Concentration inequality

In mathematics, concentration inequalities provide probability bounds on how a random variable deviates from some value (e.g. its expectation). The laws of large numbers of classical probability theory state that sums of independent random variables are, under very mild conditions, close to their expectation with a large probability. Such sums are the most basic examples of random variables concentrated around their mean. Recent results shows that such behavior is shared by other functions of independent random variables.

Markov's inequality

If X is any random variable and a > 0, then

$\Pr(|X|\geq a)\leq {\frac {{\textrm {E}}(|X|)}{a}}.$

Proof can be found here.

We can extend Markov's inequality to a strictly increasing and non-negative function $\Phi$. We have

$\Pr(X\geq a)=\Pr(\Phi (X)\geq \Phi (a))\leq {\frac {{\textrm {E}}(\Phi (X))}{\Phi (a)}}.$

Chebyshev's inequality

Chebyshev's inequality is a special case of generalized Markov's inequality when $\Phi =x^{2}$

If X is any random variable and a > 0, then

$\Pr(|X-{\textrm {E}}(X)|\geq a)\leq {\frac {{\textrm {Var}}(X)}{a^{2}}},$

Where Var(X) is the variance of X, defined as:

$\operatorname {Var} (X)=\operatorname {E} [(X-\operatorname {E} (X))^{2}].$

Asymptotic Behavior of Binomial Distribution

If a random variable X follows the binomial distribution with parameter $n$ and $p$. The probability of getting exact $k$ successes in $n$ trials is given by the probability mass function

$f(k;n,p)=\Pr(K=k)={n \choose k}p^{k}(1-p)^{n-k}$

Let $S_{n}=\sum _{i=1}^{n}X_{i}$ and $X_{i}$'s are i.i.d. Bernoulli random variables with parameter $p$. $S_{n}$ follows the binomial distribution with parameter with parameter $n$ and $p$. Central Limit Theorem suggests when $n\to \infty$, $S_{n}$ is approximately normally distributed with mean $np$ and variance $np(1-p)$, and

$\lim _{n\to \infty }\Pr[a\sigma <S_{n}-np<b\sigma ]=\int _{a}^{b}{\frac {1}{\sqrt {2\pi }}}e^{-x^{2}/2}dx$

For $p={\frac {\lambda }{n}}$, where $\lambda$ is a constant, the limit distribution of binomial distribution $B(n,p)$ is the Poisson distribution $P(\lambda )$

General Chernoff Inequality

Chernoff bound gives exponentially decreasing bounds on tail distributions of sums of independent random variables. Let $X_{i}$ denote i.i.d. random variables, satisfying $X_{i}\geq E(X_{i})-a_{i}-M$, for $1\leq i\leq n$.

$X=\sum _{i=1}^{n}X_{i}$ we have lower tail inequality:

$\Pr[X\leq E(X)-\lambda ]\leq e^{-{\frac {\lambda ^{2}}{2(Var(X)+\sum _{i=1}^{n}a_{i}^{2}+M\lambda /3)}}}$

If $X_{i}$ satisfies $X_{i}\geq E(X_{i})+a_{i}+M$, we have upper tail inequality:

$\Pr[X\leq E(X)+\lambda ]\leq e^{-{\frac {\lambda ^{2}}{2(Var(X)+\sum _{i=1}^{n}a_{i}^{2}+M\lambda /3)}}}$

If $X_{i}$ are i.i.d., $|X_{i}|\leq 1$ and $\sigma ^{2}$ is the variance of $X_{i}$. A typical version of Chernoff Inequality is:

$\Pr[|X|\geq k\sigma ]\leq 2e^{-k^{2}/4}$

Hoeffdin's Inequality

Hoeffding's inequality can be stated as follows:

If :$X_{1},\dots ,X_{n}\!$ are independent. Assume that the $X_{i}$ are almost surely bounded; that is, assume for $1\leq i\leq n$ that

$\Pr(X_{i}\in [a_{i},b_{i}])=1.\!$

Then, for the empirical mean of these variables

${\overline {X}}=(X_{1}+\cdots +X_{n})/n\!$

we have the inequalities (Hoeffding 1963, Theorem 2 ):

$\Pr({\overline {X}}-\mathrm {E} [{\overline {X}}]\geq t)\leq \exp \left(-{\frac {2t^{2}n^{2}}{\sum _{i=1}^{n}(b_{i}-a_{i})^{2}}}\right),\!$

$\Pr(|{\overline {X}}-\mathrm {E} [{\overline {X}}]|\geq t)\leq 2\exp \left(-{\frac {2t^{2}n^{2}}{\sum _{i=1}^{n}(b_{i}-a_{i})^{2}}}\right),\!$

Bennett's Inequality

Let X₁, … X_n be independent random variables, and assume (for simplicity but without loss of generality) they all have zero expected value. Further assume |X_i| ≤ a almost surely for all i, and let

$\sigma ^{2}={\frac {1}{n}}\sum _{i=1}^{n}\operatorname {Var} (X_{i}).$

Then for any t ≥ 0,

$\Pr \left(\sum _{i=1}^{n}X_{i}>t\right)\leq \exp \left(-{\frac {n\sigma ^{2}}{a^{2}}}h\left({\frac {at}{n\sigma ^{2}}}\right)\right),$

where h(u) = (1 + u)log(1 + u) – u.