Information Complexity Density and Simulation of Protocols Himanshu Tyagi Indian Institute of Science, Bangalore with Pramod Viswanath (UIUC), Shaileshh Venkatakrishnan (UIUC), and Shun Watanabe (TUAT)
Private Coin Interactive Protocols X Y 1
Private Coin Interactive Protocols π X Y 1
Private Coin Interactive Protocols π X Y Denote by Π = (Π 1 , Π 2 , Π 3 , ... ) the random transcript 1
Private Coin Interactive Protocols π X Y Denote by Π = (Π 1 , Π 2 , Π 3 , ... ) the random transcript Π 1 — X — Y Π 2 — Y, Π 1 — X Π 3 — X, Π 1 , Π 2 — Y · · · 1
Private Coin Interactive Protocols π X Y Denote by Π = (Π 1 , Π 2 , Π 3 , ... ) the random transcript Π 1 — X — Y Π 2 — Y, Π 1 — X Π 3 — X, Π 1 , Π 2 — Y · · · | π | = depth of the protocol tree 1
ǫ -Simulation of a Protocol π π sim X Y X Y Π y Π x Π Definition A protocol π sim constitutes an ǫ -simulation of π if it can produce outputs Π x and Π y at X and Y , respectively, such that � � � P XY ΠΠ − P XY Π x Π y TV ≤ ǫ. � 2
ǫ -Simulation of a Protocol π π sim X Y X Y Π y Π x Π Definition A protocol π sim constitutes an ǫ -simulation of π if it can produce outputs Π x and Π y at X and Y , respectively, such that � � � P XY ΠΠ − P XY Π x Π y TV ≤ ǫ. � We seek to characterize D ǫ ( π | P XY ) = min. length of an ǫ -simulation of π 2
ǫ -Compression of a Protocol π π com Y X X Y Π y Π x Π Definition A protocol π com constitutes an ǫ -compression of π if it can produce outputs Π x and Π y at X and Y , respectively, such that Pr (Π = Π x = Π y ) ≥ 1 − ǫ. 3
ǫ -Compression of a Protocol π π com Y X X Y Π y Π x Π Definition A protocol π com constitutes an ǫ -compression of π if it can produce outputs Π x and Π y at X and Y , respectively, such that Pr (Π = Π x = Π y ) ≥ 1 − ǫ. For deterministic protocols, compression ≡ simulation. 3
Information Complexity of π def IC ( π ) = I (Π ∧ X | Y ) + I (Π ∧ Y | X ) 4
Information Complexity of π def IC ( π ) = I (Π ∧ X | Y ) + I (Π ∧ Y | X ) Examples ◮ Π( x, y ) = x IC ( π ) = H ( X | Y ) ◮ Π( x, y ) = ( x, y ) IC ( π ) = H ( X | Y ) + H ( Y | X ) 4
Information Complexity of π def IC ( π ) = I (Π ∧ X | Y ) + I (Π ∧ Y | X ) Examples ◮ Π( x, y ) = x IC ( π ) = H ( X | Y ) ◮ Π( x, y ) = ( x, y ) IC ( π ) = H ( X | Y ) + H ( Y | X ) Theorem (Amortized Communication Complexity [BR’10] ) For coordinate-wise repetition π n of π and i.i.d. ( X n , Y n ) , 1 nD ǫ ( π n | P X n Y n ) = IC ( π ) . lim ǫ → 0 lim n →∞ 4
Information Complexity of π def IC ( π ) = I (Π ∧ X | Y ) + I (Π ∧ Y | X ) Examples ◮ Π( x, y ) = x [Slepian-Wolf ’74] IC ( π ) = H ( X | Y ) ◮ Π( x, y ) = ( x, y ) [Csiszár-Narayan ’04] IC ( π ) = H ( X | Y ) + H ( Y | X ) Theorem (Amortized Communication Complexity [BR’10] ) For coordinate-wise repetition π n of π and i.i.d. ( X n , Y n ) , 1 nD ǫ ( π n | P X n Y n ) = IC ( π ) . lim ǫ → 0 lim n →∞ 4
Questions ◮ Strong converse. Does lim n →∞ 1 n D ǫ ( π n | P X n Y n ) depend on ǫ ? ◮ Mixed protocols. What about a mixed protocol π ( n ) given by � π n h , w.p. p, π ( n ) = π n l , w.p. 1 − p. Note that IC ( π ( n ) ) = n � � p IC ( π h ) + (1 − p ) IC ( π l ) ◮ ... General distributions? Second-order asymptotics? Single-shot? 5
Questions ◮ Strong converse. Does lim n →∞ 1 n D ǫ ( π n | P X n Y n ) depend on ǫ ? ◮ Mixed protocols. What about a mixed protocol π ( n ) given by � π n h , w.p. p, π ( n ) = π n l , w.p. 1 − p. Note that IC ( π ( n ) ) = n � � p IC ( π h ) + (1 − p ) IC ( π l ) ◮ ... General distributions? Second-order asymptotics? Single-shot? Why do we care? 5
Questions ◮ Strong converse. Does lim n →∞ 1 n D ǫ ( π n | P X n Y n ) depend on ǫ ? ◮ Mixed protocols. What about a mixed protocol π ( n ) given by � π n h , w.p. p, π ( n ) = π n l , w.p. 1 − p. Note that IC ( π ( n ) ) = n � � p IC ( π h ) + (1 − p ) IC ( π l ) ◮ ... General distributions? Second-order asymptotics? Single-shot? Why do we care? 42. 5
The Tail of Information Complexity Density
Information Complexity Density = log P Π | XY ( τ | x, y ) + log P Π | XY ( τ | x, y ) def ic ( τ ; x, y ) P Π | X ( τ | x ) P Π | Y ( τ | y ) Note that E [ ic (Π; X, Y )] = IC ( π ) . 7
Information Complexity Density = log P Π | XY ( τ | x, y ) + log P Π | XY ( τ | x, y ) def ic ( τ ; x, y ) P Π | X ( τ | x ) P Π | Y ( τ | y ) Note that E [ ic (Π; X, Y )] = IC ( π ) . ǫ -Tails of ic (Π; X, Y ) are closely related to D ǫ ( π | P XY ) 7
Illustration Consider the Slepian-Wolf problem ( Π( x, y ) = x ). ◮ ic ( τ ; x, y ) = − log P X | Y ( x | y ) 8
Illustration Consider the Slepian-Wolf problem ( Π( x, y ) = x ). ◮ ic ( τ ; x, y ) = − log P X | Y ( x | y ) ◮ If Pr ( ic (Π; X, Y ) ≥ λ ) ≤ ǫ , - a random hash λ -bit hash of X constitutes an ǫ -compression. ◮ If Pr ( ic (Π; X, Y ) ≥ λ ) > ǫ , - any subset with prob. ≥ 1 − ǫ has cardinality less than λ 8
Illustration Consider the Slepian-Wolf problem ( Π( x, y ) = x ). ◮ ic ( τ ; x, y ) = − log P X | Y ( x | y ) ◮ If Pr ( ic (Π; X, Y ) ≥ λ ) ≤ ǫ , - a random hash λ -bit hash of X constitutes an ǫ -compression. ◮ If Pr ( ic (Π; X, Y ) ≥ λ ) > ǫ , - any subset with prob. ≥ 1 − ǫ has cardinality less than λ Prob. > � Prob. ≤ � Spectrum of h ( X | Y ) = − log P X | Y ( X | Y ) 8
Main Results
Lower Bound Theorem Given 0 ≤ ǫ < 1 and a protocol π , D ǫ ( π ) � sup { λ : Pr ( ic (Π; X, Y ) > λ ) ≥ ǫ } . 10
Lower Bound Theorem Given 0 ≤ ǫ < 1 and a protocol π , D ǫ ( π ) � sup { λ : Pr ( ic (Π; X, Y ) > λ ) ≥ ǫ } . Weaknesses. ◮ The fudge parameters are of the order log( spectrum width ) . ◮ Uses only the joint pmf, not the structure of the protocol. 10
Upper bound Theorem Given 0 ≤ ǫ < 1 and a bounded rounds protocol π , D ǫ ( π ) � sup { λ : Pr ( ic (Π; X, Y ) > λ ) ≤ ǫ } . Distribution of ic (Π; X, Y ) Pr( ic (Π; X, Y )) > λ ) > � Pr( ic (Π; X, Y )) > λ ) ≤ � Lower bound Upper Bound 11
Upper bound Theorem Given 0 ≤ ǫ < 1 and a bounded rounds protocol π , D ǫ ( π ) � sup { λ : Pr ( ic (Π; X, Y ) > λ ) ≤ ǫ } . Distribution of ic (Π; X, Y ) Pr( ic (Π; X, Y )) > λ ) > � Pr( ic (Π; X, Y )) > λ ) ≤ � Lower bound Upper Bound Weaknesses. ◮ The fudge parameters depend on the number of rounds. ◮ Protocol based on round-by-round compression. 11
Questions ◮ Strong converse. Does lim n →∞ 1 n D ǫ ( π n | P X n Y n ) depend on ǫ ? ◮ Mixed protocols. What about a mixed protocol π ( n ) given by � π n h , w.p. p, π ( n ) = π n l , w.p. 1 − p. Note that IC ( π ( n ) ) = n � � p IC ( π h ) + (1 − p ) IC ( π l ) 12
Questions ◮ Strong converse. Does lim n →∞ 1 n D ǫ ( π n | P X n Y n ) depend on ǫ ? Answer. No. In fact, n V ( ic (Π; X, Y )) Q − 1 ( ǫ ) + o ( √ n ) � D ǫ ( π n ) = n IC ( π ) + ◮ Mixed protocols. What about a mixed protocol π ( n ) given by � π n h , w.p. p, π ( n ) = π n l , w.p. 1 − p. Note that IC ( π ( n ) ) = n � � p IC ( π h ) + (1 − p ) IC ( π l ) 12
Questions ◮ Strong converse. Does lim n →∞ 1 n D ǫ ( π n | P X n Y n ) depend on ǫ ? Answer. No. In fact, n V ( ic (Π; X, Y )) Q − 1 ( ǫ ) + o ( √ n ) � D ǫ ( π n ) = n IC ( π ) + ◮ Mixed protocols. What about a mixed protocol π ( n ) given by � π n h , w.p. p, π ( n ) = π n l , w.p. 1 − p. Note that IC ( π ( n ) ) = n � � p IC ( π h ) + (1 − p ) IC ( π l ) Answer. 1 nD ǫ ( π ( n ) ) = IC ( π h ) lim ǫ → 0 lim sup n →∞ 12
42 Function Computation [BR ’10], [MI ’10]: 1 nD ǫ ( f n ) = IC ( f ) . ǫ → 0 lim lim n → 13
42 Function Computation [BR ’10], [MI ’10]: 1 nD ǫ ( f n ) = IC ( f ) . ǫ → 0 lim lim n → ◮ Strong converse? Our bound yields 1 nD ǫ ( f n ) ≥ H ( f ( X, Y ) | X ) + H ( f ( X, Y ) | Y ) lim n → 13
42 Function Computation [BR ’10], [MI ’10]: 1 nD ǫ ( f n ) = IC ( f ) . ǫ → 0 lim lim n → ◮ Strong converse? Our bound yields 1 nD ǫ ( f n ) ≥ H ( f ( X, Y ) | X ) + H ( f ( X, Y ) | Y ) lim n → ◮ Direct product or Arimoto converse? [BRWY ’13], [BW’14]: n IC ( f ) poly (log n ) ⇒ Pr ( F = F x = F y ) ≤ e − nc ∀ n large | π n | < 13
42 Function Computation [BR ’10], [MI ’10]: 1 nD ǫ ( f n ) = IC ( f ) . ǫ → 0 lim lim n → ◮ Strong converse? Our bound yields 1 nD ǫ ( f n ) ≥ H ( f ( X, Y ) | X ) + H ( f ( X, Y ) | Y ) lim n → ◮ Direct product or Arimoto converse? [BRWY ’13], [BW’14]: n IC ( f ) poly (log n ) ⇒ Pr ( F = F x = F y ) ≤ e − nc ∀ n large | π n | < Our bound yields a threshold of n [ H ( F | X ) + H ( F | Y )] . 13
42 Separation of D ǫ ( π ) and IC ( π ) [BBCR ’10]: D ǫ ( π ) ≤ ˜ � O ( | π | IC ( π )) [B ’12]: D ǫ ( π ) ≤ 2 O ( IC ( π )) 14
Recommend
More recommend
