Improved Lower Bounds for Coded Caching Aditya Ramamoorthy Iowa - PowerPoint PPT Presentation

Improved Lower Bounds for Coded Caching Aditya Ramamoorthy Iowa State University Joint work with Hooshang Ghasemi DIMACS Workshop on Network Coding: the Next 15 Years December 17, 2015 Aditya Ramamoorthy Improved Lower Bounds for Coded Caching 1 / 35

Conventional Content Delivery with Caching Server Shared link . . . User 1 User 2 User K-1 User K Cache 1 Cache 2 Cache K-1 Cache K Mechanism for reducing transmission rates from server to clients. ◮ Conventional approach: clients cache portions of popular content. Coding in the cache and coded transmission from server are typically not considered. Aditya Ramamoorthy Improved Lower Bounds for Coded Caching 2 / 35

Coded Caching Formulation [Maddah-Ali & Niesen ’13] W 1 . . Server contains N files . each of size F bits. W N K users each with a Shared link cache of size MF bits. The i -th user requests . . . file d i ∈ { 1 , . . . , N } . User 1 User 2 User K-1 User K M Cache 1 Cache 2 Cache K-1 Cache K Aditya Ramamoorthy Improved Lower Bounds for Coded Caching 3 / 35

Coded Caching Formulation [Maddah-Ali & Niesen ’13] W 1 Placement phase : The 1 content of the caches are . . . populated, does not depend W N on users actual requests. Shared link Delivery phase : the server 2 transmits a signal of rate RF bits over the shared link . . . User 1 User 2 User K-1 User K so that each user’s request Cache 1 Cache 2 Cache K-1 Cache K is satisfied. Aditya Ramamoorthy Improved Lower Bounds for Coded Caching 4 / 35

Coded Caching Formulation [Maddah-Ali & Niesen ’13] N files { W n } N n = 1 , W 1 . i -th user requests the file W d i , . . Cache content: Z i , W N Delivery phase signal: X d 1 ,..., d K X d 1 , d 2 ,..., d K , Decoding file for i -th user: ˆ W d 1 ,..., d K ; i , . . . User 1 User 2 User K-1 User K Probability of error: M Z 1 Z 2 Z K − 1 Z K max d 1 ,..., d K max i P ( ˆ W d 1 ,..., d K ; i � = W d i ) . Achievable Pair ( M , R ) : The pair is said to be achievable if for any ǫ > 0 there exist a file size F large enough and a ( M , R ) caching scheme with probability of at most ǫ . Aditya Ramamoorthy Improved Lower Bounds for Coded Caching 5 / 35

Coded Caching Formulation [Maddah-Ali & Niesen ’13] N files { W n } N n = 1 , W 1 i -th user requests the file W d i , . . . Cache content: Z i , W N Delivery phase signal: X d 1 ,..., d K X d 1 , d 2 ,..., d K , Decoding file for i -th user: ˆ W d 1 ,..., d K ; i , . . . User 1 User 2 User K-1 User K Probability of error: M Z K − 1 Z 1 Z 2 Z K max d 1 ,..., d K max i P ( ˆ W d 1 ,..., d K ; i � = W d i ) . Memory-rate tradeoff R ⋆ ( M ) = inf { R : ( M , R ) is achievable } . Aditya Ramamoorthy Improved Lower Bounds for Coded Caching 5 / 35

Achievable rates N = 1000 , K = 100 100 Coded Caching Rate 90 Conventional Caching Rate 80 70 Rate(R) 60 50 40 30 20 10 0 0 10 20 30 40 50 60 70 80 90 100 Cache Size(M) � � � � 1 − M 1 + KM / N , N 1 R C ( M ) = K · min , N K Aditya Ramamoorthy Improved Lower Bounds for Coded Caching 6 / 35

Achievable rates N = 1000 , K = 100 100 Coded Caching Rate 90 Conventional Caching Rate 80 70 Rate(R) 60 50 40 30 20 10 0 0 10 20 30 40 50 60 70 80 90 100 Cache Size(M) � � � � 1 − M 1 + KM / N , N 1 R C ( M ) = K · min , N K However, tight lower bounds on R C ( M ) are not known at this point. Aditya Ramamoorthy Improved Lower Bounds for Coded Caching 6 / 35

Related Work Cutset bound [Maddah-Ali & Niesen ’13]. Show that R C ( M ) / R star ( M ) ≤ 12 (multiplicative gap). Parallel works ◮ Improved bounds using Han’s inequality [Sengupta, Tandon, Clancy ’15]. Show a multiplicative gap of 8. ◮ Another approach (can be considered a special case of our work) by [Ajaykrishnan et al. 15]. ◮ Computational approach of [Tian ’15] (Arxiv preprint) for the specific case of N = K = 3. Aditya Ramamoorthy Improved Lower Bounds for Coded Caching 7 / 35

An Example: N = K = 3 and M = 1. 2 R ⋆ F + 2 MF ≥ H ( Z 1 ) + H ( X 1 , 2 , 3 ) + H ( Z 2 ) + H ( X 3 , 1 , 2 ) ≥ H ( Z 1 , X 1 , 2 , 3 ) + H ( Z 2 , X 3 , 1 , 2 ) ≥ I ( W 1 ; Z 1 , X 1 , 2 , 3 ) + H ( Z 1 , X 1 , 2 , 3 | W 1 ) + I ( W 1 ; Z 2 , X 3 , 1 , 2 ) + H ( Z 2 , X 3 , 1 , 2 | W 1 ) I ( W 1 ; Z 1 , X 1 , 2 , 3 ) = H ( W 1 ) − H ( W 1 | Z 1 , X 1 , 2 , 3 ) ≥ F ( 1 − ǫ ) Aditya Ramamoorthy Improved Lower Bounds for Coded Caching 8 / 35

An Example: N = K = 3 and M = 1. 2 R ⋆ F + 2 MF ≥ H ( Z 1 ) + H ( X 1 , 2 , 3 ) + H ( Z 2 ) + H ( X 3 , 1 , 2 ) ≥ H ( Z 1 , X 1 , 2 , 3 ) + H ( Z 2 , X 3 , 1 , 2 ) ≥ I ( W 1 ; Z 1 , X 1 , 2 , 3 ) + H ( Z 1 , X 1 , 2 , 3 | W 1 ) + I ( W 1 ; Z 2 , X 3 , 1 , 2 ) + H ( Z 2 , X 3 , 1 , 2 | W 1 ) I ( W 1 ; Z 1 , X 1 , 2 , 3 ) = H ( W 1 ) − H ( W 1 | Z 1 , X 1 , 2 , 3 ) ≥ F ( 1 − ǫ ) Aditya Ramamoorthy Improved Lower Bounds for Coded Caching 8 / 35

An Example: N = K = 3 and M = 1. 2 R ⋆ F + 2 MF ≥ H ( Z 1 ) + H ( X 1 , 2 , 3 ) + H ( Z 2 ) + H ( X 3 , 1 , 2 ) ≥ H ( Z 1 , X 1 , 2 , 3 ) + H ( Z 2 , X 3 , 1 , 2 ) ≥ I ( W 1 ; Z 1 , X 1 , 2 , 3 ) + H ( Z 1 , X 1 , 2 , 3 | W 1 ) + I ( W 1 ; Z 2 , X 3 , 1 , 2 ) + H ( Z 2 , X 3 , 1 , 2 | W 1 ) I ( W 1 ; Z 1 , X 1 , 2 , 3 ) = H ( W 1 ) − H ( W 1 | Z 1 , X 1 , 2 , 3 ) ≥ F ( 1 − ǫ ) Aditya Ramamoorthy Improved Lower Bounds for Coded Caching 8 / 35

An Example: N = K = 3 and M = 1. 2 R ⋆ F + 2 MF ≥ H ( Z 1 ) + H ( X 1 , 2 , 3 ) + H ( Z 2 ) + H ( X 3 , 1 , 2 ) ≥ H ( Z 1 , X 1 , 2 , 3 ) + H ( Z 2 , X 3 , 1 , 2 ) ≥ I ( W 1 ; Z 1 , X 1 , 2 , 3 ) + H ( Z 1 , X 1 , 2 , 3 | W 1 ) + I ( W 1 ; Z 2 , X 3 , 1 , 2 ) + H ( Z 2 , X 3 , 1 , 2 | W 1 ) I ( W 1 ; Z 1 , X 1 , 2 , 3 ) = H ( W 1 ) − H ( W 1 | Z 1 , X 1 , 2 , 3 ) ≥ F ( 1 − ǫ ) Aditya Ramamoorthy Improved Lower Bounds for Coded Caching 8 / 35

An Example: N = K = 3 and M = 1. 2 R ⋆ F + 2 MF ≥ H ( Z 1 ) + H ( X 1 , 2 , 3 ) + H ( Z 2 ) + H ( X 3 , 1 , 2 ) ≥ H ( Z 1 , X 1 , 2 , 3 ) + H ( Z 2 , X 3 , 1 , 2 ) ≥ I ( W 1 ; Z 1 , X 1 , 2 , 3 ) + H ( Z 1 , X 1 , 2 , 3 | W 1 ) + I ( W 1 ; Z 2 , X 3 , 1 , 2 ) + H ( Z 2 , X 3 , 1 , 2 | W 1 ) Writing mutual information another way ... I ( W 1 ; Z 1 , X 1 , 2 , 3 ) = H ( W 1 ) − H ( W 1 | Z 1 , X 1 , 2 , 3 ) ≥ F ( 1 − ǫ ) Aditya Ramamoorthy Improved Lower Bounds for Coded Caching 8 / 35

An Example: N = K = 3 and M = 1. 2 R ⋆ F + 2 MF ≥ H ( Z 1 ) + H ( X 1 , 2 , 3 ) + H ( Z 2 ) + H ( X 3 , 1 , 2 ) ≥ H ( Z 1 , X 1 , 2 , 3 ) + H ( Z 2 , X 3 , 1 , 2 ) ≥ I ( W 1 ; Z 1 , X 1 , 2 , 3 ) + H ( Z 1 , X 1 , 2 , 3 | W 1 ) + I ( W 1 ; Z 2 , X 3 , 1 , 2 ) + H ( Z 2 , X 3 , 1 , 2 | W 1 ) Writing mutual information another way ... I ( W 1 ; Z 1 , X 1 , 2 , 3 ) = H ( W 1 ) − H ( W 1 | Z 1 , X 1 , 2 , 3 ) ≥ F ( 1 − ǫ ) Since W 1 can be recovered from Z 1 and X 1 , 2 , 3 with ǫ -error. Aditya Ramamoorthy Improved Lower Bounds for Coded Caching 8 / 35

An Example: N = K = 3 and M = 1. ≥ I ( W 1 ; Z 1 , X 1 , 2 , 3 ) + H ( Z 1 , X 1 , 2 , 3 | W 1 ) + I ( W 1 ; Z 2 , X 3 , 1 , 2 ) + H ( Z 2 , X 3 , 1 , 2 | W 1 ) , = 2 F ( 1 − ǫ ) + H ( Z 1 , X 1 , 2 , 3 | W 1 ) + H ( Z 2 , X 3 , 1 , 2 | W 1 ) ≥ 2 F ( 1 − ǫ ) + H ( Z 1 , Z 2 , X 1 , 2 , 3 , X 3 , 1 , 2 | W 1 ) = 2 F ( 1 − ǫ ) + I ( W 2 , W 3 ; Z 1 , Z 2 , X 1 , 2 , 3 , X 3 , 1 , 2 | W 1 ) + H ( Z 1 , Z 2 , X 1 , 2 , 3 , X 3 , 1 , 2 | W 1 , W 2 , W 3 ) ≥ 2 F ( 1 − ǫ ) + 2 F ( 1 − ǫ ) = 4 F ( 1 − ǫ ) Aditya Ramamoorthy Improved Lower Bounds for Coded Caching 9 / 35

An Example: N = K = 3 and M = 1. ≥ I ( W 1 ; Z 1 , X 1 , 2 , 3 ) + H ( Z 1 , X 1 , 2 , 3 | W 1 ) + I ( W 1 ; Z 2 , X 3 , 1 , 2 ) + H ( Z 2 , X 3 , 1 , 2 | W 1 ) , = 2 F ( 1 − ǫ ) + H ( Z 1 , X 1 , 2 , 3 | W 1 ) + H ( Z 2 , X 3 , 1 , 2 | W 1 ) ≥ 2 F ( 1 − ǫ ) + H ( Z 1 , Z 2 , X 1 , 2 , 3 , X 3 , 1 , 2 | W 1 ) = 2 F ( 1 − ǫ ) + I ( W 2 , W 3 ; Z 1 , Z 2 , X 1 , 2 , 3 , X 3 , 1 , 2 | W 1 ) + H ( Z 1 , Z 2 , X 1 , 2 , 3 , X 3 , 1 , 2 | W 1 , W 2 , W 3 ) ≥ 2 F ( 1 − ǫ ) + 2 F ( 1 − ǫ ) = 4 F ( 1 − ǫ ) Aditya Ramamoorthy Improved Lower Bounds for Coded Caching 9 / 35

An Example: N = K = 3 and M = 1. ≥ I ( W 1 ; Z 1 , X 1 , 2 , 3 ) + H ( Z 1 , X 1 , 2 , 3 | W 1 ) + I ( W 1 ; Z 2 , X 3 , 1 , 2 ) + H ( Z 2 , X 3 , 1 , 2 | W 1 ) , = 2 F ( 1 − ǫ ) + H ( Z 1 , X 1 , 2 , 3 | W 1 ) + H ( Z 2 , X 3 , 1 , 2 | W 1 ) ≥ 2 F ( 1 − ǫ ) + H ( Z 1 , Z 2 , X 1 , 2 , 3 , X 3 , 1 , 2 | W 1 ) = 2 F ( 1 − ǫ ) + I ( W 2 , W 3 ; Z 1 , Z 2 , X 1 , 2 , 3 , X 3 , 1 , 2 | W 1 ) + H ( Z 1 , Z 2 , X 1 , 2 , 3 , X 3 , 1 , 2 | W 1 , W 2 , W 3 ) ≥ 2 F ( 1 − ǫ ) + 2 F ( 1 − ǫ ) = 4 F ( 1 − ǫ ) Aditya Ramamoorthy Improved Lower Bounds for Coded Caching 9 / 35

An Example: N = K = 3 and M = 1. ≥ I ( W 1 ; Z 1 , X 1 , 2 , 3 ) + H ( Z 1 , X 1 , 2 , 3 | W 1 ) + I ( W 1 ; Z 2 , X 3 , 1 , 2 ) + H ( Z 2 , X 3 , 1 , 2 | W 1 ) , = 2 F ( 1 − ǫ ) + H ( Z 1 , X 1 , 2 , 3 | W 1 ) + H ( Z 2 , X 3 , 1 , 2 | W 1 ) ≥ 2 F ( 1 − ǫ ) + H ( Z 1 , Z 2 , X 1 , 2 , 3 , X 3 , 1 , 2 | W 1 ) = 2 F ( 1 − ǫ ) + I ( W 2 , W 3 ; Z 1 , Z 2 , X 1 , 2 , 3 , X 3 , 1 , 2 | W 1 ) + H ( Z 1 , Z 2 , X 1 , 2 , 3 , X 3 , 1 , 2 | W 1 , W 2 , W 3 ) ≥ 2 F ( 1 − ǫ ) + 2 F ( 1 − ǫ ) = 4 F ( 1 − ǫ ) Aditya Ramamoorthy Improved Lower Bounds for Coded Caching 9 / 35