communication complexity, information complexity and applications to privacy toniann pitassi...
DESCRIPTION
Communication Complexity, Information Complexity and Applications to Privacy Toniann Pitassi University of Toronto. m 1. m 2. m 3. 2-Party Communication Complexity [Yao]. 2-party communication: each party has a dataset. Goal is to compute a function f(D A ,D B ). m k-1. m k. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Communication Complexity, Information Complexity and Applications to Privacy Toniann Pitassi University of Toronto](https://reader036.vdocuments.net/reader036/viewer/2022081512/5681681e550346895dddadcc/html5/thumbnails/1.jpg)
Communication Complexity, Information Complexity and
Applications to Privacy
Toniann PitassiUniversity of Toronto
![Page 2: Communication Complexity, Information Complexity and Applications to Privacy Toniann Pitassi University of Toronto](https://reader036.vdocuments.net/reader036/viewer/2022081512/5681681e550346895dddadcc/html5/thumbnails/2.jpg)
2-Party Communication Complexity
[Yao]2-party communication: each party has a dataset.
Goal is to compute a function f(DA,DB)m1
m2
m3
mk-1
mk
DA
x1
x2
xn
DB
y1
y2
ym
f(DA,DB) f(DA,DB)
Communication complexity of a protocol for f is the number of bits exchanged between A and B.
In this talk, all protocols are assumed to be randomized.
![Page 3: Communication Complexity, Information Complexity and Applications to Privacy Toniann Pitassi University of Toronto](https://reader036.vdocuments.net/reader036/viewer/2022081512/5681681e550346895dddadcc/html5/thumbnails/3.jpg)
Deterministic Protocols• A deterministic protocol Π specifies:– Function of board contents:• if the protocol is over• if YES, the output• if NO, which player writes next
– Function of board contents and input available to player P:• what P writes
• Cost of Π = max number of bits written on the board over all inputs
![Page 4: Communication Complexity, Information Complexity and Applications to Privacy Toniann Pitassi University of Toronto](https://reader036.vdocuments.net/reader036/viewer/2022081512/5681681e550346895dddadcc/html5/thumbnails/4.jpg)
Randomized Protocols• In a randomized protocol Π, what
player P writes is also a function of the (private and/or public) random string available to P
• Protocol allowed to err with probability ε over choice of random strings
• The cost of Π = max number of bits written on the board, over inputs and random strings
![Page 5: Communication Complexity, Information Complexity and Applications to Privacy Toniann Pitassi University of Toronto](https://reader036.vdocuments.net/reader036/viewer/2022081512/5681681e550346895dddadcc/html5/thumbnails/5.jpg)
Communication Complexity
• Focus on randomized communication complexity: CC(F,ε) = the communication cost of computing F with error ε.
• A distributional flavor of randomized communication complexity: CC(F,μ,ε) = the communication cost of computing F with error ε with respect to μ.
• Yao’s minimax: CC(F,ε)=maxμ CC(F,μ,ε). 5
![Page 6: Communication Complexity, Information Complexity and Applications to Privacy Toniann Pitassi University of Toronto](https://reader036.vdocuments.net/reader036/viewer/2022081512/5681681e550346895dddadcc/html5/thumbnails/6.jpg)
Stunning variety of applications of CC Lower Bounds
1. Lower Bounds for Streaming Algorithms2. Data Structure Lower Bounds3. Proof Complexity Lower Bounds4. Game Theory5. Circuit Complexity Lower Bounds6. Quantum Computation7. Differential Privacy8. ……
![Page 7: Communication Complexity, Information Complexity and Applications to Privacy Toniann Pitassi University of Toronto](https://reader036.vdocuments.net/reader036/viewer/2022081512/5681681e550346895dddadcc/html5/thumbnails/7.jpg)
2-Party Information Complexity2-party communication: each party has a
dataset. Goal is to compute a function f(DA,DB)m1
m2
m3
mk-1
mk
DA
x1
x2
xn
DB
y1
y2
ym
f(DA,DB) f(DA,DB)
Information complexity of a protocol for f is the amount of information the players reveal to each other / or to an eavesdropper (Eve)
![Page 8: Communication Complexity, Information Complexity and Applications to Privacy Toniann Pitassi University of Toronto](https://reader036.vdocuments.net/reader036/viewer/2022081512/5681681e550346895dddadcc/html5/thumbnails/8.jpg)
Information Complexity[Chakrabarti,Shi,Wirth,Yao ‘01],
[Bar-Yossef,Jayram,Kumar,Sivakumar ‘04]
Entropy: H(X) = Σx p(x) log (1/p(x)Conditional entropy: H(X|Y) = Σy H(X|Y=y)
p(Y=y)Mutual Information: I(X;Y) = H(X) - H(X|Y)
External IC: information about XY revealed to EveICext (π,μ) = I(XY;π)
ICext (f,μ,ε) = maxπ ICext(π,μ)Internal IC: information revealed to Alice and Bob
ICint (π,μ) = I(X;π|Y) + I(Y;π|X)ICint (f,μ,ε) = maxπ ICint (π,μ)
![Page 9: Communication Complexity, Information Complexity and Applications to Privacy Toniann Pitassi University of Toronto](https://reader036.vdocuments.net/reader036/viewer/2022081512/5681681e550346895dddadcc/html5/thumbnails/9.jpg)
Why study information complexity?
• Intrinsically interesting quantity• Related to longstanding questions in
complexity theory (direct sum conjecture)
• Very useful when studying privacy, and quantum computation
![Page 10: Communication Complexity, Information Complexity and Applications to Privacy Toniann Pitassi University of Toronto](https://reader036.vdocuments.net/reader036/viewer/2022081512/5681681e550346895dddadcc/html5/thumbnails/10.jpg)
Simple Facts about Information Complexity
• External information cost is greater than internal: ICext(π,μ) ≥ ICint (π,μ) ICext(π) = I(XY;π)
= I(X;π) + I(Y;π | X) ≥ I(X;π|Y) + I(Y; π | X)= ICint (π)
• Information complexity lower bounds imply Communication Complexity lower bounds:CC(f,μ,ε) ≥ ICext(f,μ,ε) ≥ ICint (f,μ,ε)
![Page 11: Communication Complexity, Information Complexity and Applications to Privacy Toniann Pitassi University of Toronto](https://reader036.vdocuments.net/reader036/viewer/2022081512/5681681e550346895dddadcc/html5/thumbnails/11.jpg)
Do CC Lower Bounds imply IC Lower Bounds? (i.e., CC=IC?)
• For constant-round protocols, IC and CC are basically equal [CSWY, JRS]
• Open for general protocols. • Significant step for general case by [BBCR]
![Page 12: Communication Complexity, Information Complexity and Applications to Privacy Toniann Pitassi University of Toronto](https://reader036.vdocuments.net/reader036/viewer/2022081512/5681681e550346895dddadcc/html5/thumbnails/12.jpg)
Compressing Interactive Communication[Barak,Braverman,Chen,Rao]
Theorem 1For any distribution μ, any C-bit protocol of
internal IC I can be simulated by a new protocol using O(√(CI) logC) bits.
Theorem 2For any product distribution μ, any C-bit
protocol of internal IC I can be simulated by a protocol using O(I logC) bits.
![Page 13: Communication Complexity, Information Complexity and Applications to Privacy Toniann Pitassi University of Toronto](https://reader036.vdocuments.net/reader036/viewer/2022081512/5681681e550346895dddadcc/html5/thumbnails/13.jpg)
Connection to the Direct Sum Problem
Does it take m times the amount of resources to solve m instances?
• Direct Sum Question for CC: CC(fm) ≥ m CC(f) for every f and every
distribution? - Each copy should have error ε
• For search problems, the direct sum problem is equivalent to separating NC1 from P !
![Page 14: Communication Complexity, Information Complexity and Applications to Privacy Toniann Pitassi University of Toronto](https://reader036.vdocuments.net/reader036/viewer/2022081512/5681681e550346895dddadcc/html5/thumbnails/14.jpg)
Connection to the Direct Sum Problem, 2
• The direct sum property holds for information
complexity: Lemma [Direct Sum for IC]: IC(fm) ≥ m IC(f)
• Best general direct sum theorem known for cc: Theorem [Barak,Braverman,Chen,Rao]:
CC(fm) ≥ √m CC(f) ignoring polylog factors
• The direct sum property for cc is equivalent to IC=CC! Theorem [Braverman,Rao]:
IC(f,μ,ε) = limn ∞ CC(Fn, μn,ε)/n
![Page 15: Communication Complexity, Information Complexity and Applications to Privacy Toniann Pitassi University of Toronto](https://reader036.vdocuments.net/reader036/viewer/2022081512/5681681e550346895dddadcc/html5/thumbnails/15.jpg)
Methods for Proving CC and IC Lower Bounds
Jain and Klauck initiated the formal study of CC lower bound methods: all formalizable as solutions to (different) LPs
• Discrepancy Method, Smooth Discrepancy Method
• Rectangle Bound, Smooth Rectangle Bound
• Partition Bound
![Page 16: Communication Complexity, Information Complexity and Applications to Privacy Toniann Pitassi University of Toronto](https://reader036.vdocuments.net/reader036/viewer/2022081512/5681681e550346895dddadcc/html5/thumbnails/16.jpg)
The Partition Bound [Jain, Klauck]
Min Σz,R wz,R
∀ (x,y) Σ R, (x,y) ϵ R wf(x,y),R ≥ 1-ε∀ (x,y) ΣR, (x,y) in R Σz wz,R = 1∀ z,R w z,R ≥ 0
![Page 17: Communication Complexity, Information Complexity and Applications to Privacy Toniann Pitassi University of Toronto](https://reader036.vdocuments.net/reader036/viewer/2022081512/5681681e550346895dddadcc/html5/thumbnails/17.jpg)
Relationships
The Partition bound is greater than or equal to all known CC lower bounds methods, including:
• Discrepancy• Generalized Discrepancy• Rectangle• Smooth Rectangle
![Page 18: Communication Complexity, Information Complexity and Applications to Privacy Toniann Pitassi University of Toronto](https://reader036.vdocuments.net/reader036/viewer/2022081512/5681681e550346895dddadcc/html5/thumbnails/18.jpg)
• [KLLR] define the relaxed partition bound. The relaxed partition bound is greater than or equal to all known CC lower bound methods (except the partition bound).
• They show that the relaxed Partition bound is equivalent to designing a zero-communication protocol with error exp(-I)
• Given a protocol for f with ICint = I, they construct a zero-communication protocol st (i) non-abort probability is exp(-I), and (ii) if it does not abort, it computes f correctly whp
All known CC Lower Bound Methods Imply IC Lower Bounds!
[Kerenidis, Laplante, Lerays, Roland Xiao ‘12]
![Page 19: Communication Complexity, Information Complexity and Applications to Privacy Toniann Pitassi University of Toronto](https://reader036.vdocuments.net/reader036/viewer/2022081512/5681681e550346895dddadcc/html5/thumbnails/19.jpg)
Applications of Information Complexity
• Differential Privacy
• PAR
![Page 20: Communication Complexity, Information Complexity and Applications to Privacy Toniann Pitassi University of Toronto](https://reader036.vdocuments.net/reader036/viewer/2022081512/5681681e550346895dddadcc/html5/thumbnails/20.jpg)
Applications of Information Complexity
• Differential Privacy
• PAR
![Page 21: Communication Complexity, Information Complexity and Applications to Privacy Toniann Pitassi University of Toronto](https://reader036.vdocuments.net/reader036/viewer/2022081512/5681681e550346895dddadcc/html5/thumbnails/21.jpg)
Differential Privacy: The Basic Scenario[Dwork, McSherry, Nissim, Smith 06]
• Database with rows x1..xn
• Each row corresponds to an individual in the database
• Columns correspond to fields, such as “name”, “zip code”; some fields contain sensitive information.
Goal: Compute and release information about a sensitive database without revealing information about any individual
Sanitizer
OutputData
![Page 22: Communication Complexity, Information Complexity and Applications to Privacy Toniann Pitassi University of Toronto](https://reader036.vdocuments.net/reader036/viewer/2022081512/5681681e550346895dddadcc/html5/thumbnails/22.jpg)
Differential Privacy [Dwork,McSherry,Nissim,Smith 2006]
Y
Pr [response]
ratio bounded
Q = space of queries; Y = output space; X = row space
Mechanism M: Xn x Q Y is -differentially private if: for all q in Q, for all adjacent x, x’ in Xn, the distributions M(x,q), M(x’,q) are similar: ∀ y in Y, q in Q:
e -𝜀 ≤ Pr[M(x,q) =y] ≤ eε
Pr[M(x’,q)=y]Note: Randomness is crucial
![Page 23: Communication Complexity, Information Complexity and Applications to Privacy Toniann Pitassi University of Toronto](https://reader036.vdocuments.net/reader036/viewer/2022081512/5681681e550346895dddadcc/html5/thumbnails/23.jpg)
23
Achieving DP: Add Laplacian Noise
f = maxD,D’ |f(D) – f(D’)|
0 b 2b 3b 4b 5b-b-2b-3b-4b
Theorem: To achieve -differential privacy, add symmetric noise [Lap(b)] with b = f/.P(y) ∽ exp(-|y - q(x)|/b)
=exp( - | y – q(x’)| / f )
Pr [M(x, q) = y]Pr [(M(x’, q) = y]
exp( - | y – q(x)| / f ) ≤ exp().
![Page 24: Communication Complexity, Information Complexity and Applications to Privacy Toniann Pitassi University of Toronto](https://reader036.vdocuments.net/reader036/viewer/2022081512/5681681e550346895dddadcc/html5/thumbnails/24.jpg)
Differentially Private Communication Complexity: A Distributed View
Andrews,Mironov,P,Reingold,Talwar,Vadhan
Goal: compute a joint function while maintaining privacy for any individual, with respect to both the outside world and the other database owners.
Multiple databases, each with private data.
D1 D2
D3
D4 D5
F(D1,D2,..,D5)
![Page 25: Communication Complexity, Information Complexity and Applications to Privacy Toniann Pitassi University of Toronto](https://reader036.vdocuments.net/reader036/viewer/2022081512/5681681e550346895dddadcc/html5/thumbnails/25.jpg)
2-Party Differentially Private CC2-party (& multiparty) DP privacy: each party
has a dataset; want to compute a joint function f(DA,DB) m1
m2
m3
mk-1
mk
DA
x1
x2
xn
DB
y1
y2
ym
ZA f(DA,DB) ZB f(DA,DB)
A’s view should be a differentially private function of DB (even if A deviates from protocol), and vice-versa
![Page 26: Communication Complexity, Information Complexity and Applications to Privacy Toniann Pitassi University of Toronto](https://reader036.vdocuments.net/reader036/viewer/2022081512/5681681e550346895dddadcc/html5/thumbnails/26.jpg)
Two-Party Differential PrivacyLet P(x,y) be a 2-party protocol. P is ε-DP if: (1) for all y, for every pair x, x’ that are
neighbors, and for every transcript π, Pr[P(x,y) = π ] ≤ exp(ε) Pr[P(x’,y) = π ](2) symmetrically, for all x, for every pair of
neighbors y,y’ and for every transcript πPr[P(x,y)=π ] ≤ exp(ε) Pr[P(x,y’) = π]
• Privacy and accuracy are the important parameters
![Page 27: Communication Complexity, Information Complexity and Applications to Privacy Toniann Pitassi University of Toronto](https://reader036.vdocuments.net/reader036/viewer/2022081512/5681681e550346895dddadcc/html5/thumbnails/27.jpg)
Examples1. Ones(x,y) = the number of ones in xy Ones(00001111,10101010) = 8.
CC(Ones) = logn. There is a low error DP protocol.
2. Hamming Distance HD(x,y) = the number of positions i where xi ≠ yi.
HD(00001111, 10101010) = 4
CC(HD)=n. No low error DP protocol
Is this a coincidence? Is there a connection between low cc and low-error DP protocols?
![Page 28: Communication Complexity, Information Complexity and Applications to Privacy Toniann Pitassi University of Toronto](https://reader036.vdocuments.net/reader036/viewer/2022081512/5681681e550346895dddadcc/html5/thumbnails/28.jpg)
Information Cost and DP Protocols[McGregor, Mironov, P,Reingold,Talwar,Vadhan]
Lemma. If π has ε-DP, then for every distribution μ on XY, IC(π,μ,ε) ≤ 3εn
Proof sketch: For every z,z’, by ε-DP, exp(-2εn) ≤ Pr[π(z) = π]/Pr[π(z’)=π] ≤ exp(2εn)
I(π(Z); X) = H(π(Z)) – H(π(Z) | Z) = Exp{z,π} log[ Pr[π(Z)=π | Z=z] / Pr[π(Z)=π] ] ≤ 2 (log ε) εn
DP Partition Theorem. Let P be an ε-DP protocol for a partial function with error at most γ. Then log prtγ(f) ≤ 3 ε n
![Page 29: Communication Complexity, Information Complexity and Applications to Privacy Toniann Pitassi University of Toronto](https://reader036.vdocuments.net/reader036/viewer/2022081512/5681681e550346895dddadcc/html5/thumbnails/29.jpg)
Lower Bound:Hamming Distance[McGregor, Mironov, P,Reingold,Talwar,Vadhan]
Gap Hamming: GHD(x,y) = 1 if HD(x,y) > n/2 + √n 0 if HD(x,y) < n/2 – √n
Theorem. Any ε-DP protocol for Hamming distance must incur an additive error Ω(√n).
Note: This lower bound is tight.Proof sketch: [Chakrabarti-Regev 2012] prove: CC(GHD,μ,1/3) = Ω (n). Proof shows GHD has a smooth rectangle bound of
2Ω(n). By Jain-Klauck, this implies that the partition
bound for GHD is at least as large. Thus proof follows by DP Partition Theorem.
![Page 30: Communication Complexity, Information Complexity and Applications to Privacy Toniann Pitassi University of Toronto](https://reader036.vdocuments.net/reader036/viewer/2022081512/5681681e550346895dddadcc/html5/thumbnails/30.jpg)
Implications of Lower bound for Hamming Distance
1. Separation between ε-DP protocols and computational ε-DP protocols [MPRV]:Hamming distance has an O(1) error computational ε-DP protocol, but any ε-DP protocol has error √n. We also exhibit another function with linear separation. (Any ε-DP protocol has error Ωn)
2. Pan Privacy: Our lower bound for Hamming Distance implies lower bounds for pan-private streaming algorithms
![Page 31: Communication Complexity, Information Complexity and Applications to Privacy Toniann Pitassi University of Toronto](https://reader036.vdocuments.net/reader036/viewer/2022081512/5681681e550346895dddadcc/html5/thumbnails/31.jpg)
Pan-Private Streaming Model [Dwork,P,Rothblum, Naor,Yekhanin]
• Data is a stream of items; each item belongs to a user. Sanitizer sees each item and updates internal state. Generates output at end of the stream (single pass).
state
Pan-Privacy: For every two adjacent streams, at any single point in time, the internal state (and final output) are differentially private.
![Page 32: Communication Complexity, Information Complexity and Applications to Privacy Toniann Pitassi University of Toronto](https://reader036.vdocuments.net/reader036/viewer/2022081512/5681681e550346895dddadcc/html5/thumbnails/32.jpg)
What statistics have pan-private algorithms?
We give pan-private streaming algorithms for:• Stream density / number of distinct
elements• t-cropped mean: mean, over users, of min(t,
#appearances)• Fraction of users appearing exactly k times • Fraction of users appearing exactly 0 times
modulo k • Fraction of heavy-hitters, users appearing at
least k times
![Page 33: Communication Complexity, Information Complexity and Applications to Privacy Toniann Pitassi University of Toronto](https://reader036.vdocuments.net/reader036/viewer/2022081512/5681681e550346895dddadcc/html5/thumbnails/33.jpg)
Pan Privacy lower bounds via ε-DP lower bounds• Lower Bounds for ε-DP communication
protocols imply pan privacy lower bounds for density estimation (via Hamming distance lower bound).
• Lower bounds also hold for multi-pass pan-private models
• Analogy: 2-party communication complexity lower bounds imply lower bounds in streaming model.
![Page 34: Communication Complexity, Information Complexity and Applications to Privacy Toniann Pitassi University of Toronto](https://reader036.vdocuments.net/reader036/viewer/2022081512/5681681e550346895dddadcc/html5/thumbnails/34.jpg)
DP Protocols and CompressionSo back to Ones(x,y) and HD(x,y)...is DP the same as
compressible?
Theorem. [BBCR] (Low Icost implies compression) For every product distribution μ, and protocol P, there exists a
protocol Q (β-approximating P) with comm. complexity ∼ Icostμ(P) x polylog(CC(P))/β
Corollary. (DP protocols can be compressed) Let P be an ε-DP protocol P. Then there exists a protocol Q of cost
3εn polylog(CC(P))/β and error β.
DP almost implies low cc, except for this annoying polylog(CC(P)) factor
Moreover, the low cc protocol can often be made DP (if the number of rounds is bounded.)
![Page 35: Communication Complexity, Information Complexity and Applications to Privacy Toniann Pitassi University of Toronto](https://reader036.vdocuments.net/reader036/viewer/2022081512/5681681e550346895dddadcc/html5/thumbnails/35.jpg)
Differential Privacy andCompression
• We have seen that DP protocols have low information cost
• By BBCR this implies they can be compressed (and thus have low comm complexity)
What about the other direction? Can functions with low cc be made DP?
Yes! (with some caveats..the error is proportional not only to the cc, but also the number of rounds.)Proof uses the exponential mechanism [MT]
![Page 36: Communication Complexity, Information Complexity and Applications to Privacy Toniann Pitassi University of Toronto](https://reader036.vdocuments.net/reader036/viewer/2022081512/5681681e550346895dddadcc/html5/thumbnails/36.jpg)
Applications of Information Complexity
• Differential Privacy
• PAR
![Page 37: Communication Complexity, Information Complexity and Applications to Privacy Toniann Pitassi University of Toronto](https://reader036.vdocuments.net/reader036/viewer/2022081512/5681681e550346895dddadcc/html5/thumbnails/37.jpg)
37
Approximate Privacy in Mechanism Design
• Traditional goal of mechanism design: Incent agents to reveal private information that is needed to compute optimal results.
• Complementary, newly important goal: Enable agents not to reveal private information that is not needed to compute optimal results.
• Example (Naor-Pinkas-Sumner, EC ’99): It’s undesirable for the auctioneer to learn the winning bid in a 2nd–price Vickrey auction.
![Page 38: Communication Complexity, Information Complexity and Applications to Privacy Toniann Pitassi University of Toronto](https://reader036.vdocuments.net/reader036/viewer/2022081512/5681681e550346895dddadcc/html5/thumbnails/38.jpg)
38
Perfect Privacy [Kushilevitz ’92]
• Protocol P for f is perfectly private iff for all x,x’,y,y’ f(x,y)=f(x’,y’) R(x,y)=R(x’,y’)
• f is perfectly privately computable iff M(f) has no forbidden submatrix
f(x1, x2) = f(x’1, x2) = f(x’1, x’2) = a, but f(x1, x’2) ≠ a
x1
x’1
X2 X’2
![Page 39: Communication Complexity, Information Complexity and Applications to Privacy Toniann Pitassi University of Toronto](https://reader036.vdocuments.net/reader036/viewer/2022081512/5681681e550346895dddadcc/html5/thumbnails/39.jpg)
39
Example 1: Millionaires’ Problem(not perfectly privately computable)
0
1
2
3
0 1 2 3
millionaire 1
millionaire 2
A(f)
f(x1, x2) = 1 if x1 ≥ x2 ; else f(x1, x2) = 2
![Page 40: Communication Complexity, Information Complexity and Applications to Privacy Toniann Pitassi University of Toronto](https://reader036.vdocuments.net/reader036/viewer/2022081512/5681681e550346895dddadcc/html5/thumbnails/40.jpg)
40
Example 2: Vickrey Auction[Brandt, Sandholm]
2, 1
winnerprice
2, 01, 0
1, 1
1, 2 2, 2
1, 3
0
1
2
3
bidder 1
bidder 2 0 1 2 3
RI (2, 0)
•The ascending-price, English auction protocol is the unique perfectly private protocol
•However the communication cost is exponential !!
![Page 41: Communication Complexity, Information Complexity and Applications to Privacy Toniann Pitassi University of Toronto](https://reader036.vdocuments.net/reader036/viewer/2022081512/5681681e550346895dddadcc/html5/thumbnails/41.jpg)
41
Worst-case PAR[Feigenbaum, Jaggard,Schapira ‘10]
• Worst-case privacy approximation ratio of a protocol π for f:
PAR(f,π) = max x,y | P(x,y)|/ |R(x,y)|,
P(x,y): set of all pairs (x’,y’) st f(x,y)=f’(x’,y’)R(x,y): rectangle containing (x,y) induced by π
• Worst-case PAR of f:
PAR(f) = min π PAR(f,π)
![Page 42: Communication Complexity, Information Complexity and Applications to Privacy Toniann Pitassi University of Toronto](https://reader036.vdocuments.net/reader036/viewer/2022081512/5681681e550346895dddadcc/html5/thumbnails/42.jpg)
42
Average-case PAR[Feigenbaum, Jaggard, Schapira ‘10]
(1) Average-case PAR of π:
AvgPAR1(f,π) = log E(x,y) |P(x,y)|/|R(x,y)|AvgPAR1(f) = minπ AvgPAR(f,π)
(2) Alternative definition:
AvgPAR2(f,π) = I(XY; π | f) = E(x,y) log |P(x,y)/|R(x,y)|AvgPAR2(f) = minπ AvgPAR2(f,π)
• 1 is log of Expectation, 2 is Expectation of log.• For boolean functions, AvgPAR2(f) is basically the same
as Icost(f) (differs by at most 1).
![Page 43: Communication Complexity, Information Complexity and Applications to Privacy Toniann Pitassi University of Toronto](https://reader036.vdocuments.net/reader036/viewer/2022081512/5681681e550346895dddadcc/html5/thumbnails/43.jpg)
43
New Results[Ada,Chattopadhyay,Cook,Fontes,P ‘12]
(1) Using the fact that AvgPAR1 ≥ AvgPAR2, together with known IC lower bounds:Theorem AvgPAR2 of set intersection is Ω(n)
(2) We prove strong tradeoffs for both worst-case PAR and avgPAR for Vickrey auctions.
(3) Using compression [BBCR], it follows that any deterministic, low AvgPAR1 protocols can be compressed. Thus binary search protocol for millionaires implies a polylogn randomized protocol.
![Page 44: Communication Complexity, Information Complexity and Applications to Privacy Toniann Pitassi University of Toronto](https://reader036.vdocuments.net/reader036/viewer/2022081512/5681681e550346895dddadcc/html5/thumbnails/44.jpg)
Important Open Questions
• IC=CC?• IC in the multiparty NOF setting• IC lower bounds for search
problems Very important for proof complexity and circuit complexity
• Other applications of ICData structures? Game Theory?
![Page 45: Communication Complexity, Information Complexity and Applications to Privacy Toniann Pitassi University of Toronto](https://reader036.vdocuments.net/reader036/viewer/2022081512/5681681e550346895dddadcc/html5/thumbnails/45.jpg)
Thanks!