ,ℓ(ˆ xi )}

∆ s.t. I(X; ˆ X) = β − → Fidelity constraint

k

• i=1

2−ℓ(ˆ

xi ) ≤ 1 −

→ Kraft’s inequality ℓ(ˆ xi) ∈ Z+ − → Feasibility constraints

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Relaxed Problem

◮ We allow codeword lengths to be real-valued

min

{ˆ pi ,ℓ(ˆ xi )}

E[L2] 2E[L] + E[L] − → The long term AoI s.t. H( ˆ X) = β − → Fidelity constraint

k

• i=1

2−ℓ(ˆ

xi ) ≤ 1 −

→ Kraft’s inequality ℓ(ˆ xi) ∈ R+ − → Feasibility constraints

◮ As H( ˆ

X|X) = 0, the constraint on the mutual information I(X; ˆ X) = β, is equivalent to H( ˆ X) = β

◮ This problem is NP-hard and the optimal solution can be found by

searching over all possible partitions

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Further Relaxed Problem

◮ We relax the pmf constraint

◮ Originally the pmfs are limited only to the pmfs that can be generated from

the partitions of n original updates to k partial updates

◮ Here, we allow all possible pmfs for the partial updates

◮ Thus, we write the further relaxed problem as

min

{ˆ pi ,ℓ(ˆ xi )}

E[L2] 2E[L] + E[L] − → The long term AoI s.t. H( ˆ X) = β − → Fidelity constraint

k

• i=1

2−ℓ(ˆ

xi ) ≤ 1 −

→ Kraft’s inequality

k

• i=1

ˆ pi = 1 − → Feasibility constraint ˆ pi ≥ 0, ℓ(ˆ xi) ∈ R+ − → Feasibility constraints

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Fractional Programming Method

◮ We define p(λ) as

p(λ) : min

{ˆ pi ,ℓ(ˆ xi )}

E[L2] 2 + E[L]2 − λE[L] s.t. H( ˆ X) = β

k

• i=1

2−ℓ(ˆ

xi ) ≤ 1 k

• i=1

ˆ pi = 1 ˆ pi ≥ 0, ℓ(ˆ xi) ∈ R+

◮ This approach was introduced in [2] and has been used in [3], [4] ◮ p(λ) decreases with λ. The optimal solution is obtained when p(λ) = 0 ◮ The optimal age is equal to λ, i.e., ∆∗ = λ [2] W. Dinkelbach. On nonlinear fractional programming. Management Science, 13(7):435607, March 1967. [3] Y. Sun, Y. Polyanskiy, and E. Uysal-Biyikoglu. Remote estimation of the Wiener process over a channel with random delay. In IEEE ISIT, June 2017. [4] A. Arafa, J. Yang, and S. Ulukus. Age-minimal online policies for energy harvesting sensors with random battery recharges. In IEEE ICC, May 2018.

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The Overall Solution

◮ We apply an alternating minimization method where

◮ for a given set of pmf, we find the age-optimal real valued codeword lengths ◮ for a given set of update lengths, we update the pmf

◮ We repeat this procedure until the first order optimality conditions are met ◮ Since the overall optimization problem is not convex, the solution may not

be globally optimal

◮ This method is especially useful when n is large ◮ When n is small, the optimal solution can be found by searching over all

possible partitions

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Numerical Results

◮ We use Zipf(s, n) as the pmf for the original updates,

PX(xi) = i−s n

j=1 j−s ,

i = 1, 2, . . . , n

◮ For the first example, we use Zipf(0.5, 8) ◮ We vary the entropy constraint β and find the corresponding optimum age

with real-valued codeword lengths for k ∈ {3, 4, 5, 6}

0.5 1 1.5 2 2.5 1.5 2 2.5 3 3.5 4

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Numerical Results

◮ For the second example, we again use Zipf(0.5, 8) as the pmf for X ◮ We find the age-optimal pmf and the corresponding age-optimal

real-valued codeword lengths when k = 3

◮ We vary the entropy constraint β ∈ {0.82, 1.43, 1.58}

1 2 3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1 2 3 0.5 1 1.5 2 2.5

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Numerical Results

◮ We use the proposed alternating minimization algorithm to find the pmf

and the corresponding age-optimal codeword lengths for k = 10

◮ We use the same initial pmf for different entropy constraints ◮ We vary the entropy constraint β ∈ {1.6, 2.4, 3.2}

5 10 15 20 2 2.5 3 3.5 4 4.5 5 5 10 15 20 1.5 2 2.5 3

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Conclusion

◮ We study the problem of generating partial updates, and finding their

corresponding real-valued codeword lengths

◮ in order to minimize the average age experienced by the receiver ◮ while maintaining a desired level of mutual information between the original

and partial updates

◮ This problem is NP hard due to the partition of the original updates ◮ We relax the problem and develop an alternating minimization based

iterative algorithm that

◮ generates a pmf for the partial updates ◮ finds the age-optimal real-valued codeword length for each update

◮ We observe that there is a trade-off between the attained average age and

the mutual information between the original and partial updates

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