14 research outputs found
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
Upper Bounds on the Runtime of the Univariate Marginal Distribution Algorithm on OneMax
A runtime analysis of the Univariate Marginal Distribution Algorithm (UMDA)
is presented on the OneMax function for wide ranges of its parameters and
. If for some constant and
, a general bound on the expected runtime
is obtained. This bound crucially assumes that all marginal probabilities of
the algorithm are confined to the interval . If for a constant and , the
behavior of the algorithm changes and the bound on the expected runtime becomes
, which typically even holds if the borders on the marginal
probabilities are omitted.
The results supplement the recently derived lower bound
by Krejca and Witt (FOGA 2017) and turn out as
tight for the two very different values and . They also improve the previously best known upper bound by Dang and Lehre (GECCO 2015).Comment: Version 4: added illustrations and experiments; improved presentation
in Section 2.2; to appear in Algorithmica; the final publication is available
at Springer via http://dx.doi.org/10.1007/s00453-018-0463-
From Understanding Genetic Drift to a Smart-Restart Parameter-less Compact Genetic Algorithm
One of the key difficulties in using estimation-of-distribution algorithms is
choosing the population size(s) appropriately: Too small values lead to genetic
drift, which can cause enormous difficulties. In the regime with no genetic
drift, however, often the runtime is roughly proportional to the population
size, which renders large population sizes inefficient.
Based on a recent quantitative analysis which population sizes lead to
genetic drift, we propose a parameter-less version of the compact genetic
algorithm that automatically finds a suitable population size without spending
too much time in situations unfavorable due to genetic drift.
We prove a mathematical runtime guarantee for this algorithm and conduct an
extensive experimental analysis on four classic benchmark problems both without
and with additive centered Gaussian posterior noise. The former shows that
under a natural assumption, our algorithm has a performance very similar to the
one obtainable from the best problem-specific population size. The latter
confirms that missing the right population size in the original cGA can be
detrimental and that previous theory-based suggestions for the population size
can be far away from the right values; it also shows that our algorithm as well
as a previously proposed parameter-less variant of the cGA based on parallel
runs avoid such pitfalls. Comparing the two parameter-less approaches, ours
profits from its ability to abort runs which are likely to be stuck in a
genetic drift situation.Comment: 4 figures. Extended version of a paper appearing at GECCO 202
Neural Architecture Search by Estimation of Network Structure Distributions
The influence of deep learning is continuously expanding across different
domains, and its new applications are ubiquitous. The question of neural
network design thus increases in importance, as traditional empirical
approaches are reaching their limits. Manual design of network architectures
from scratch relies heavily on trial and error, while using existing pretrained
models can introduce redundancies or vulnerabilities. Automated neural
architecture design is able to overcome these problems, but the most successful
algorithms operate on significantly constrained design spaces, assuming the
target network to consist of identical repeating blocks. While such approach
allows for faster search, it does so at the cost of expressivity. We instead
propose an alternative probabilistic representation of a whole neural network
structure under the assumption of independence between layer types. Our matrix
of probabilities is equivalent to the population of models, but allows for
discovery of structural irregularities, while being simple to interpret and
analyze. We construct an architecture search algorithm, inspired by the
estimation of distribution algorithms, to take advantage of this
representation. The probability matrix is tuned towards generating
high-performance models by repeatedly sampling the architectures and evaluating
the corresponding networks, while gradually increasing the model depth. Our
algorithm is shown to discover non-regular models which cannot be expressed via
blocks, but are competitive both in accuracy and computational cost, while not
utilizing complex dataflows or advanced training techniques, as well as
remaining conceptually simple and highly extensible.Comment: 16 pages, 4 figures, 3 table
Runtime analysis of the univariate marginal distribution algorithm under low selective pressure and prior noise
We perform a rigorous runtime analysis for the Univariate Marginal
Distribution Algorithm on the LeadingOnes function, a well-known benchmark
function in the theory community of evolutionary computation with a high
correlation between decision variables. For a problem instance of size , the
currently best known upper bound on the expected runtime is
(Dang and Lehre, GECCO 2015), while a
lower bound necessary to understand how the algorithm copes with variable
dependencies is still missing. Motivated by this, we show that the algorithm
requires a runtime with high probability and in expectation
if the selective pressure is low; otherwise, we obtain a lower bound of
on the expected runtime.
Furthermore, we for the first time consider the algorithm on the function under
a prior noise model and obtain an expected runtime for the
optimal parameter settings. In the end, our theoretical results are accompanied
by empirical findings, not only matching with rigorous analyses but also
providing new insights into the behaviour of the algorithm.Comment: To appear at GECCO 2019, Prague, Czech Republi
On the limitations of the univariate marginal distribution algorithm to deception and where bivariate EDAs might help
We introduce a new benchmark problem called Deceptive Leading Blocks (DLB) to
rigorously study the runtime of the Univariate Marginal Distribution Algorithm
(UMDA) in the presence of epistasis and deception. We show that simple
Evolutionary Algorithms (EAs) outperform the UMDA unless the selective pressure
is extremely high, where and are the parent and
offspring population sizes, respectively. More precisely, we show that the UMDA
with a parent population size of has an expected runtime
of on the DLB problem assuming any selective pressure
, as opposed to the expected runtime
of for the non-elitist
with . These results illustrate
inherent limitations of univariate EDAs against deception and epistasis, which
are common characteristics of real-world problems. In contrast, empirical
evidence reveals the efficiency of the bi-variate MIMIC algorithm on the DLB
problem. Our results suggest that one should consider EDAs with more complex
probabilistic models when optimising problems with some degree of epistasis and
deception.Comment: To appear in the 15th ACM/SIGEVO Workshop on Foundations of Genetic
Algorithms (FOGA XV), Potsdam, German
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