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 1 / 16
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. 2 / 16
Information Update Model transmitter ^ X i X i 101 : : : source processing encoder receiver ◮ 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 3 / 16
Partial Updates ◮ The source generates i.i.d. status updates ◮ from a set X = { x 1 , x 2 , . . . , x n } with a pmf P X ( x i ) = { p 1 , p 2 , . . . , p n } ◮ 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 one partial update from the set ˆ X ◮ The pmf of the partial updates is equal to � P ˆ X (ˆ x i ) = { ˆ p i | ˆ p i = p i , S i = { j | g ( x j ) = ˆ x i , j = 1 , . . . , n } , i = 1 , . . . , k } i ∈ S i 4 / 16
Partial Updates: Example original partial updates pmf updates pmf g ( a ) 0 : 5 0 : 5 a a g ( b ) b 0 : 25 b 0 : 25 g ( c ) 0 : 125 c f c; d g 0 : 25 0 : 125 d 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 x i ) with lengths ℓ (ˆ x i ) to each partial update by using a binary alphabet ◮ The first and second moments of the codeword lengths are k � E [ L ] = X (ˆ x i ) ℓ (ˆ x i ) P ˆ i =1 k E [ L 2 ] = � x i ) 2 P ˆ X (ˆ x i ) ℓ (ˆ i =1 ◮ If update ˆ x i is transmitted, it takes ℓ (ˆ x i ) units of time to deliver this partial update to the receiver 6 / 16
Average Age Analysis a ( t ) t s 1 s 2 s 3 s 4 r T ◮ We define ∆ T as the average AoI in the time interval [0 , T ], which is � T ∆ T = 1 a ( t ) dt T 0 ◮ The long term average AoI ∆ is equal to T →∞ ∆ T = E [ S 2 ] ∆ = lim 2 E [ S ] + E [ S ] , where E [ S ] = E [ L ] and E [ S 2 ] = E [ L 2 ] 7 / 16
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 p i ,ℓ (ˆ x i ) } I ( X ; ˆ X ) = β − s.t. → Fidelity constraint k x i ) ≤ 1 − � 2 − ℓ (ˆ → Kraft’s inequality i =1 x i ) ∈ Z + − ℓ (ˆ → Feasibility constraints 8 / 16
Relaxed Problem ◮ We allow codeword lengths to be real-valued E [ L 2 ] 2 E [ L ] + E [ L ] − min → The long term AoI { ˆ p i ,ℓ (ˆ x i ) } H ( ˆ X ) = β − s.t. → Fidelity constraint k x i ) ≤ 1 − 2 − ℓ (ˆ � → Kraft’s inequality i =1 x i ) ∈ R + − ℓ (ˆ → 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 9 / 16
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 E [ L 2 ] 2 E [ L ] + E [ L ] − min → The long term AoI { ˆ p i ,ℓ (ˆ x i ) } H ( ˆ X ) = β − s.t. → Fidelity constraint k x i ) ≤ 1 − � 2 − ℓ (ˆ → Kraft’s inequality i =1 k � p i = 1 − ˆ → Feasibility constraint i =1 x i ) ∈ R + − ˆ p i ≥ 0 , ℓ (ˆ → Feasibility constraints 10 / 16
Fractional Programming Method ◮ We define p ( λ ) as E [ L 2 ] + E [ L ] 2 − λ E [ L ] p ( λ ) : min 2 { ˆ p i ,ℓ (ˆ x i ) } H ( ˆ s.t. X ) = β k x i ) ≤ 1 � 2 − ℓ (ˆ i =1 k � ˆ p i = 1 i =1 x i ) ∈ R + p i ≥ 0 , ˆ ℓ (ˆ ◮ 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. 11 / 16
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 12 / 16
Numerical Results ◮ We use Zipf( s , n ) as the pmf for the original updates, i − s P X ( x i ) = j =1 j − s , i = 1 , 2 , . . . , n � 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 } 4 3.5 3 2.5 2 1.5 0.5 1 1.5 2 2.5 13 / 16
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 } 2.5 0.8 2 0.7 0.6 1.5 0.5 0.4 1 0.3 0.2 0.5 0.1 0 0 1 2 3 1 2 3 14 / 16
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 4.5 3 4 2.5 3.5 3 2 2.5 1.5 2 5 10 15 20 5 10 15 20 15 / 16
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 16 / 16
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