132 research outputs found
Runtime Analysis for Self-adaptive Mutation Rates
We propose and analyze a self-adaptive version of the
evolutionary algorithm in which the current mutation rate is part of the
individual and thus also subject to mutation. A rigorous runtime analysis on
the OneMax benchmark function reveals that a simple local mutation scheme for
the rate leads to an expected optimization time (number of fitness evaluations)
of when is at least for
some constant . For all values of , this
performance is asymptotically best possible among all -parallel
mutation-based unbiased black-box algorithms.
Our result shows that self-adaptation in evolutionary computation can find
complex optimal parameter settings on the fly. At the same time, it proves that
a relatively complicated self-adjusting scheme for the mutation rate proposed
by Doerr, Gie{\ss}en, Witt, and Yang~(GECCO~2017) can be replaced by our simple
endogenous scheme.
On the technical side, the paper contributes new tools for the analysis of
two-dimensional drift processes arising in the analysis of dynamic parameter
choices in EAs, including bounds on occupation probabilities in processes with
non-constant drift
How Well Does the Metropolis Algorithm Cope With Local Optima?
The Metropolis algorithm (MA) is a classic stochastic local search heuristic.
It avoids getting stuck in local optima by occasionally accepting inferior
solutions. To better and in a rigorous manner understand this ability, we
conduct a mathematical runtime analysis of the MA on the CLIFF benchmark. Apart
from one local optimum, cliff functions are monotonically increasing towards
the global optimum. Consequently, to optimize a cliff function, the MA only
once needs to accept an inferior solution. Despite seemingly being an ideal
benchmark for the MA to profit from its main working principle, our
mathematical runtime analysis shows that this hope does not come true. Even
with the optimal temperature (the only parameter of the MA), the MA optimizes
most cliff functions less efficiently than simple elitist evolutionary
algorithms (EAs), which can only leave the local optimum by generating a
superior solution possibly far away. This result suggests that our
understanding of why the MA is often very successful in practice is not yet
complete. Our work also suggests to equip the MA with global mutation
operators, an idea supported by our preliminary experiments.Comment: To appear in the proceedings of GECCO 2023. With appendix containing
all proofs. 28 page
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
Drift Analysis with Fitness Levels for Elitist Evolutionary Algorithms
The fitness level method is a popular tool for analyzing the computation time
of elitist evolutionary algorithms. Its idea is to divide the search space into
multiple fitness levels and estimate lower and upper bounds on the computation
time using transition probabilities between fitness levels. However, the lower
bound generated from this method is often not tight. To improve the lower
bound, this paper rigorously studies an open question about the fitness level
method: what are the tightest lower and upper time bounds that can be
constructed based on fitness levels? To answer this question, drift analysis
with fitness levels is developed, and the tightest bound problem is formulated
as a constrained multi-objective optimization problem subject to fitness level
constraints. The tightest metric bounds from fitness levels are constructed and
proven for the first time. Then the metric bounds are converted into linear
bounds, where existing linear bounds are special cases. This paper establishes
a general framework that can cover various linear bounds from trivial to best
coefficients. It is generic and promising, as it can be used not only to draw
the same bounds as existing ones, but also to draw tighter bounds, especially
on fitness landscapes where shortcuts exist. This is demonstrated in the case
study of the (1+1) EA maximizing the TwoPath function
Runtime analysis of non-elitist populations: from classical optimisation to partial information
Although widely applied in optimisation, relatively little has been proven rigorously about the role and behaviour of populations in randomised search processes. This paper presents a new method to prove upper bounds on the expected optimisation time of population-based randomised search heuristics that use non-elitist selection mechanisms and unary variation operators. Our results follow from a detailed drift analysis of the population dynamics in these heuristics. This analysis shows that the optimisation time depends on the relationship between the strength of the selective pressure and the degree of variation introduced by the variation operator. Given limited variation, a surprisingly weak selective pressure suffices to optimise many functions in expected polynomial time. We derive upper bounds on the expected optimisation time of non-elitist Evolutionary Algorithms (EA) using various selection mechanisms, including fitness proportionate selection. We show that EAs using fitness proportionate selection can optimise standard benchmark functions in expected polynomial time given a sufficiently low mutation rate.
As a second contribution, we consider an optimisation scenario with partial information, where fitness values of solutions are only partially available. We prove that non-elitist EAs under a set of specific conditions can optimise benchmark functions in expected polynomial time, even when vanishingly little information about the fitness values of individual solutions or populations is available. To our knowledge, this is the first runtime analysis of randomised search heuristics under partial information
The First Proven Performance Guarantees for the Non-Dominated Sorting Genetic Algorithm II (NSGA-II) on a Combinatorial Optimization Problem
The Non-dominated Sorting Genetic Algorithm-II (NSGA-II) is one of the most
prominent algorithms to solve multi-objective optimization problems. Recently,
the first mathematical runtime guarantees have been obtained for this
algorithm, however only for synthetic benchmark problems.
In this work, we give the first proven performance guarantees for a classic
optimization problem, the NP-complete bi-objective minimum spanning tree
problem. More specifically, we show that the NSGA-II with population size computes all extremal points of the Pareto front in
an expected number of iterations, where
is the number of vertices, the number of edges, and is the
maximum edge weight in the problem instance. This result confirms, via
mathematical means, the good performance of the NSGA-II observed empirically.
It also shows that mathematical analyses of this algorithm are not only
possible for synthetic benchmark problems, but also for more complex
combinatorial optimization problems.
As a side result, we also obtain a new analysis of the performance of the
global SEMO algorithm on the bi-objective minimum spanning tree problem, which
improves the previous best result by a factor of , the number of extremal
points of the Pareto front, a set that can be as large as . The
main reason for this improvement is our observation that both multi-objective
evolutionary algorithms find the different extremal points in parallel rather
than sequentially, as assumed in the previous proofs.Comment: Author-generated version of a paper appearing in the proceedings of
IJCAI 202
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