# L4-Intro to Differential Privacy

## 拉普拉斯机制（Laplace Mechanism）

### 定义1

$\Delta^{(f)}=\max_{X,X'}||f(X)-f(X')||_1$

### 定义2

$p(x)=\frac{1}{2b}\exp(-\frac{|x|}{b})$

### 定义3

$M(X)=f(X)+(Y_1,\dots,Y_k)$

### 定理4

\begin{align} \frac{p_X(z)}{p_Y(z)} & = \frac{\prod^k_{i=1}\exp(-\frac{\epsilon|f(X)_i-z_i|}{\Delta})} {\prod^k_{i=1}\exp(-\frac{\epsilon|f(Y)_i-z_i|}{\Delta})} \\ & = \prod^k_{i=1}\exp(-\frac{\epsilon(|f(X)_i-z_i|-|f(X)_i-z_i|)}{\Delta}) \\ & \le \prod^k_{i=1}\exp(-\frac{\epsilon|f(X)_i-f(X)_i|}{\Delta}) \\ & = \exp(\frac{\epsilon\sum^k_{i=1}|f(X)_i-f(X)_i|}{\Delta}) \\ & = \exp(\frac{\epsilon||f(X)-f(X)||_1}{\Delta}) \\ & \le \exp(\epsilon) \end{align}

## 直方图（Histograms）

### 公理5.

$\mathbf{Pr}[|Y|\ge tb]=\exp(-t)$

## 差分隐私的性质（Properties of Differential Privacy）

### 定理6.

\begin{align} \mathbf{Pr}[F(M(X))\in T] & = \mathbf{E}_{f\sim F}[\mathbf{Pr}[M(X)\in f^{-1}(M)]] \\ & \le\mathbf{E}_{f\sim F}[e^{\epsilon}\mathbf{Pr}[M(X')\in f^{-1}(M)]] \\ & = e^{\epsilon}\mathbf{Pr}[F(M(X'))\in T] \end{align}

### 定理7.

\begin{align} \mathbf{Pr}[M(X)\in T] & \le \exp(k\epsilon)\mathbf{Pr}[M(X')\in T] \end{align}

\begin{align} \mathbf{Pr}[M(X^{(0)})\in T] & \le e^\epsilon \mathbf{Pr}[M(X^{(1)})\in T] \\ & \le e^{2\epsilon} \mathbf{Pr}[M(X^{(2)})\in T] \\ & \cdots\\ & \le e^{k\epsilon} \mathbf{Pr}[M(X^{(k)})\in T] \\ \end{align}

## 基本组合性（ (Basic) Composition ）

\begin{align} \frac{\mathbf{Pr}[M(X)=y]}{\mathbf{Pr}[M(X')=y]} & = \prod^k_{i=1}\frac{\mathbf{Pr}[M_i(X)=y_i|(M_1(X),\dots,M_{i-1}(X))=(y_1,\dots,y_k)]} {\mathbf{Pr}[M_i(X')=y_i|(M_1(X'),\dots,M_{i-1}(X'))=(y_1,\dots,y_k)]} \\ & \le \prod^k_{i=1} \exp(\epsilon) \\ & = \exp(k\epsilon) \end{align}

posted @ 2021-03-07 21:15  Uzuki  阅读(537)  评论(0编辑  收藏  举报