随机算法 (Fall 2011)/Markov's Inequality

When applying probabilistic analysis, we often want a bound in form of [math]\displaystyle{ \Pr[X\ge t]\lt \epsilon }[/math] for some random variable [math]\displaystyle{ X }[/math] (think that [math]\displaystyle{ X }[/math] is a cost such as running time of a randomized algorithm). We call this a tail bound, or a tail inequality.

Besides directly computing the probability [math]\displaystyle{ \Pr[X\ge t] }[/math], we want to have some general way of estimating tail probabilities from some measurable information regarding the random variables.

Markov's Inequality

One of the most natural information about a random variable is its expectation, which is the first moment of the random variable. Markov's inequality draws a tail bound for a random variable from its expectation.

Theorem (Markov's Inequality)

Let [math]\displaystyle{ X }[/math] be a random variable assuming only nonnegative values. Then, for all [math]\displaystyle{ t\gt 0 }[/math], [math]\displaystyle{ \begin{align} \Pr[X\ge t]\le \frac{\mathbf{E}[X]}{t}. \end{align} }[/math]

Proof. Let [math]\displaystyle{ Y }[/math] be the indicator such that [math]\displaystyle{ \begin{align} Y &= \begin{cases} 1 & \mbox{if }X\ge t,\\ 0 & \mbox{otherwise.} \end{cases} \end{align} }[/math] It holds that [math]\displaystyle{ Y\le\frac{X}{t} }[/math]. Since [math]\displaystyle{ Y }[/math] is 0-1 valued, [math]\displaystyle{ \mathbf{E}[Y]=\Pr[Y=1]=\Pr[X\ge t] }[/math]. Therefore, [math]\displaystyle{ \Pr[X\ge t] = \mathbf{E}[Y] \le \mathbf{E}\left[\frac{X}{t}\right] =\frac{\mathbf{E}[X]}{t}. }[/math] [math]\displaystyle{ \square }[/math]

Example (from Las Vegas to Monte Carlo)

Let [math]\displaystyle{ A }[/math] be a Las Vegas randomized algorithm for a decision problem [math]\displaystyle{ f }[/math], whose expected running time is within [math]\displaystyle{ T(n) }[/math] on any input of size [math]\displaystyle{ n }[/math]. We transform [math]\displaystyle{ A }[/math] to a Monte Carlo randomized algorithm [math]\displaystyle{ B }[/math] with bounded one-sided error as follows:

[math]\displaystyle{ B(x) }[/math]:

• Run [math]\displaystyle{ A(x) }[/math] for [math]\displaystyle{ 2T(n) }[/math] long where [math]\displaystyle{ n }[/math] is the size of [math]\displaystyle{ x }[/math].
• If [math]\displaystyle{ A(x) }[/math] returned within [math]\displaystyle{ 2T(n) }[/math] time, then return what [math]\displaystyle{ A(x) }[/math] just returned, else return 1.

Since [math]\displaystyle{ A }[/math] is Las Vegas, its output is always correct, thus [math]\displaystyle{ B(x) }[/math] only errs when it returns 1, thus the error is one-sided. The error probability is bounded by the probability that [math]\displaystyle{ A(x) }[/math] runs longer than [math]\displaystyle{ 2T(n) }[/math]. Since the expected running time of [math]\displaystyle{ A(x) }[/math] is at most [math]\displaystyle{ T(n) }[/math], due to Markov's inequality,

[math]\displaystyle{ \Pr[\mbox{the running time of }A(x)\ge2T(n)]\le\frac{\mathbf{E}[\mbox{running time of }A(x)]}{2T(n)}\le\frac{1}{2}, }[/math]

thus the error probability is bounded.

This easy reduction implies that ZPP[math]\displaystyle{ \subseteq }[/math]RP.

Generalization

For any random variable [math]\displaystyle{ X }[/math], for an arbitrary non-negative real function [math]\displaystyle{ h }[/math], the [math]\displaystyle{ h(X) }[/math] is a non-negative random variable. Applying Markov's inequality, we directly have that

[math]\displaystyle{ \Pr[h(X)\ge t]\le\frac{\mathbf{E}[h(X)]}{t}. }[/math]

This trivial application of Markov's inequality gives us a powerful tool for proving tail inequalities. With the function [math]\displaystyle{ h }[/math] which extracts more information about the random variable, we can prove sharper tail inequalities.