Laplace Noise Mechanism Reminder: Counting queries Given a predicate - PowerPoint PPT Presentation

Laplace Noise Mechanism

Reminder: Counting queries Given a predicate q : X ! { 0 , 1 } (e.g., “smoker?”), we define the corresponding (normalized) counting query n q ( X ) = 1 X q ( x i ) . n i =1 A workload Q of counting queries is given by predicates q 1 , . . . , q k . 0 q 1 ( X ) 1 . A 2 [0 , 1] k . . Q ( X ) = B C . @ q k ( X ) E.g., “smoker?”, “smoker and over 30?”, “smoker and heart disease?”, etc. " smoker , heart disease ? " 30 over , 9

Answering counting queries with Randomized Response k counting How queries can I w/ Dp e answer . ? RR using whole mechanism ) query ) instances RR ( one E- DP by K of per with ( q ) is - Dp composition than All get c- IL queries error answers we I - of with prob n → kIe8l . Exercise : Do the long as as £42 details ! 10

Sensitivity - f } e. g. text QQ The ` 1 sensitivity of f : X n ! R k is k X X ⇠ X 0 k f ( X ) � f ( X 0 ) k 1 = max | f ( X ) i � f ( X 0 ) i | ∆ 1 f = max % neighbouring X ⇠ X 0 → IR i =1 " f :D , if then ns.f-zinqylfkl - f 't 'll e.g Measure of how much a person can influence f . 11

Sensitivity of a workload of counting queries - ( g ! It , ) - alt ) for workload Suppose flx ) a - - of k counting queries Give bound , Q D upper an on gilt ) - t ki = In c- soil 's D , q . ) gil qilxj . . Ei !9i"?ia%"" ' 1a em a. - - . 12

Laplace noise mechanism The Laplace noise mechanism M Lap (for a function f : X n ! R k ) outputs M Lap ( X ) = f ( X ) + Z , are independent . Z , Z where Z 2 R k is sampled from Lap (0 , ∆ 1 f ε ). . . . . . Laplo , - dimensional from Zi is a one Lap ( µ, b ) is the Laplace distribution on R k with expectation µ 2 R k and scale b > 0, and has pdf k ! 1 1 � 1 (2 b ) k e �k z � µ k 1 / b = X p ( z ) = (2 b ) k exp | z i � µ i | b µ wpise-DpTy i =1 13

Privacy of the Laplace noise mechanism THE ;fCX nhapHH.FI/=flHltZ For any f , M Lap is ε -DP. ~ Laplftx 't Let X ∼ X 0 , and let p be the pdf of M ( X ), and p 0 the pdf of M ( X 0 ). . At ) lap Lap ◆ k ◆ k ✓ 1 E ✓ 1 E e � ε k z � f ( X 0 ) k 1 / ∆ 1 f e � ε k z � f ( X ) k 1 / ∆ 1 f p 0 ( z ) = p ( z ) = 2 ∆ 1 f 2 ∆ 1 f → Ethel Claim: enough to show max z 2 R k p ( z ) p 0 ( z ) ≤ e ε ' Henk dZ Else - 1PM IPCMLAPIXIES ) p' czidz ' c- S ) e ' IPCMCX ' ) ee fgptztdz = = . 14

Privacy of the Laplace noise mechanism ' ¥7 , Need Kzn ee ; Y ◆ k E ◆ k ✓ 1 ✓ 1 e � ε k z � f ( X ) k 1 / ∆ 1 f p 0 ( z ) = e � ε k z � f ( X 0 ) k 1 / ∆ 1 f p ( z ) = 2 ∆ 1 f 2 ∆ 1 f , ) ) = exp ( Ff ( UZ - HNK - TH FI ' ) " - Hz e ee , ' IZ ) my p - fit 'm , EHFIX ) ← Triangle inequality - fix 'll ! - fl till , k fit ) - fl X' Ill t HZ E HZ , ¥i 15

Histograms " ' mi . Suppose the query workload Q = { q 1 , . . . , q k } “partitions” the data. • ∀ x ∈ X : at most one of q 1 ( x ) , . . . , q k ( x ) equals 1. a • e.g., “votes for the Liberal Party per riding” Xu 's X of 4 . . . . . What is the sensitivity? F . .tn } ' ' 94 . . ti - = . . . . . , E E , ÷! Iq - gilt 'll - QH 'll ! - K ) " QIN fray finger - - . , values query E 2 because c- In each . change by 16

Accuracy of the Laplace noise mechanism " K Generalizes mechanism to " - norm If Z ∈ R k is a Laplace random variable from Lap ( µ, b ), then, for every i Hardt , Talwar P ( | Z i − µ i | ≥ t ) = e � t / b . - , diet ) - Lap ( fit ) Map ( H at llk.pk/i-fCXIilza1z?zlPllkaplxt.-fctHzH Rl ? - debit error z d) else -1¥ E k . e Minar is , ft kn , then E B - Q ( H f- ( x ) . if E. g , n z Hulking - k counting if answers to i.e. 17 queries he ,