Upper Bounds on the Running Time of the Univariate Marginal Distribution Algorithm on OneMax
The Univariate Marginal Distribution Algorithm (UMDA) is a randomized search heuristic that builds a stochastic model of the underlying optimization problem by repeatedly sampling λ solutions and adjusting the model according to the best μ samples. We present a running time analysis of the UMDA on t...
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| Published in: | Algorithmica Vol. 81; no. 2; pp. 632 - 667 |
|---|---|
| Main Author: | |
| Format: | Journal Article |
| Language: | English |
| Published: |
New York
Springer US
15.02.2019
Springer Nature B.V |
| Subjects: | |
| ISSN: | 0178-4617, 1432-0541 |
| Online Access: | Get full text |
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| Summary: | The Univariate Marginal Distribution Algorithm (UMDA) is a randomized search heuristic that builds a stochastic model of the underlying optimization problem by repeatedly sampling
λ
solutions and adjusting the model according to the best
μ
samples. We present a running time analysis of the UMDA on the classical OneMax benchmark function for wide ranges of the parameters
μ
and
λ
. If
μ
≥
c
log
n
for some constant
c
>
0
and
λ
=
(
1
+
Θ
(
1
)
)
μ
, we obtain a general bound
O
(
μ
n
)
on the expected running time. This bound crucially assumes that all marginal probabilities of the algorithm are confined to the interval
[
1
/
n
,
1
-
1
/
n
]
. If
μ
≥
c
′
n
log
n
for a constant
c
′
>
0
and
λ
=
(
1
+
Θ
(
1
)
)
μ
, the behavior of the algorithm changes and the bound on the expected running time becomes
O
(
μ
n
)
, which typically holds even if the borders on the marginal probabilities are omitted. The results supplement the recently derived lower bound
Ω
(
μ
n
+
n
log
n
)
by Krejca and Witt (Proceedings of FOGA 2017, ACM Press, New York, pp 65–79,
2017
) and turn out to be tight for the two very different choices
μ
=
c
log
n
and
μ
=
c
′
n
log
n
. They also improve the previously best known upper bound
O
(
n
log
n
log
log
n
)
by Dang and Lehre (Proceedings of GECCO ’15, ACM Press, New York, pp 513–518,
2015
) that was established for
μ
=
c
log
n
and
λ
=
(
1
+
Θ
(
1
)
)
μ
. |
|---|---|
| Bibliography: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0178-4617 1432-0541 |
| DOI: | 10.1007/s00453-018-0463-0 |