Hoeffding's inequality

In probability theory, Hoeffding's inequality provides an upper bound on the probability that the sum of bounded independent random variables deviates from its expected value by more than a certain amount. Hoeffding's inequality was proven by Wassily Hoeffding in 1963.^[1]

Hoeffding's inequality is a special case of the Azuma–Hoeffding inequality and McDiarmid's inequality. It is similar to the Chernoff bound, but tends to be less sharp, in particular when the variance of the random variables is small.^[2] It is similar to, but incomparable with, one of Bernstein's inequalities.

Statement

Let X₁, ..., X_n be independent random variables such that ${\displaystyle a_{i}\leq X_{i}\leq b_{i}}$ almost surely. Consider the sum of these random variables,

${\displaystyle S_{n}=X_{1}+\cdots +X_{n}.}$

Then Hoeffding's theorem states that, for all t > 0,^[3]

${\displaystyle {\begin{aligned}\operatorname {P} \left(S_{n}-\mathrm {E} \left[S_{n}\right]\geq t\right)&\leq \exp \left(-{\frac {2t^{2}}{\sum _{i=1}^{n}(b_{i}-a_{i})^{2}}}\right)\\\operatorname {P} \left(\left|S_{n}-\mathrm {E} \left[S_{n}\right]\right|\geq t\right)&\leq 2\exp \left(-{\frac {2t^{2}}{\sum _{i=1}^{n}(b_{i}-a_{i})^{2}}}\right)\end{aligned}}}$

Here E[S_n] is the expected value of S_n.

Note that the inequalities also hold when the X_i have been obtained using sampling without replacement; in this case the random variables are not independent anymore. A proof of this statement can be found in Hoeffding's paper. For slightly better bounds in the case of sampling without replacement, see for instance the paper by Serfling (1974).

Generalization

Let ${\displaystyle Y_{1},\dots ,Y_{n}}$ be independent observations such that ${\displaystyle \operatorname {E} (Y_{i})=0}$ and ${\displaystyle a_{i}\leq Y_{i}\leq b_{i}}$. Let ${\displaystyle \epsilon >0}$. Then, for any ${\displaystyle t>0}$,^[4]$${\displaystyle P\left(\sum _{i=1}^{n}Y_{i}\geq \epsilon \right)\leq \exp \left(-t\epsilon +\sum _{i=1}^{n}t^{2}(b_{i}-a_{i})^{2}/8\right)}$$

Special Case: Bernoulli RVs

Suppose ${\displaystyle a_{i}=0}$ and ${\displaystyle b_{i}=1}$ for all i. This can occur when X_i are independent Bernoulli random variables, though they need not be identically distributed. Then we get the inequality^[5]

${\displaystyle {\begin{aligned}\operatorname {P} \left(S_{n}-\mathrm {E} \left[S_{n}\right]\geq t\right)&\leq \exp(-2t^{2}/n)\\\operatorname {P} \left(\left|S_{n}-\mathrm {E} \left[S_{n}\right]\right|\geq t\right)&\leq 2\exp(-2t^{2}/n)\end{aligned}}}$

or equivalently,

$${\displaystyle {\begin{aligned}\operatorname {P} \left((S_{n}-\mathrm {E} \left[S_{n}\right])/n\geq t\right)&\leq \exp(-2nt^{2})\\\operatorname {P} \left(\left|(S_{n}-\mathrm {E} \left[S_{n}\right])/n\right|\geq t\right)&\leq 2\exp(-2nt^{2})\end{aligned}}}$$

for all ${\displaystyle t\geq 0}$. This is a version of the additive Chernoff bound which is more general, since it allows for random variables that take values between zero and one, but also weaker, since the Chernoff bound gives a better tail bound when the random variables have small variance.

General case of bounded from above random variables

Hoeffding's inequality can be extended to the case of bounded from above random variables.^[6]

Let X₁, ..., X_n be independent random variables such that ${\displaystyle \mathrm {E} X_{i}=0}$ and ${\displaystyle X_{i}\leq b_{i}}$ almost surely. Denote by

${\displaystyle {\begin{aligned}C_{i}^{2}=\left\{{\begin{array}{ll}\mathrm {E} X_{i}^{2},&\mathrm {if} \ \mathrm {E} X_{i}^{2}\geq b_{i}^{2},\\\displaystyle {\frac {1}{4}}\left(b_{i}+{\frac {\mathrm {E} X_{i}^{2}}{b_{i}}}\right)^{2},&{\textrm {otherwise}}.\end{array}}\right.\end{aligned}}}$

Hoeffding's inequality for bounded from above random variables states that for all ${\displaystyle t\geq 0}$,

${\displaystyle \mathrm {P} \left(\left|\sum _{i=1}^{n}X_{i}\right|\geq t\right)\leq 2\exp \left(-{\frac {t^{2}}{2\sum _{i=1}^{n}C_{i}^{2}}}\right).}$

In particular, if ${\displaystyle \mathrm {E} X_{i}^{2}\geq b_{i}^{2}}$ for all ${\displaystyle i}$, then for all ${\displaystyle t\geq 0}$,

${\displaystyle \mathrm {P} \left(\left|\sum _{i=1}^{n}X_{i}\right|\geq t\right)\leq 2\exp \left(-{\frac {t^{2}}{2\sum _{i=1}^{n}\mathrm {E} X_{i}^{2}}}\right).}$

General case of sub-Gaussian random variables

The proof of Hoeffding's inequality can be generalized to any sub-Gaussian distribution. Recall that a random variable X is called sub-Gaussian,^[7] if

${\displaystyle \mathrm {P} (|X|\geq t)\leq 2e^{-ct^{2}},}$

for some ${\displaystyle c>0}$. For any bounded variable X, ${\displaystyle \mathrm {P} (|X|\geq t)=0\leq 2e^{-ct^{2}}}$ for ${\displaystyle t>T}$ for some sufficiently large T. Then ${\displaystyle 2e^{-cT^{2}}\leq 2e^{-ct^{2}}}$ for all ${\displaystyle t\leq T}$ so taking ${\displaystyle c=\log(2)/T^{2}}$ yields

${\displaystyle \mathrm {P} (|X|\geq t)\leq 1\leq 2e^{-cT^{2}}\leq 2e^{-ct^{2}},}$

for ${\displaystyle t\leq T}$. So every bounded variable is sub-Gaussian.

For a random variable X, the following norm is finite if and only if X is sub-Gaussian:

${\displaystyle \Vert X\Vert _{\psi _{2}}:=\inf \left\{c\geq 0:\mathrm {E} \left(e^{X^{2}/c^{2}}\right)\leq 2\right\}.}$

Then let X₁, ..., X_n be independent sub-Gaussian random variables, the general version of the Hoeffding's inequality states that:

${\displaystyle \mathrm {P} \left(\left|\sum _{i=1}^{n}(X_{i}-E[X_{i}])\right|\geq t\right)\leq 2\exp \left(-{\frac {ct^{2}}{\sum _{i=1}^{n}\Vert X_{i}\Vert _{\psi _{2}}^{2}}}\right),}$

where c > 0 is an absolute constant.^[8]

Equivalently, we can define sub-Gaussian distributions by variance proxy, defined as follows. If there exists some ${\displaystyle s^{2}}$ such that ${\displaystyle \operatorname {E} [e^{(X-\operatorname {E} [X])t}]\leq e^{\frac {s^{2}t^{2}}{2}}}$ for all ${\displaystyle t}$, then ${\displaystyle s^{2}}$ is called a variance proxy, and the smallest such ${\displaystyle s^{2}}$ is called the optimal variance proxy, and denoted by ${\displaystyle \Vert X\Vert _{\mathrm {vp} }^{2}}$. In this form, Hoeffding's inequality states$${\displaystyle \mathrm {P} \left(\left|\sum _{i=1}^{n}(X_{i}-E[X_{i}])\right|\geq t\right)\leq 2\exp \left(-{\frac {t^{2}}{2\sum _{i=1}^{n}\|X_{i}\|_{\mathrm {vp} }^{2}}}\right),}$$

Proof

We quote:

Hoeffding's lemma—Suppose X is a real random variable such that ${\displaystyle X\in \left[a,b\right]}$ almost surely. Then

${\displaystyle \mathrm {E} \left[e^{s\left(X-\mathrm {E} \left[X\right]\right)}\right]\leq \exp \left({\tfrac {1}{8}}s^{2}(b-a)^{2}\right).}$

The proof of Hoeffding's inequality then follows similarly to concentration inequalities like Chernoff bounds.^[9]

Proof

As in the theorem statement, suppose X₁, ..., X_n are n independent random variables such that ${\displaystyle X_{i}\in [a_{i},b_{i}]}$ almost surely for all i, and let ${\displaystyle S_{n}=X_{1}+\cdots +X_{n}}$.

Then for s, t > 0, Markov's inequality and the independence of X_i implies:

${\displaystyle {\begin{aligned}\operatorname {P} \left(S_{n}-\mathrm {E} \left[S_{n}\right]\geq t\right)&=\operatorname {P} \left(\exp(s(S_{n}-\mathrm {E} \left[S_{n}\right]))\geq \exp(st)\right)\\&\leq \exp(-st)\mathrm {E} \left[\exp(s(S_{n}-\mathrm {E} \left[S_{n}\right]))\right]\\&=\exp(-st)\prod _{i=1}^{n}\mathrm {E} \left[\exp(s(X_{i}-\mathrm {E} \left[X_{i}\right]))\right]\\&\leq \exp(-st)\prod _{i=1}^{n}\exp {\Big (}{\frac {s^{2}(b_{i}-a_{i})^{2}}{8}}{\Big )}\\&=\exp \left(-st+{\tfrac {1}{8}}s^{2}\sum _{i=1}^{n}(b_{i}-a_{i})^{2}\right)\end{aligned}}}$

This upper bound is the best for the value of s minimizing the value inside the exponential. This can be done easily by optimizing a quadratic, giving

${\displaystyle s={\frac {4t}{\sum _{i=1}^{n}(b_{i}-a_{i})^{2}}}.}$

Writing the above bound for this value of s, we get the desired bound:

${\displaystyle \operatorname {P} \left(S_{n}-\mathrm {E} \left[S_{n}\right]\geq t\right)\leq \exp \left(-{\frac {2t^{2}}{\sum _{i=1}^{n}(b_{i}-a_{i})^{2}}}\right).}$

This proof easily generalizes to the case for sub-Gaussian distributions with variance proxy.

Usage

Confidence intervals

Hoeffding's inequality can be used to derive confidence intervals. We consider a coin that shows heads with probability p and tails with probability 1 − p. We toss the coin n times, generating n samples ${\displaystyle X_{1},\ldots ,X_{n}}$ (which are i.i.d Bernoulli random variables). The expected number of times the coin comes up heads is pn. Furthermore, the probability that the coin comes up heads at least k times can be exactly quantified by the following expression:

${\displaystyle \operatorname {P} (H(n)\geq k)=\sum _{i=k}^{n}{\binom {n}{i}}p^{i}(1-p)^{n-i},}$

where H(n) is the number of heads in n coin tosses.

When k = (p + ε)n for some ε > 0, Hoeffding's inequality bounds this probability by a term that is exponentially small in ε²n:

${\displaystyle \operatorname {P} (H(n)-pn>\varepsilon n)\leq \exp \left(-2\varepsilon ^{2}n\right).}$

Since this bound holds on both sides of the mean, Hoeffding's inequality implies that the number of heads that we see is concentrated around its mean, with exponentially small tail.

${\displaystyle \operatorname {P} \left(|H(n)-pn|>\varepsilon n\right)\leq 2\exp \left(-2\varepsilon ^{2}n\right).}$

Thinking of ${\displaystyle {\overline {X}}={\frac {1}{n}}H(n)}$ as the "observed" mean, this probability can be interpreted as the level of significance ${\displaystyle \alpha }$ (probability of making an error) for a confidence interval around ${\displaystyle p}$ of size 2ɛ:

${\displaystyle \alpha =\operatorname {P} (\ {\overline {X}}\notin [p-\varepsilon ,p+\varepsilon ])\leq 2e^{-2\varepsilon ^{2}n}}$

Finding n for opposite inequality sign in the above, i.e. n that violates inequality but not equality above, gives us:

${\displaystyle n\geq {\frac {\log(2/\alpha )}{2\varepsilon ^{2}}}}$

Therefore, we require at least ${\displaystyle \textstyle {\frac {\log(2/\alpha )}{2\varepsilon ^{2}}}}$ samples to acquire a ${\displaystyle \textstyle (1-\alpha )}$-confidence interval ${\displaystyle \textstyle p\pm \varepsilon }$.

Hence, the cost of acquiring the confidence interval is sublinear in terms of confidence level and quadratic in terms of precision. Note that there are more efficient methods of estimating a confidence interval.

See also

• Concentration inequality – a summary of tail-bounds on random variables.
• Hoeffding's lemma
• Bernstein inequalities (probability theory)