Azuma's inequality

In probability theory, the Azuma–Hoeffding inequality (named after Kazuoki Azuma and Wassily Hoeffding) gives a concentration result for the values of martingales that have bounded differences.

Suppose { X_k : k = 0, 1, 2, 3, ... } is a martingale (or super-martingale) and

|X_k - X_{k-1}| < c_k, \,

almost surely. Then for all positive integers N and all positive reals t,

P(X_N - X_0 \geq t) \leq \exp\left ({-t^2 \over 2 \sum_{k=1}^N c_k^2} \right).

And symmetrically (when X_k is a sub-martingale):

P(X_N - X_0 \leq -t) \leq \exp\left ({-t^2 \over 2 \sum_{k=1}^N c_k^2} \right).

If X is a martingale, using both inequalities above and applying the union bound allows one to obtain a two-sided bound:

P(|X_N - X_0| \geq t) \leq 2\exp\left ({-t^2 \over 2 \sum_{k=1}^N c_k^2} \right).

Azuma's inequality applied to the Doob martingale gives the method of bounded differences (MOBD) which is common in the analysis of randomized algorithms.

Simple example of Azuma's inequality for coin flips

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 $X_i = \sum_{j=1}^i F_j$ yields a martingale with |X_k − X_k−1| ≤ 1, allowing us to apply Azuma's inequality. Specifically, we get

\Pr[X_{n}>t]\leq \exp \left({\frac {-t^{2}}{2n}}\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 $t={\sqrt {2n\ln n}}$ we get:

\Pr[X_{n}>{\sqrt {2n\ln n}}]\leq 1/n,

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

Remark

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 18 of his 1963 paper).

References

Alon, N.; Spencer, J. (1992). The Probabilistic Method. New York: Wiley.
Azuma, K. (1967). "Weighted Sums of Certain Dependent Random Variables" (PDF). Tôhoku Mathematical Journal. 19 (3): 357–367. doi:10.2748/tmj/1178243286. MR 0221571.
Bernstein, Sergei N. (1937). На определенных модификациях неравенства Чебишева [On certain modifications of Chebyshev's inequality]. Doklady Akademii Nauk SSSR (in Russian). 17 (6): 275–277. (vol. 4, item 22 in the collected works)
McDiarmid, C. (1989). "On the method of bounded differences". Surveys in Combinatorics. London Math. Soc. Lectures Notes 141. Cambridge: Cambridge Univ. Press. pp. 148–188. MR 1036755.
Hoeffding, W. (1963). "Probability inequalities for sums of bounded random variables". Journal of the American Statistical Association. 58 (301): 13–30. doi:10.2307/2282952. MR 0144363.
Godbole, A. P.; Hitczenko, P. (1998). "Beyond the method of bounded differences". DIMACS Series in Discrete Mathematics and Theoretical Computer Science. 41: 43–58. MR 1630408.

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Azuma's inequality

Simple example of Azuma's inequality for coin flips

Remark

See also

References