week7 laplace oct14 - archive.ece.cmu.eduece734/fall2016/recitations/week7_laplace_oct14.pdf ·...
TRANSCRIPT
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Administrative
• HW2 will be graded and statistics posted by the end of today
• HW3 due next Friday• Recitation next Friday cancelled due to the mid-‐semester break
• Office hours next Friday cancelled because I’m out of town– Find me by next Monday for in-‐person questions on the homework
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HW2 Statistics
Mean 86Median 86
Standard Deviation 5.5Max 94Min 73
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18734 Recitation
Differential Privacy and theLaplace Mechanism
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Database Sanitization
Noise injection
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Differential Privacy
Does smoking increase your chances of getting cancer?
Control
Experimental
Does smoking increase your chances of getting cancer?
Control
Experimental
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Differential Privacy: Definition
xi X’i
Dataset X
Function K runs on X to produce K(X)
Dataset X’
K is ε-‐differentially privacy if for any two neighbor datasets X, X’ that differ in just one element at position i:
For all outcomes v,
𝑒"# ≤ %& () * +,)%& () *. +,)
≤ 𝑒#
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Approximate Differential Privacy: Definition
xi X’i
Dataset X
Function K runs on X to produce K(X)
Dataset X’
K has ε-‐differentially privacy if for any two neighbor datasets X, X’ that differ in just one element at position i:
For all outcomes v,
1 − 𝜀 ≈ 𝑒"# ≤ %& ) * +,%& ) *. +,
≤ 𝑒# ≈ 1 + 𝜀
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Laplace Mechanism
8Slide: Adam Smith
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Laplace Distribution
9
PDF =
Source: Wikipedia
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Laplace Distribution
10
Change of notation from previous slide:μà 0b à λ
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Laplace Mechanism: Proof Idea
11
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Laplace Mechanism: Proof
• To Prove: Pr[A(x)∈ S ] ≤ eε Pr[A(x’)∈ S ]
– A(x) = f(x) + Laplace(GSf/ε)– A(x’) = f(x’) + Laplace(GSf/ε)
• Steps:– Distribution of A(x): Laplace(f(x), GSf/ε)– Distribution of A(x’): Laplace(f(x’), GSf/ε)
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Laplace Mechanism: Proof