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.
is a martingale (or super-martingale) and almost surely.
, And symmetrically (when Xk is a sub-martingale): If X is a martingale, using both inequalities above and applying the union bound allows one to obtain a two-sided bound: 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.
, to use Azuma's inequality, one need to choose
which might be a waste of information on the boundedness of
However, this issue can be resolved and one can obtain a tighter probability bound with the following general form of Azuma's inequality.
be a martingale (or supermartingale) with respect to filtration
Assume there are predictable processes
, Since a submartingale is a supermartingale with signs reversed, we have if instead
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: We will prove the supermartingale case only as the rest are self-evident.
By Doob decomposition, we could decompose supermartingale
is a nonincreasing predictable sequence (Note that if
, we have Applying Chernoff bound to
is a predictable process; (iv)
; by applying Hoeffding's lemma[note 1], we have Repeating this step, one could get Note that the minimum is achieved at
is nonincreasing, so event
, and therefore Note that by setting
, 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 Fi be a sequence of independent and identically distributed random coin flips (i.e., let Fi be equally likely to be −1 or 1 independent of the other values of Fi).
yields a martingale with |Xk − Xk−1| ≤ 1, allowing us to apply Azuma's inequality.
Specifically, we get For example, if we set t proportional to n, then this tells us that although the maximum possible value of Xn scales linearly with n, the probability that the sum scales linearly with n decreases exponentially fast with n. If we set
we get: which means that the probability of deviating more than
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).