Randomized Algorithms (Spring 2010)/Martingales

Martingales

Martingales and Azuma's Inequality

Azuma's Inequality: Let [math]\displaystyle{ X_0,X_1,\ldots }[/math] be a martingale such that, for all [math]\displaystyle{ k\ge 1 }[/math], [math]\displaystyle{ |X_{k}-X_{k-1}|\le c_k, }[/math] Then [math]\displaystyle{ \begin{align} \Pr\left[|X_n-X_0|\ge t\right]\le 2\exp\left(-\frac{t^2}{2\sum_{k=1}^nc_k^2}\right). \end{align} }[/math]

Corollary: Let [math]\displaystyle{ X_0,X_1,\ldots }[/math] be a martingale such that, for all [math]\displaystyle{ k\ge 1 }[/math], [math]\displaystyle{ |X_{k}-X_{k-1}|\le c, }[/math] Then [math]\displaystyle{ \begin{align} \Pr\left[|X_n-X_0|\ge ct\sqrt{n}\right]\le 2 e^{-t^2/2}. \end{align} }[/math]

Generalizations

Azuma's Inequality (general version): Let [math]\displaystyle{ Y_0,Y_1,\ldots }[/math] be a martingale with respect to the sequence [math]\displaystyle{ X_0,X_1,\ldots }[/math] such that, for all [math]\displaystyle{ k\ge 1 }[/math], [math]\displaystyle{ |Y_{k}-Y_{k-1}|\le c_k, }[/math] Then [math]\displaystyle{ \begin{align} \Pr\left[|Y_n-Y_0|\ge t\right]\le 2\exp\left(-\frac{t^2}{2\sum_{k=1}^nc_k^2}\right). \end{align} }[/math]

The Method of Bounded Differences

Theorem (The method of averaged bounded differences): Let [math]\displaystyle{ X=(X_0,X_1,\ldots, X_n) }[/math] be arbitrary random variables and let [math]\displaystyle{ f }[/math] be a function of [math]\displaystyle{ X_0,X_1,\ldots, X_n }[/math] satisfying that, for all [math]\displaystyle{ 1\le i\le n }[/math], [math]\displaystyle{ |\mathbf{E}[f(X)\mid X_1,\ldots,X_i]-\mathbf{E}[f(X)\mid X_1,\ldots,X_{i-1}]|\le c_i, }[/math] Then [math]\displaystyle{ \begin{align} \Pr\left[|f(X)-\mathbf{E}[f(X)]|\ge t\right]\le 2\exp\left(-\frac{t^2}{2\sum_{i=1}^nc_i^2}\right). \end{align} }[/math]

Corollary (Method of bounded differences): Let [math]\displaystyle{ X=(X_0,X_1,\ldots, X_n) }[/math] be independent random variables and let [math]\displaystyle{ f }[/math] be a function satisfying the Lipschitz condition. Then [math]\displaystyle{ \begin{align} \Pr\left[|f(X)-\mathbf{E}[f(X)]|\ge t\right]\le 2\exp\left(-\frac{t^2}{2n}\right). \end{align} }[/math]