Azuma's inequality

The proof shares similar idea of the proof for the general form of Azuma's inequality listed below. Actually, this can be viewed as a direct corollary of the general form of Azuma's inequality.

Note that the vanilla Azuma's inequality requires symmetric bounds on martingale increments, i.e. $-c\_t\; \backslash leq\; X\_t\; -\; X\_\{t-1\}\; \backslash leq\; c\_t$. So, if known bound is asymmetric, e.g. $a\_t\; \backslash leq\; X\_t\; -\; X\_\{t-1\}\; \backslash leq\; b\_t$, to use Azuma's inequality, one need to choose $c\_t\; =\; \backslash max(|a\_t|,\; |b\_t|)$ which might be a waste of information on the boundedness of $X\_t\; -\; X\_\{t-1\}$. However, this issue can be resolved and one can obtain a tighter probability bound with the following general form of Azuma's inequality.

Let $\backslash left\backslash \{X\_0,\; X\_1,\; \backslash cdots\; \backslash right\backslash \}$ be a martingale (or supermartingale) with respect to filtration $\backslash left\backslash \{\backslash mathcal\{F\}\_0,\; \backslash mathcal\{F\}\_1,\; \backslash cdots\; \backslash right\backslash \}$. Assume there are predictable processes $\backslash left\backslash \{A\_0,\; A\_1,\; \backslash cdots\backslash right\backslash \}$ and $\backslash left\backslash \{\; B\_0,\; B\_1,\; \backslash dots\; \backslash right\backslash \}$ with respect to $\backslash left\backslash \{\; \backslash mathcal\{F\}\_0,\; \backslash mathcal\{F\}\_1,\; \backslash cdots\; \backslash right\backslash \}$, i.e. for all $t$, $A\_t,\; B\_t$ are $\backslash mathcal\{F\}\_\{t-1\}$-measurable, and constants

:$$

A_t \leq X_t - X_{t-1} \leq B_t \quad \text{and} \quad B_t - A_t \leq c_t

almost surely. Then for all $\backslash epsilon>0$,

:$$

\text{P}(X_n - X_0 \geq \epsilon) \leq \exp \left( - \frac{2\epsilon^2}{ \sum_{t=1}^{n} c_t^2 } \right).

Since a submartingale is a supermartingale with signs reversed, we have if instead $\backslash left\backslash \{X\_0,\; X\_1,\; \backslash dots\; \backslash right\backslash \}$ is a martingale (or submartingale),

:$$

\text{P}(X_n - X_0 \leq -\epsilon) \leq \exp \left(- \frac{2\epsilon^2}{ \sum_{t=1}^{n} c_t^2 } \right).

If $\backslash left\backslash \{X\_0,\; X\_1,\; \backslash dots\; \backslash right\backslash \}$ is a martingale, since it is both a supermartingale and submartingale, by applying union bound to the two inequalities above, we could obtain the two-sided bound:

:$$

\text{P}(|X_n - X_0| \geq \epsilon) \leq 2\exp \left(- \frac{2\epsilon^2}{ \sum_{t=1}^{n} c_t^2 } \right).

We will prove the supermartingale case only as the rest are self-evident. By Doob decomposition, we could decompose supermartingale $\backslash left\backslash \{X\_t\backslash right\backslash \}$ as $X\_t\; =\; Y\_t\; +\; Z\_t$ where $\backslash left\backslash \{Y\_t,\; \backslash mathcal\{F\}\_t\backslash right\backslash \}$ is a martingale and $\backslash left\backslash \{Z\_t,\; \backslash mathcal\{F\}\_t\backslash right\backslash \}$ is a nonincreasing predictable sequence (Note that if $\backslash left\backslash \{X\_t\backslash right\backslash \}$ itself is a martingale, then $Z\_t\; =\; 0$). From $A\_t\; \backslash leq\; X\_t\; -\; X\_\{t-1\}\; \backslash leq\; B\_t$, we have

:$$

-(Z_t - Z_{t-1}) + A_t

\leq

Y_t - Y_{t-1}

\leq

-(Z_t - Z_{t-1}) + B_t

Applying Chernoff bound to $Y\_n\; -\; Y\_0$, we have for $\backslash epsilon>0$,

:$\backslash begin\{align\}$

\text{P}(Y_n-Y_0 \geq \epsilon)

& \leq \underset{s>0}{\min} \ e^{-s\epsilon} \mathbb{E} [e^{s (Y_n-Y_0) }] \\

& = \underset{s>0}{\min} \ e^{-s\epsilon} \mathbb{E} \left[\exp \left( s \sum_{t=1}^{n}(Y_t-Y_{t-1}) \right) \right] \\

& = \underset{s>0}{\min} \ e^{-s\epsilon} \mathbb{E} \left[\exp \left( s \sum_{t=1}^{n-1}(Y_t-Y_{t-1}) \right) \mathbb{E} \left[\exp \left( s(Y_n-Y_{n-1}) \right) \mid \mathcal{F}_{n-1} \right] \right]

\end{align}

For the inner expectation term, since

(i) $\backslash mathbb\{E\}[Y\_t\; -\; Y\_\{t-1\}\; \backslash mid\; \backslash mathcal\{F\}\_\{t-1\}]=0$ as $\backslash left\backslash \{Y\_t\backslash right\backslash \}$ is a martingale;

(ii) $-(Z\_t\; -\; Z\_\{t-1\})\; +\; A\_t\; \backslash leq\; Y\_t\; -\; Y\_\{t-1\}\; \backslash leq\; -(Z\_t\; -\; Z\_\{t-1\})\; +\; B\_t$;

(iii) $-(Z\_t\; -\; Z\_\{t-1\})\; +\; A\_t$ and $-(Z\_t\; -\; Z\_\{t-1\})\; +\; B\_t$ are both $\backslash mathcal\{F\}\_\{t-1\}$-measurable as $\backslash left\backslash \{Z\_t\backslash right\backslash \}$ is a predictable process;

(iv) $B\_t\; -\; A\_t\; \backslash leq\; c\_t$;

by applying Hoeffding's lemma{{NoteTag|It is not a direct application of Hoeffding's lemma though. The statement of Hoeffding's lemma handles the total expectation, but it also holds for the case when the expectation is conditional expectation and the bounds are measurable with respect to the sigma-field the conditional expectation is conditioned on. The proof is the same as for the classical Hoeffding's lemma.}}, we have

:$$

\mathbb{E} \left[\exp \left( s(Y_t-Y_{t-1}) \right) \mid \mathcal{F}_{t-1} \right]

\leq

\exp \left(\frac{s^2 (B_t - A_t)^2}{8} \right)

\leq

\exp \left(\frac{s^2 c_t^2}{8} \right).

Repeating this step, one could get

:$$

\text{P}(Y_n-Y_0 \geq \epsilon)

\leq

\underset{s>0}{\min} \ e^{-s\epsilon} \exp \left(\frac{s^2 \sum_{t=1}^{n}c_t^2}{8}\right).

Note that the minimum is achieved at $s\; =\; \backslash frac\{4\; \backslash epsilon\}\{\backslash sum\_\{t=1\}^\{n\}c\_t^2\}$, so we have

:$$

\text{P}(Y_n-Y_0 \geq \epsilon)

\leq

\exp \left(-\frac{2 \epsilon^2}{\sum_{t=1}^{n}c_t^2}\right).

Finally, since $X\_n\; -\; X\_0\; =\; (Y\_n\; -\; Y\_0)\; +\; (Z\_n\; -\; Z\_0)$ and $Z\_n\; -\; Z\_0\; \backslash leq\; 0$ as $\backslash left\backslash \{Z\_n\; \backslash right\backslash \}$ is nonincreasing, so event $\backslash left\backslash \{X\_n\; -\; X\_0\; \backslash geq\; \backslash epsilon\backslash right\backslash \}$ implies $\backslash left\backslash \{Y\_n\; -\; Y\_0\; \backslash geq\; \backslash epsilon\backslash right\backslash \}$, and therefore

:$$

\text{P}(X_n-X_0 \geq \epsilon)

\leq

\text{P}(Y_n-Y_0 \geq \epsilon)

\leq

\exp \left(-\frac{2 \epsilon^2}{\sum_{t=1}^{n}c_t^2}\right). \square

Note that by setting $A\_t\; =\; -c\_t,\; B\_t\; =\; c\_t$, we could obtain the vanilla Azuma's inequality.

Note that for either submartingale or supermartingale, only one side of Azuma's inequality holds. We can't say much about how fast a submartingale with bounded increments rises (or a supermartingale falls).

This general form of Azuma's inequality applied to the Doob martingale gives McDiarmid's inequality which is common in the analysis of randomized algorithms.

Let F_i be a sequence of independent and identically distributed random coin flips (i.e., let F_i be equally likely to be −1 or 1 independent of the other values of F_i). Defining $X\_i\; =\; \backslash sum\_\{j=1\}^i\; F\_j$ yields a martingale with |X_k − X_k−1| ≤ 1, allowing us to apply Azuma's inequality. Specifically, we get

:$\backslash operatorname\{P\}(X\_n\; >\; t)\; \backslash leq\; \backslash exp\backslash left(\backslash frac\{-t^2\}\{2\; n\}\backslash right).$

For example, if we set t proportional to n, then this tells us that although the maximum possible value of X_n scales linearly with n, the probability that the sum scales linearly with n decreases exponentially fast with n.

If we set $t=\backslash sqrt\{2\; n\; \backslash ln\; n\}$ we get:

:$\backslash operatorname\{P\}\backslash left(X\_n\; >\; \backslash sqrt\{2\; n\; \backslash ln\; n\}\backslash right)\; \backslash leq\; \backslash frac1n,$

which means that the probability of deviating more than $\backslash sqrt\{2\; n\; \backslash ln\; n\}$ approaches 0 as n goes to infinity.

A similar inequality was proved under weaker assumptions by Sergei Bernstein in 1937.

Hoeffding proved this result for independent variables rather than martingale differences, and also observed that slight modifications of his argument establish the result for martingale differences (see page 9 of his 1963 paper).

• Concentration inequality - a summary of tail-bounds on random variables.

{{NoteFoot}}

• {{cite book|first1=N. |last1=Alon |first2= J. |last2=Spencer|title=The Probabilistic Method|publisher= Wiley|location=New York|year= 1992}}
• {{cite journal|doi=10.2748/tmj/1178243286|first=K. |last=Azuma|title=Weighted Sums of Certain Dependent Random Variables|journal=Tôhoku Mathematical Journal|volume=19|pages= 357–367 |year=1967|issue=3|url=http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.tmj/1178243286|format=PDF|mr=0221571|doi-access=free}}
• {{cite journal| last=Bernstein | first=Sergei N. | authorlink=Sergei Natanovich Bernstein | year=1937 |trans-title=On certain modifications of Chebyshev's inequality | journal=Doklady Akademii Nauk SSSR | volume=17 | issue=6 | pages=275–277 |script-title=ru:О некоторых модификациях неравенства Чебышёва|language=Russian }} (vol. 4, item 22 in the collected works)
• {{cite book| first=C. |last=McDiarmid|chapter= On the method of bounded differences|title=Surveys in Combinatorics|series= London Math. Soc. Lectures Notes 141|publisher= Cambridge Univ. Press|location= Cambridge |year=1989|pages=148–188|mr=1036755}}
• {{Cite journal|doi=10.2307/2282952|first1=W. |last1=Hoeffding|title=Probability inequalities for sums of bounded random variables|journal=Journal of the American Statistical Association|volume=58|issue=301|pages= 13–30|year= 1963|mr= 0144363 |jstor=2282952 |url=http://www.lib.ncsu.edu/resolver/1840.4/2170 }}
• {{Cite book|first1=A. P. |last1=Godbole |first2= P. |last2=Hitczenko |title=Microsurveys in Discrete Probability |chapter=Beyond the method of bounded differences |series=DIMACS Series in Discrete Mathematics and Theoretical Computer Science |volume= 41|pages=43–58|year= 1998 |mr=1630408 |doi=10.1090/dimacs/041/03 |isbn=9780821808276 }}

Category:Probabilistic inequalities

Category:Martingale theory