4,686 research outputs found
General Upper Bounds on the Runtime of Parallel Evolutionary Algorithms
We present a general method for analyzing the runtime of parallel evolutionary algorithms with spatially structured populations. Based on the fitness-level method, it yields upper bounds on the expected parallel runtime. This allows for a rigorous estimate of the speedup gained by parallelization. Tailored results are given for common migration topologies: ring graphs, torus graphs, hypercubes, and the complete graph. Example applications for pseudo-Boolean optimization show that our method is easy to apply and that it gives powerful results. In our examples the performance guarantees improve with the density of the topology. Surprisingly, even sparse topologies such as ring graphs lead to a significant speedup for many functions while not increasing the total number of function evaluations by more than a constant factor. We also identify which number of processors lead to the best guaranteed speedups, thus giving hints on how to parameterize parallel evolutionary algorithms
Optimal Parameter Choices Through Self-Adjustment: Applying the 1/5-th Rule in Discrete Settings
While evolutionary algorithms are known to be very successful for a broad
range of applications, the algorithm designer is often left with many
algorithmic choices, for example, the size of the population, the mutation
rates, and the crossover rates of the algorithm. These parameters are known to
have a crucial influence on the optimization time, and thus need to be chosen
carefully, a task that often requires substantial efforts. Moreover, the
optimal parameters can change during the optimization process. It is therefore
of great interest to design mechanisms that dynamically choose best-possible
parameters. An example for such an update mechanism is the one-fifth success
rule for step-size adaption in evolutionary strategies. While in continuous
domains this principle is well understood also from a mathematical point of
view, no comparable theory is available for problems in discrete domains.
In this work we show that the one-fifth success rule can be effective also in
discrete settings. We regard the ~GA proposed in
[Doerr/Doerr/Ebel: From black-box complexity to designing new genetic
algorithms, TCS 2015]. We prove that if its population size is chosen according
to the one-fifth success rule then the expected optimization time on
\textsc{OneMax} is linear. This is better than what \emph{any} static
population size can achieve and is asymptotically optimal also among
all adaptive parameter choices.Comment: This is the full version of a paper that is to appear at GECCO 201
OneMax in Black-Box Models with Several Restrictions
Black-box complexity studies lower bounds for the efficiency of
general-purpose black-box optimization algorithms such as evolutionary
algorithms and other search heuristics. Different models exist, each one being
designed to analyze a different aspect of typical heuristics such as the memory
size or the variation operators in use. While most of the previous works focus
on one particular such aspect, we consider in this work how the combination of
several algorithmic restrictions influence the black-box complexity. Our
testbed are so-called OneMax functions, a classical set of test functions that
is intimately related to classic coin-weighing problems and to the board game
Mastermind.
We analyze in particular the combined memory-restricted ranking-based
black-box complexity of OneMax for different memory sizes. While its isolated
memory-restricted as well as its ranking-based black-box complexity for bit
strings of length is only of order , the combined model does not
allow for algorithms being faster than linear in , as can be seen by
standard information-theoretic considerations. We show that this linear bound
is indeed asymptotically tight. Similar results are obtained for other memory-
and offspring-sizes. Our results also apply to the (Monte Carlo) complexity of
OneMax in the recently introduced elitist model, in which only the best-so-far
solution can be kept in the memory. Finally, we also provide improved lower
bounds for the complexity of OneMax in the regarded models.
Our result enlivens the quest for natural evolutionary algorithms optimizing
OneMax in iterations.Comment: This is the full version of a paper accepted to GECCO 201
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
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