The Exponential Mechanism (and maybe some mechanism design) Frank McSherry and Kunal Talwar Microsoft Research, Silicon Valley Campus

Transcription

1 The Exponential Mechanism (and maybe some mechanism design) Frank McSherry and Kunal Talwar Microsoft Research, Silicon Valley Campus

2 Intentionally Blank Slide. 1

3 Differential Privacy Context: A data set d D N and mechanism M : D N R. Evaluating M(d) shouldn t give specific info about tuples in d. Source of much definitional anxiety for some 30-odd years. What is specific info? Can we prevent everything/anything? Definition: A mechanism M gives ɛ-differential privacy if: For d, d D N differing on at most one datum, and any S R, P r[m(d) S] exp(ɛ) P r[m(d ) S]. Changing one tuple can not change the output distribution much. Relative change in the probability of any event (subset S of R). 1

4 Previous Constructions Simple scheme: Apply f : D N R to data, return noisy result. K f (DB) f(db) + Noise. Database ƒ ` Noise? Theorem: Using Laplace(σ, 0) gives ( f/σ)-differential privacy, f = max max f(db Me) f(db + Me). DB Me For many statistical properties: f is small, small noise benign. 2

5 Problems with Perturbation Pricing: Inputs are n bids in [0, 1]. Output is a price p [0, 1]. Want to make lots of money, but we don t want to reveal bids. Problem: Perturbing the true answer by some noise may fail. 1. The function may have high sensitivity. (eg: Pricing) 2. Perturbations may not actually be useful. (eg: Pricing) Moreover: Additive perturbations also fail when 3. Outputs are not numbers. (eg: strings, trees, etc...) 3

6 A General Mechanism Previously a query was f : D N R, mapping data to result. Implicit assumption that results r near f(d) are nearly as good. Now, a query is q : (D N R) R. Score of result r for data d. Eg: Given bids and a price, revenue is q(d, r) = r #(i : d i > r) Definition: Let E ɛ q(d) output r with probability exp(ɛq(d, r)). 4

7 A General Mechanism Previously a query was f : D N R, mapping data to result. Implicit assumption that results r near f(d) are nearly as good. Now, a query is q : (D N R) R. Score of result r for data d. Eg: Given bids and a price, revenue is q(d, r) = r #(i : d i > r) Definition: Let E ɛ q(d) output r with probability exp(ɛq(d, r)). 4

8 A General Mechanism Previously a query was f : D N R, mapping data to result. Implicit assumption that results r near f(d) are nearly as good. Now, a query is q : (D N R) R. Score of result r for data d. Eg: Given bids and a price, revenue is q(d, r) = r #(i : d i > r) Definition: Let E ɛ q(d) output r with probability exp(ɛq(d, r)). 4

9 Two Exciting Properties Privacy: E ɛ q gives (2ɛ q)-differential privacy, where we define q = max r max d d q(d, r) q(d, r). Proof: Density, normalization alter by factors of at most exp(ɛ q). Utility: For S R, write µ(s) for its base measure. (pre-e ɛ q). Lem: Let S t = {r : q(d, r) > OP T t}. Pr(S 2t ) < exp( t)/µ(s t ). Proof: LHS < Pr(S 2t )/ Pr(S t ) < exp( t)µ(s 2t )/µ(s t ) < RHS. Thm: E[q(d, E q (d))] > OP T 3t, for those t > ln(op T/tµ(S t )). Proof: Pr(OP T 2t) 1 exp( t)/µ(s t ) 1 t/op T. Multiply. 5

10 Two Exciting Properties Privacy: E ɛ q gives (2ɛ q)-differential privacy, where we define q = max r max d d q(d, r) q(d, r). Proof: Density, normalization alter by factors of at most exp(ɛ q). Utility: For S R, write µ(s) for its base measure. (pre-e ɛ q). Lem: Let S t = {r : q(d, r) > OP T t}. Pr(S 2t ) < exp( t)/µ(s t ). Proof: LHS < Pr(S 2t )/ Pr(S t ) < exp( t)µ(s 2t )/µ(s t ) < RHS. Thm: E[q(d, E q (d))] > OP T 3t, for those t > ln(op T/tµ(S t )). Proof: Pr(OP T 2t) 1 exp( t)/µ(s t ) 1 t/op T. Multiply. 5

11 Two Exciting Properties Privacy: E ɛ q gives (2ɛ q)-differential privacy, where we define q = max r max d d q(d, r) q(d, r). Proof: Density, normalization alter by factors of at most exp(ɛ q). Utility: For S R, write µ(s) for its base measure. (pre-e ɛ q). Lem: Let S t = {r : q(d, r) > OP T t}. Pr(S 2t ) exp( t)/µ(s t ). Proof: LHS Pr(S 2t )/ Pr(S t ) exp( t)µ(s 2t )/µ(s t ) RHS. Thm: E[q(d, E q (d))] OP T 3t, for those t ln(op T/tµ(S t )). Proof: Pr(OP T 2t) 1 exp( t)/µ(s t ) 1 t/op T. Multiply. 5

12 Two Exciting Properties Privacy: E ɛ q gives (2ɛ q)-differential privacy, where we define q = max r max d d q(d, r) q(d, r). Proof: Density, normalization alter by factors of at most exp(ɛ q). Utility: For S R, write µ(s) for its base measure. (pre-e ɛ q). Lem: Let S t = {r : q(d, r) > OP T t}. Pr(S 2t ) exp( t)/µ(s t ). Proof: LHS Pr(S 2t )/ Pr(S t ) exp( t)µ(s 2t )/µ(s t ) RHS. Thm: E[q(d, E ɛ q(d))] OP T 3t, for those t ln(op T/tµ(S t )). Proof: Pr(OP T 2t) 1 exp( t)/µ(s t ) 1 t/op T. Multiply. 5

13 Applications to Pricing Every bidder gives a demand curve: d i : [0, 1] R +. (rd i (r) 1) Theorem: Taking q(d, r) = r i d i (r), then the mechanism Eq ɛ gives (2ɛ)-differential privacy, and has expected revenue at least OP T 3 ln(e + ɛ 2 OP T m)/ɛ, where m is the number of items sold at the optimal price. Proof: Grind t = ln(e+ɛ 2 OP T m) through the previous theorem. Argue that µ(s t ) is not small. (near-opt r gives near-opt q(d, r)). 6

14 Game Theory Implications Differential Privacy implies many game-theoretic properties: P r[m(d) S] exp(ɛ) P r[m(d )] S]. ɛ-dominance: For any utility function g : R R +, E[g(M(d))] exp(ɛ) E[g(M(d ))]. Collusion Resilient: For d t d, (ie: differing on t data) P r[m(d) S] exp(ɛt) P r[m(d )] S]. Repeatability: For M = (M 1, M 2,... M t ) with dependencies, P r[m(d) S] exp ( ) ɛ i P r[m(d )] S]. Truthful whp [CKMT]: M can be implemented so that: For all d, t, with prob exp( 2ɛt), M(d) = M(d ) for all d t d. i t 7

15 Conclusions, Future Direction Stuff we did: General mechanism E ɛ q, more robust, awesome than previously. Applications to Auctions/Pricing of various and new flavors. Neat non-truthful solution concept. Cool consequences. Stuff we didn t do / did badly: Computational questions of sampling from E ɛ q efficiently. Going beyond auctions/pricing to other mechanism problems. Thanks! Questions? 8