Performance Limits of Coded Caching under Heterogeneous Settings Xiaojun Lin Associate Professor, Purdue University

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1 Performance Limits of Coded Caching under Heterogeneous Settings Xiaojun Lin Associate Professor, Purdue University Joint work with Jinbei Zhang (SJTU), Chih-Chun Wang (Purdue) and Xinbing Wang (SJTU)

2 The Importance of Caching Server cache cache cache User 1 User 2 User 3 Data traffic continues to grow at significant rates A major fraction (60-80%) of traffic will be generated by multimedia content, such as video Caching is important for reducing backhaul requirement in serving large volumes of content that multiple users are interested in 2

3 Traditional (Uncoded) Caching: Individual cache size needs to be large N=3 files (unit-size): A = (A 1, A 2, A 3 ) B = (B 1, B 2, B 3 ) C = (C 1, C 2, C 3 ) Back-haul Requirement: Uncoded Caching K 1 M N = 2 Individual caching gain K=3 users Cache size M=1 (A 1, B 1, C 1 ) Broadcast channel Server A 2, A 3 B 2, B 3 C 2, C 3 (A 1, B 1, C 1 ) (A 1, B 1, C 1 ) User 1 wants A User 2 wants B User 3 wants C 3

4 Coded Caching: Global Caching Gains N=3 files: A = (A 1, A 2, A 3 ) B = (B 1, B 2, B 3 ) C = (C 1, C 2, C 3 ) Back-haul Requirement: Uncoded Caching K 1 M N = 2 Coded Caching [1] K 1 M N KM N = 1 K=3 users Cache size M=1 Global caching gain A 2 B 1 A 3 C 1 B 3 C 2 (A 1, B 1, C 1 ) (A 2, B 2, C 2 ) (A 3, B 3, C 3 ) A 2 B 1 A 3 C 1 B 3 C 2 User 1 wants A Broadcast channel Server User 2 wants B [1] Fundamental Limits of Caching, M. Maddah-Ali and U. Niesen, IEEE Trans. Inf. Theory, User 3 wants C 4

5 Homogeneous vs Heterogeneous Settings [Maddah-Ali and Niesen 14] shows that the worst-case transmission rate K 1 M N 1 1+ KM N is at most a constant factor (12x) away from the (information-theoretic) minimum possible Generalized to Decentralized/probabilistic caching schemes [Maddah-Ali and Niesen `14] Hierarchical caching [Karamchandani et al `14 ] Online caching [Pedarsani et al `13] These studies assume a homogeneous setting where all files are equally important and are with the same parameters 5

6 Homogeneous vs Heterogeneous Settings In practice, heterogeneity arises naturally In homogeneous settings, all files are cached uniformly In heterogeneous settings: Should more popular files be cached more aggressively [Niesen and Maddah-Ali `14, Ji et al `14, Hachem et al `14]? Should larger files be cached more aggressively? 6

7 Our Contribution Coded caching needs to be adapted in different ways to different aspects of heterogeneity Heterogeneous popularity: Only files above a popularity threshold are cached However, all popular files are cached uniformly (similar to [Ji et al 14]) We show constant-factor bounds that are independent of the popularity distribution Heterogeneous file-sizes (Roughly) quadratically more content is cached for larger files We show logarithmic-factor bounds While the new achievable schemes are quite intuitive, the corresponding lower bounds are more involved and reveal useful insights 7

8 Outline Coded Caching under Arbitrary Popularity Distributions System Model Achievable Bounds and Intuitions Lower Bounds Coded Caching under Distinct File Sizes System Model Achievable Bounds and Intuitions Lower Bounds Conclusion and Discussions 8

9 Network Model: Heterogeneous Popularity Server with a broadcast channel K users: cache size M N (unit-size) files: F = F 1,, F N Popularity (decreasing): P = p 1,, p N Random request W i W i = f i1,, f ik, f ik F 2 File Request User 1 File 1 1 File Placement 2 User 2... Server File N File Transmission 2 File Request User K Rate for serving W i is r(w i ) Expected rate: N K R K, F, P = r(w i )P(W i ) i=1 9

10 Average-Case vs. Worst-Case Expected rate: Worst-case rate [Maddah-Ali and Niesen `14]: Obviously: N K i=1 N K i=1 r(w i )P(W i ) max r(w i) i=1,,n K r(w i )P W i max i=1,,n K r(w i) However, constant-factor results do NOT carry over from the worst case to the average case max r(w i) i=1,,n K max i=1,,n r (W i ) K c N K i=1 N K i=1 r(w i )P(W i ) r (W i )P(W i ) c 10

11 Related Work on the Average Case U. Niesen, and M.A. Maddah-Ali, Coded Caching with Nonuniform Demands, arxiv: v2 [cs.it], Mar Divide the files into groups The gap between the lower bound and the achievable (upper) bound increases with # of groups (unbounded) J. Hachem, N. Karamchandani and S. Diggavi, Multi-level Coded Caching, arxiv: [cs.it], Apr Popularity has multiple levels The gap increases with # of levels (unbounded) M. Ji, A. Tulino, J. Llorca and G. Caire, On the Average Performance of Caching and Coded Multicasting with Random Demands, arxiv: v2 [cs.it], Jul Zipf popularity distribution p i 1 i α The gap increases with 1 α 1 when α > 1 (unbounded) 11

12 Our Main Results Constant-factor gap between the lower bound (R lb ) and the achievable (upper) bound (R ub ) of the expected backhual transmission rate: R ub 87R lb +2 The achievable bound (R ub ) is attained by a simple coded caching scheme similar to [Ji et al 14] Perform coded caching only among the most popular N 1 files However, all N 1 popular files are treated uniformly 1 KM Popularity The key step is to show a matching lower bound N 1 Arbitrary Popularity Distribution! File Index 12

13 Main Intuition: An Insensitivity Property The best worst-case rate for serving N files can be achieved by uniform caching [Maddah-Ali and Niesen 14] N M N M 1 Uncoded caching Coded caching M 1 N M N K (1 ) (1 ) 1 N KM 1 M N M N whenever K >> N/M Key Insight: Beyond K=N/M, the above rate is independent of the number of users K Due to its global caching gain, coded caching significantly reduce the threshold for this insensitivity to arise N M N K 13

14 Main Intuition: Average Case Consider the following scheme: Only perform coded caching among most popular files 1 to N 1 The average transmission rate for the popular files will be upperbounded by the worst-case rate: M 1 N1 M N1 K ' (1 ) (1 ) 1 N K' M 1 1 M N1 M N 1? Popularity N 1 File Index whenever K >> N 1 /M If these files are indeed very popular, K will be large. Thus, the N expected rate will likely be close to this upper bound 1 1 M Once a file is popular, its popularity does not matter! 14

15 Achievable Bound An achievable rate: N 1 [ 1] Kpi M i N Popular The minimum occurs at 1 M Kp N 1 Unpopular 1 p N Proposition 1: Assume M 2. There exists an achievable scheme whose average transmission rate satisfies: R K, F, P R ub = N 1 M KM 1 KM Popularity + Kp i i>n 1 where N 1 satisfies p N1 1 KM and p N 1 +1 < 1 KM. 15 N 1 File Index

16 Outline Coded Caching under Arbitrary Popularity Distributions System Model Achievable Bounds and Intuitions Lower Bounds Coded Caching under Distinct File Sizes System Model Achievable Bounds and Intuitions Lower Bounds Conclusion and Discussions 16

17 Lower Bound: Statement Proposition 2: Assume that M 2. Let N 1 be the least popular file with popularity no smaller than 1 KM, i.e., p N1 1 KM and p N 1 +1 < 1 KM For all possible coded caching schemes, the average transmission rate is lower bounded by R K, F, P R lb = max{ KM Popularity N 1 N 1 M 1 1, + 58 [ Kp i 2] + } i>n 1 File Index Lower bound for serving popular files 1 to N 1 Lower bound for serving unpopular files N 1 +1 to N 17

18 Lower Bound: Challenges Need to show: For popular files, popularity does not matter Reduce to uniform popularity 1/KM and use stochastic dominance With uniform popularity for all popular files, their worst-case rate and average-case rate are on the same order [Niesen and Maddah-Ali `14] For unpopular files, it is a good idea not to use any caching 1 KM Popularity N 1 File Index 18

19 Popular Files: Popularity Does Not Matter F, P F 1, p 1 F N1, p N1 F N1 +1, p N1 +1 F N, p N R K, F, P Remove all unpopular files Reduce all popularity to the lowest value p N1 F 1, p 1 F N1, p N F N1 (empty file),1 1 1, P 1 p i R K, F 1, P 1 i=1 F 1, P 2 F 1, p N1 F N1, p N1 (empty file),1 N 1 p N1 R K, F 1, P 2 System 2 19

20 System 2 K, F 1, P 2 : Worst-case vs. Average-Case Each of the K user request one of the N 1 popular files with equal probability p N1 1 KM The average number of users requesting popular files is KN 1 p N1 N 1 M Property 1: with reasonable probability, the number of users K r requesting popular files is no smaller than N 1 M P K r N 1 M 0.5. More precisely, Property 2: with reasonable probability, the number of distinct files K d requested is no smaller than 0.5K r. More precisely P K d 1 2 K r K r 0.56 P K d 1 2 N 1 M 0.28 denoted as K 3 20

21 Popular Files: Further Reduction from System 2 System 2 R K, F 1, P 2 F 1, p N1 F N1, p N1 (empty file),1 N 1 p N1 If K d K 3, i.e., the number of distinct files requested is no smaller than K 3 (This happens with probability 0.28) Users 1 2 K If K d < K 3, 1 2 K Exactly K 3 users requests K 3 distinct files. No files are requested Each pattern is equally likely. For the left hand side: Minimizing the average case will be equivalent to minimizing the worst case R 1 8 N 1 M 1 + R K, F, P N 1 M N 1 M

22 Lower Bound: Unpopular Files R K, F, P max{ 1 29 N 1 M 1 1, + 58 [ Kp i 2] + } i>n 1 Bound for serving popular files Bound for serving unpopular files F 1, p 1 F N1, p N1 F N1 +1, p N1 +1 F i, p i F N, p N Remove all popular files F j, p j N,1 i=n 1 +1 p i F N1 +1, p N1 +1 F i, p i F j, p j F N, p N R K, F, P R K, F 3, P 3 Merge any two unpopular files into one Lower Rate N,1 i=n 1 +1 p i F N1 +1, p N1 +1 V k, p i + p j F N, p N 22

23 Unpopular Files: Reduction to System 2 N,1 i=n 1 +1 p i F N1 +1, p N1 +1 V k, p i + p j F N, p N Merge files until the sum popularity is just above 1/KM 2 > v KM i 1, KM N 2 ( i>n 1 p i )KM /2 N,1 2 i=1 p i V 1, v 1 V 2, v 2 V N2, v N2 R K, F 4, P 4 Reduce all popularity to 1/KM,1 N 2 KM 1 V 1, KM 1 V 1, KM 1 V N2, KM R K, F 4, P 5 This is exactly like System 2! R K, F, P 1 29 N 2 M i>n1 2Kp i

24 Constant Factor We have shown the lower bound: R K, F, P R lb = max{ 1 29 Recall the achievable bound Constant-factor: N 1 M 1 + R K, F, P R ub = N 1 M 1 + R ub 87R lb +2, 1 58 [ Kp i i>n 1 2] + } + Kp i i>n 1 Arbitrary Popularity Distribution! 24

25 Numerical Comparison LFU [Lee et al 01]: Cache the M most popular contents (No coding) Uniform-caching [Maddah-Ali and Niesen 14]: Randomly cache M portion of every content, regardless of popularity N Group-caching [Niesen and Maddah-Ali, 14]: Divide the files into groups with similar popularity; perform coded caching within each group Include an additional cache-allocation optimization RLFU (Random LFU) [Ji et al 14]: Assume Zipf popularity distribution; perform coded caching among the most popular files 1 to N 1 Numerically optimize N 1 based on some upper bound For uniform-caching, group-caching and RLFU, we plot the upper bound on the average transmission rate 25

26 Upper Bound Numerical Results: Zipf Distribution p i 1 iα and α = 1.1 N = 5000, K = 500, α = 1.1 Uniform caching LFU (no coding) Group-caching [Maddah-Ali and Niesen 14] Proposed scheme RLFU [Ji et al 14] (Numerically optimized N 1 ) Note: For RLFU, Group-Caching, Uniform-Caching, we plot the upper bound 26

27 Non-Zipf Distribution Zipf-Mandelbrot law distribution p i 1 (i+r) α, α = 1.4,r = 2, N = 5000, K = 500 Uniform caching LFU (no coding) Group-caching [Maddah-Ali and Niesen 14] Proposed scheme 27

28 Summary: Arbitrary Popularity Distributions We study the expected transmission rate of coded caching under arbitrary popularity distributions We obtain achievable bounds that differ from the information-theoretic lower bound by at most a constant factor (except for a small additive term) Threshold Structure: Perform coded caching only among popular files p N1 1 KM However, all popular files are cached uniformly Similar to [Ji et al 14], but use a different N 1 28

29 Outline Coded Caching under Arbitrary Popularity Distributions System Model Achievable Bounds and Intuitions Lower Bounds Coded Caching under Distinct File-Sizes System Model Achievable Bounds and Intuitions Lower Bounds Conclusion and Discussions 29

30 Network Model: Distinct File Sizes Server with a broadcast channel K users: cache size M N files: F = F 1,, F N File-size (nonincreasing): F i = F i F i F j, if i j Request pattern: W i = f i1,, f ik, f ik F Rate for W i is r(w i ) 2 File Request User 1 File 1 1 File Placement 2 User 2... Server File N File Transmission 2 File Request User K Worst-case rate: R = max r(w i ) i 30

31 Power-of-2 Simplification File-sizes differ by power-of-2 factors F 1 File Size l-th type: F l = F 1 /2 l 1 N l files of type l F 1 /2 F 1 /4 File Index T distinct types The total number of files TX N l = N l=1 N 1 N 2 N 3 N T ¹T = minft; log 2 Kg Files of type l > T can be virtually neglected Their sizes F 1 /K 31

32 log(rate) Our Main Results Logarithmic-factor gap between the lower bound (R LB2 ) and the achievable (upper) bound (R UB2 ) for the worst-case transmission rate: R UB2 32 log 2 K R LB The achievable bound (R UB2 ) is attained by caching larger files more aggressively Quadratically more content is cached for larger files The key step is to show a tighter lower bound, which involves careful use of entropy inequalities Number of types UB1 K/(log K) 2 UB2 LB2 LB1 32

33 Recall Insensitivity under Unit File-size The worst-case rate with uniform file size of 1 [Maddah-Ali & Niesen `14] is given by K 1 M N 1 1+ KM N N M 1 N M, when K >> N M and M<<N Each user caches every bit of each file with probability q = M N Worst-case rate N M = 1 q When the uniform file-size is F, these numbers become: Caching probability q = M NF, Worst-case rate: KF 1 M NF 1 1+ KM NF NF2 = F, when q << 1 and K >> 1/q M q 33

34 Two Achievable Schemes (UB1 vs UB2) Achievable Scheme 1 (UB1) All files are cached with an equal probability q Linearly more content is cached for larger files Cache constraint: T q l=1 N l F l = M with T = min(t, log 2 K) R UB1 O(max l F l q ) Achievable Scheme 2 (UB2) Let the caching probability q l = F l /c Quadratically more content is cached for larger files Cache constraint: ¹TX l=1 q l N l F l = M R UB2 ¹TX l=1 ¹TX l=1 N l F 2 l F l q l + KF ¹ T +1 = cm = O( F 1 P T ¹ l=1 N lf l ) M = T ¹ c + KF T ¹ +1 P T ¹ l=1 ( ¹T + 1) N lfl 2 M 34

35 log(rate) Two Achievable Schemes (UB1 vs UB2) Achievable Scheme 1 (UB1) Achievable Scheme 2 (UB2) R UB1 O( F 1 P T ¹ l=1 N lf l ) M R UB2 ( ¹ T + 1) P ¹ T l=1 N lf 2 l M Consider T = log 2 K, N l+1 = 4N l R UB1 K N 1F 2 1 8M Critical to cache quadratically more content for larger files! R UB2 ( T ¹ + 1) T ¹ N 1Fl 2 M UB1 UB1 K/(log K) 2 UB2 UB2 Number of types 35

36 Outline Coded Caching under Arbitrary Popularity Distributions System Model Achievable Bounds and Intuitions Lower Bounds Coded Caching under Distinct File-Sizes System Model Achievable Bounds and Intuitions Lower Bounds Conclusion and Discussions 36

37 Lower Bound: Uniform File-size NF/(2M) users 2M/F Request patterns 1 2 NF/(2M) NF/(2M)+1 NF/M N-NF/(2M)+1 N In order to serve all request patterns, we must have [Maddah-Ali & Niesen `14] 2M F R (F) + NF 2M M NF R (F) NF 2 4M = F 4q

38 Lower Bound: First Try L users P l2 N l L Request patterns X l2 N l Files of types in Φ In order to serve all request patterns, we must have l Φ L N l R F + L M N l Maximizing over L and Φ l Φ R l Φ N l F 2 l F max Φ 4M l Φ N l F l (LB1)

39 log(rate) LB1 vs UB2 Lower Bound 1 (LB1) Achievable Bound 2 (UB2) R R LB1 = max Φ 2 l Φ N l F l 4M l Φ N l R R UB2 = ( ¹ T + 1) P ¹ T l=1 N lf 2 l M Consider N l+1 = 4N l R LB1 N 1 F 2 1 4M R UB2 ( T ¹ + 1) T ¹ N 1F1 2 M UB1 LB1 fails to account for heterogeneous caching probabilities! UB2 (log K) 2 LB1 Number of types 39

40 log(rate) An Improved Lower Bound (LB2) Proposition 3: Under Assumption 1: 2M F 1 and N lf l 2M are integers for all 1 l T Assumption 2: We must have T l=1 N l F l 2M K R F T l=1 N l F l 2 4M UB1 UB2 (LB2) log K Compared with UB2: P T ¹ R UB2 ( T ¹ l=1 + 1) N lfl 2 M Number of types LB2 LB1 40

41 Intuition for LB2 (Two types: F 2 = F 1 /2) R F 2 T N l F l l=1 = N 2 1F 1 4M 2 4M +N 2F 2 4M Assumption 2 ensures that s 1 +s 2 = N 1F 1 + N 2F 2 2M 2M (LB2) K s 1 = N 1F 1 2M users (U 1 ) s 2 = N 2F 2 2M users (U 2 ) patterns patterns N 1 files of type 1 (F 1 ) N 1 files of R (F) N1F1 2 type 1 (F 1 4M ) N 2 files of type 2 (F 2 ) 41

42 Proof: (Two types: F2 = F1/2) s 1 = N 1F 1, s 2M 2 = N 2F 2 2M H M 1 = 0.5H F 1 = N 1F 1 2 H M 2 = 0.5H F 2 = N 2F 2 2 I R D1 ; F 1 + H IMR D1 ; F 1 1 I N 1F 1 R D1 M 1 ; F 1 = H(F 1 ), I R 2 D 2 ; F 1 N 1F 1 2 I R D1 R D2 ; F 2 F 1 I + R D1 H R M D2 2 ; F 2 F 1 I R N 2F D1 R 2 D2, M 2 ; F 2 F 1 = H(F 2 ) 2 42

43 Proof: (Two types: F2 = F1/2) s 1 = N 1F 1, s 2M 2 = N 2F 2 2M From 2N 1 R F N s 1 F 1 + N 2F M R F N F 1 F 1 + N 2F R F N 2 1F 1 4M + N 2F 2 F 1 8M R F N 2 1F 1 4M + N 2 2F 2 4M (LB2) 43

44 Beyond Power-of-2 File Size Original file-set: F = F 1,, F N, F i = F i Upper-quantized version F UB = F i UB F i UB = F 1 2 log 2 F 1 F i Lower-quantized version N File Index F LB = F i LB F i LB = F 1 2 log 2 F 1 F i 1 R F LB R F R F UB Under certain conditions (Assumption 2), R F LB is in a constant gap with R F UB 44

45 Comparisons With Assumptions 1 & 2 General Result (without Assumptions 1 and 2): R UB2 (32log 2 K + 22)R LB2 45

46 Conclusions Heterogeneous popularity: A simple threshold-based policy (similar to [Ji et al 14]): files above a popularity threshold are cached uniformly We show constant-factor gap that is independent of the popularity distribution Heterogeneous file-sizes Quadratically more content is cached for larger files We show logarithmic-factor gap While the new achievable schemes are quite intuitive, the corresponding lower bounds more involved and reveal useful insights 46

47 Potential Future Directions Refinements: Combining heterogeneous popularity and file-sizes? Reduce the logarithmic factor? Other aspects of heterogeneity: Heterogeneous cache size? The role of cache location? Wireless environments (HetNet): Is coded caching still helpful? Practical considerations: Low-complexity coding/transmission schemes Real-time transmissions [Niesen and Maddah-Ali `15] 47

48 Thank you! J. Zhang, X. Lin and X. Wang, Coded Caching under Arbitrary Popularity Distributions,'' in ITA Workshop, UCSD, February J. Zhang, X. Lin, C.-C. Wang and X. Wang, Coded Caching for Files with Distinct File Sizes, in ISIT, June 2015 (to appear). 48