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 $n$ 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

A Chernoff bound gives exponentially decreasing bounds on tail distributions of sums of independent random variables.^[1] Let $X_i$ denote independent but not necessarily identical 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 $X_i \leq E(X_i)+a_i+M$ , we have upper tail inequality:

\Pr[X \geq 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^2/4n}

for

0 \leq k\leq 2\sigma

Hoeffding'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} = \frac{X_1 + \cdots + X_n}{n}

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

\Pr(\overline{X} - \mathrm{E}[\overline{X}] \geq t) \leq \exp \left( - \frac{2t^2n^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^2n^2}{\sum_{i=1}^n (b_i - a_i)^2} \right),\!

Bennett's inequality

Bennett's inequality was proved by George Bennett of the University of New South Wales in 1962.^[3]

Let $X 1, \dots X n$ be independent random variables, and assume (for simplicity but without loss of generality) they all have zero expected value. Further assume $| X i | \leq a$ almost surely for all $i$ , and let

\sigma^2 = \frac1n \sum_{i=1}^n \operatorname{Var}(X_i).

Then for any $t \geq 0$ ,

\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 $h (u) = (1 + u)log(1 + u) - u$ ,^[4] see also Fan et al. (2012) ^[5] for martingale version of Bennett's inequality and its improvement.

Bernstein's inequality

Bernstein inequalities give bounds on the probability that the sum of random variables deviates from its mean. In the simplest case, let X₁, ..., X_n be independent Bernoulli random variables taking values +1 and −1 with probability 1/2, then for every positive $\varepsilon$ ,

\mathbf{P} \left\{\left|\;\frac{1}{n}\sum_{i=1}^n X_i\;\right| > \varepsilon \right\} \leq 2\exp \left\{ - \frac{n\varepsilon^2}{ 2 (1 + \varepsilon/3) } \right\}.

Efron–Stein inequality

The Efron–Stein inequality (or influence inequality, or MG bound on variance) bounds the variance of a general function.

Suppose that $X_1 \dots X_n$ , $X_1' \dots X_n'$ are independent with $X_i'$ and $X_i$ having the same distribution for all $i$ .

Let $X = (X_1,\dots , X_n), X^{(i)} = (X_1, \dots , X_{i-1}, X_i',X_{i+1}, \dots , X_n).$ Then

\mathrm{Var}(f(X)) \leq \frac{1}{2} \sum_{i=1}^{n} E[(f(X)-f(X^{(i)}))^2].

References

↑ Chung, Fan. "Old and New Concentration Inequalities". Old and New Concentration Inequalities. Retrieved 2010.
↑ Wassily Hoeffding, Probability inequalities for sums of bounded random variables, Journal of the American Statistical Association 58 (301): 13–30, March 1963. (JSTOR)
↑ Bennett, G. (1962). "Probability Inequalities for the Sum of Independent Random Variables". Journal of the American Statistical Association 57 (297): 33–45. doi:10.2307/2282438. JSTOR 2282438.
↑ Devroye, Luc; Lugosi, Gábor (2001). Combinatorial methods in density estimation. Springer. p. 11. ISBN 978-0-387-95117-1.
↑ Fan, X.; Grama, I.; Liu, Q. (2012). "Hoeffding's inequality for supermartingales". Stochastic Processes and their Applications 122: 3545–3559. doi:10.1016/j.spa.2012.06.009.