30,545 research outputs found
Level-Based Analysis of the Univariate Marginal Distribution Algorithm
Estimation of Distribution Algorithms (EDAs) are stochastic heuristics that
search for optimal solutions by learning and sampling from probabilistic
models. Despite their popularity in real-world applications, there is little
rigorous understanding of their performance. Even for the Univariate Marginal
Distribution Algorithm (UMDA) -- a simple population-based EDA assuming
independence between decision variables -- the optimisation time on the linear
problem OneMax was until recently undetermined. The incomplete theoretical
understanding of EDAs is mainly due to lack of appropriate analytical tools.
We show that the recently developed level-based theorem for non-elitist
populations combined with anti-concentration results yield upper bounds on the
expected optimisation time of the UMDA. This approach results in the bound
on two problems, LeadingOnes and
BinVal, for population sizes , where and
are parameters of the algorithm. We also prove that the UMDA with
population sizes optimises
OneMax in expected time , and for larger population
sizes , in expected time
. The facility and generality of our arguments
suggest that this is a promising approach to derive bounds on the expected
optimisation time of EDAs.Comment: To appear in Algorithmica Journa
Improved Runtime Bounds for the Univariate Marginal Distribution Algorithm via Anti-Concentration
Unlike traditional evolutionary algorithms which produce offspring via
genetic operators, Estimation of Distribution Algorithms (EDAs) sample
solutions from probabilistic models which are learned from selected
individuals. It is hoped that EDAs may improve optimisation performance on
epistatic fitness landscapes by learning variable interactions. However, hardly
any rigorous results are available to support claims about the performance of
EDAs, even for fitness functions without epistasis. The expected runtime of the
Univariate Marginal Distribution Algorithm (UMDA) on OneMax was recently shown
to be in by Dang and Lehre
(GECCO 2015). Later, Krejca and Witt (FOGA 2017) proved the lower bound
via an involved drift analysis.
We prove a bound, given some restrictions
on the population size. This implies the tight bound when , matching the runtime
of classical EAs. Our analysis uses the level-based theorem and
anti-concentration properties of the Poisson-Binomial distribution. We expect
that these generic methods will facilitate further analysis of EDAs.Comment: 19 pages, 1 figur
Level-Based Analysis of the Population-Based Incremental Learning Algorithm
The Population-Based Incremental Learning (PBIL) algorithm uses a convex
combination of the current model and the empirical model to construct the next
model, which is then sampled to generate offspring. The Univariate Marginal
Distribution Algorithm (UMDA) is a special case of the PBIL, where the current
model is ignored. Dang and Lehre (GECCO 2015) showed that UMDA can optimise
LeadingOnes efficiently. The question still remained open if the PBIL performs
equally well. Here, by applying the level-based theorem in addition to
Dvoretzky--Kiefer--Wolfowitz inequality, we show that the PBIL optimises
function LeadingOnes in expected time for a population size , which matches the bound
of the UMDA. Finally, we show that the result carries over to BinVal, giving
the fist runtime result for the PBIL on the BinVal problem.Comment: To appea
- …