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

Gaussian Noise Mechanism

Sensitivity, again

The `2 sensitivity of f : X n ! Rk is ∆2f = max

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Gaussian noise mechanism

The Gaussian noise mechanism MGauss (for a function f : X n ! Rk) outputs MGauss(X) = f (X) + Z, where Z 2 Rk is sampled from N ⇣ 0, (∆2f )2

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· I ⌘ . N(µ, Σ) is the Gaussian distribution on Rk with expectation µ 2 Rk and covariance matrix Σ. When Σ = 2I, it has pdf p(z) = 1 (2⇡)k/2k ekzµk2

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Approximate Differential Privacy

Problem: Gaussian tails drop off too fast! MGauss is not "-DP for any " < 1. It satisfies a relaxed privacy definition. Definition A mechanism M is ", -differentially private if, for any two neighbouring datasets X, X 0, and any set of outputs S P(M(X) 2 S)  eεP(M(X 0) 2 S) +

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Privacy of the Gaussian noise mechanism

MGauss(X) = f (X) + Z, Z ⇠ N ✓ 0, (∆2f )2 ⇢ · I ◆ For any > 0, MGauss is (", )-DP for " =

pρ 2 (p⇢ +

p 2 ln(1/)). Claim: enough to show that, for T = {z 2 Rk : p(z)

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Privacy of the Gaussian noise mechanism

ln p(z) p0(z) = ⇢ · kf (X) f (X 0)k2 2(∆2f )2 + ⇢ · hz f (X), f (X) f (X 0)i (∆2f )2

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Privacy of the Gaussian noise mechanism

• 1. For any v 2 Rk, and Z ⇠ N(0, 2I),

hZ, vi ⇠ N(0, 2kvk2

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• 2. Z ⇠ N(0, 2), then

P(Z > t) < et2/(2σ2). Then P(MGauss(X) 2 T)  P ⇢ · hZ, f (X) f (X 0)i 2(∆2f )2 > p 2⇢ ln(1/) 2 !

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Accuracy of the Gaussian noise mechanism

Z ⇠ N(µ, 2), then P(|Z µ| > t) < 2et2/(2σ2).

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