Coded Caching under Arbitrary Popularity Distributions Jinbei Zhang

Coded Caching under Arbitrary Popularity Distributions Jinbei Zhang, Xiaojun Lin, Xinbing Wang

Caching and Coded-Caching • Caching is important for reducing backhaul requirement when serving content that many users are interested in • The cache size at each user needs to be reasonable large compared to the amount of “popular” content Server • Coded caching can further reduce the backhaul requirement even when each individual cache size is small, but the global cache size is substantial [Maddah-Ali and cache Neisen ’ 14] User 1 cache User 2 User 3 2

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

Coded Caching: Global Caching Gains N=3 files: Server Broadcast channel K=3 users Back-haul Requirement: Cache size M=1 • Uncoded Caching • Coded Caching [1] Global caching gain [1] Fundamental Limits of Caching, M. Maddah-Ali and U. Niesen, IEEE Trans. Inf. Theory, 2014. 4

Average-Case vs. Worst-Case • Expected rate: • Worst-case rate [Maddah-Ali and Niesen `14]: • Obviously: • However, constant-factor results do NOT carry over from the worst case to the average case 5

Related Work on the Average Case • 6

The Open Question Can we find a coded caching scheme whose average-case performance is at most a constant factor away from the minimum independently of any popularity distributions? 7

Our Main Results • Popularity File Index Arbitrary Popularity Distribution! 8

Network Model 9

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] Uncoded caching Coded caching K 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 threshold for this insensitivity to arise 10

Main Intuition: Average Case • Consider the following scheme: Popularity – 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: File Index whenever K’ >> N 1/M • If these files are indeed very popular, K’ will be large. Thus, the expected rate will likely be close to this upper bound Once a file is “popular”, its popularity does not matter! 11

Achievable Bound Popularity • “Popular” “Unpopular” File Index 12

Lower Bound: Statement • Popularity Decrease File Index Merge File Index 13

Popular Files: Popularity Does Not Matter … … … Remove all unpopular files System 2 14

Popular Files: Further Reduction from System 2 … Users 1 2 … … … K … No files are requested For the left hand side: Minimizing the average case will be equivalent to minimizing the worst case 16

Lower Bound: Unpopular Files Impact of “popular files” … … Impact of “unpopular files” … … … Remove all popular files … Merge any two unpopular files into one … Lower Rate 17

Unpopular Files: Reduction to System 2 … … Merge files until the sum popularity is just above 1/KM … Reduce all popularity to 1/KM … This is exactly like System 2! 18

Constant Factor • Arbitrary Popularity Distribution! 19

Numerical Comparison 20

Upper Bound 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 21