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the target variable. The coefficient
α
k
is the step size and defines the speed of the
update.
Example 3.7
For
α
k
¼
1
k
,(
3.8
) is the step-size-like form of the average calculation,
that is,
x
1
þ x
2
þ ...
x
n
1
n
¼ X
n
¼ X
n
1
þ
ð
x
n
X
n
1
Þ
,
n
where
X
n
1
is recursively calculated in the same way:
1
k þ
1
X
kþ
1
¼ X
k
þ
ð
x
kþ
1
X
k
Þ
,
k ¼
1,
...
,
n
1,
X
1
¼ x
1
:
1
k
, we can immediately see that in the average calculation, the
contribution
x
k
of the
k
th update step decreases. The average thus captures changes
in the behavior of the target variable relatively poorly - it is in a certain sense “lazy”
(which however is necessary in order for it to converge). But it is not so lazy that it
cannot asymptotically capture any fluctuations in the target variable, that is, even in
the
k
th step for a large
k,
From
α
k
¼
α
k
is never “too small.” Mathematically, this is expressed
in the well-known property
1
k¼
1
1
k
¼1
.
■
Both properties can be generalized, and the conditions for the convergence
of (
3.8
) are
1
k¼
1
α
k
¼1
,
1
2
n¼
1
α
k
< 1:
ð
3
:
9
Þ
The first condition ensures that the step remains large enough to capture
movements. The second condition guarantees that ultimately, the steps become
small enough to ensure convergence. For the average, because
1
k¼
1
2
1
k
< 1
, the
second property of the convergence conditions (
3.9
) is also satisfied.
As well as the average, we will now consider a further important case, namely,
the constant step size
. Here - conversely to the average - the contribution of
x
k
in the
k
th update step is always the highest, and the contributions of the preceding
values
x
l
,
α
k
¼ α
k
decrease (even exponentially) as
l
reduces. In this way, the
constant step size adapts particularly quickly to changes in the target variable,
that is, it is very “agile” (but also therefore less stable). In fact, it even violates the
second convergence condition, since
1
l
<
2
1
2
n¼
1
α
¼ α
1
¼1
. Constant step sizes
n¼
1
are used primarily for nonstationary problems, where violation of the second
convergence condition is not critical and may even be desirable. A practical
