Gaussian Noise Mechanism Sensitivity, again The ` 2 sensitivity of - PowerPoint PPT Presentation

Gaussian Noise Mechanism

⇒ Sensitivity, again The ` 2 sensitivity of f : X n ! R k is ! 1 / 2 k X X ⇠ X 0 k f ( X ) � f ( X 0 ) k 2 = max | f ( X ) i � f ( X 0 ) i | 2 ∆ 2 f = max X ⇠ X 0 i =1 E IRK 112-14 E 112-42 tf . z D2 f EA , f- 17

Sensitivity of a workload of counting queries, again counting . % queries g. are - ( . . . . . % ! " ) bae FL QCH - gun " EE ' ) - ifaf ( II Hill ) - gilt 'll QQ n - - ±i I m 18

Gaussian noise mechanism The Gaussian noise mechanism M Gauss (for a function f : X n ! R k ) outputs are independent Zi . , Zk . . . . M Gauss ( X ) = f ( X ) + Z , Gaussian s a parameter , to be decided is g where Z 2 R k is sampled from N ⇣ 0 , ( ∆ 2 f ) 2 ⌘ · I . " , ) ( ρ Ts identity matrix ' l N ( µ, Σ ) is the Gaussian distribution on R k with expectation µ 2 R k and covariance matrix Σ . When Σ = � 2 I , it has pdf k ! 1 1 � 1 2 / (2 σ 2 ) = (2 ⇡ ) k / 2 � k e �k z � µ k 2 X | z i � µ i | 2 p ( z ) = (2 ⇡ ) k / 2 � k exp 2 � 2 i =1 19

Approximate Di ff erential Privacy w ' " \ Problem: Gaussian tails drop o ff too fast! M Gauss is not " -DP for any " < 1 . - ratio be too large It satisfies a relaxed privacy definition. Definition I ( A mechanism M is " , � -di ff erentially private if, for any two neighbouring datasets X , X 0 , and any set of outputs S P ( M ( X ) 2 S )  e ε P ( M ( X 0 ) 2 S ) + � a- We will ask th JK that do not allow that so we , " name mechanism " and shame 20

Privacy of the Gaussian noise mechanism , di To get CE DP - " erk 2 ' " → #I* 0 , ( ∆ 2 f ) 2 ✓ ◆ M Gauss ( X ) = f ( X ) + Z , Z ⇠ N · I . ⇢ p ρ 2 ( p ⇢ + IT p For any � > 0, M Gauss is ( " , � )-DP for " = 2 ln(1 / � )). → 2 of Ngau , ft ) i pdf llqauss ( XY of Pdf play Xxl : plz ) Claim: enough to show that, for T = { z 2 R k : p ( z ) p 0 ( z ) > e ε } , P ( M ( X ) 2 T )  � . Gauss Isn bad set of PL Nam , CHESNEY k too much ) " I reveal outputs SE IR c- SIT ) t = Plllaaussltl c- S ) Hts ) P( Mams , K ) feels , ,piZId ← plMaansd 2- Id Z tf Is * P' ± t f ) EST ) ' ee plllaaus , It 21 =

Privacy of the Gaussian noise mechanism - ¥449 - Sz :eIf¥ " > e } UNHR tu , Y II ; T . , - p 0 ( z ) = ⇢ · k f ( X ) � f ( X 0 ) k 2 + ⇢ · h z � f ( X ) , f ( X ) � f ( X 0 ) i ln p ( z ) 2 2( ∆ 2 f ) 2 ( ∆ 2 f ) 2 ' ) ) - fix - fit ) , HH gtz ± § + T er - { + ME E - fix 'll - > raw } , f- CX ) - HH : gtz 1- c. { z - ( Dzf ) ' 22

Privacy of the Gaussian noise mechanism ' Iii Zi Eui 1. For any v 2 R k , and Z ⇠ N (0 , � 2 I ), h Z , v i ⇠ N (0 , � 2 k v k 2 2 ). P ( Z > t ) < e � t 2 / (2 σ 2 ) . 2. Z ⇠ N (0 , � 2 ), then I ) Z - NIO , + Z - HX ) Maansslx ) Then - - D ! ⇢ · h Z , f ( X ) � f ( X 0 ) i p 2 ⇢ ln(1 / � ) P ( M Gauss ( X ) 2 T )  P > . 2( ∆ 2 f ) 2 2 HtlH-f¥ are £ ^5 ' NCO ,r2 ) ' -6ft $ Meg > Eto ) < of PIG ' ) G~N( 0,8 r 't g 23

Accuracy of the Gaussian noise mechanism P ( | Z � µ | > t ) < 2 e � t 2 / (2 σ 2 ) . Z ⇠ N ( µ, � 2 ), then - queries , with set sat . counting g k for Exercise : we have 6,81 - DP satisfies Naans , , PL mot error 22 ) Ep no , Fg it E d 24