25 research outputs found
How to Escape Local Optima in Black Box Optimisation: When Non-elitism Outperforms Elitism
Escaping local optima is one of the major obstacles to function optimisation. Using the metaphor of a fitness landscape, local optima correspond to hills separated by fitness valleys that have to be overcome. We define a class of fitness valleys of tunable difficulty by considering their length, representing the Hamming path between the two optima and their depth, the drop in fitness. For this function class we present a runtime comparison between stochastic search algorithms using different search strategies. The ((Formula presented.)) EA is a simple and well-studied evolutionary algorithm that has to jump across the valley to a point of higher fitness because it does not accept worsening moves (elitism). In contrast, the Metropolis algorithm and the Strong Selection Weak Mutation (SSWM) algorithm, a famous process in population genetics, are both able to cross the fitness valley by accepting worsening moves. We show that the runtime of the ((Formula presented.)) EA depends critically on the length of the valley while the runtimes of the non-elitist algorithms depend crucially on the depth of the valley. Moreover, we show that both SSWM and Metropolis can also efficiently optimise a rugged function consisting of consecutive valleys
When non-elitism outperforms elitism for crossing fitness valleys
Crossing fitness valleys is one of the major obstacles to function optimization. In this paper we investigate how the structure of the fitness valley, namely its depth d and length ℓ, influence the runtime of different strategies for crossing these valleys. We present a runtime comparison between the (1+1) EA and two non-elitist nature-inspired algorithms, Strong Selection Weak Mutation (SSWM) and the Metropolis algorithm. While the (1+1) EA has to jump across the valley to a point of higher fitness because it does not accept decreasing moves, the non-elitist algorithms may cross the valley by accepting worsening moves. We show that while the runtime of the (1+1) EA algorithm depends critically on the length of the valley, the runtimes of the non-elitist algorithms depend crucially only on the depth of the valley. In particular, the expected runtime of both SSWM and Metropolis is polynomial in ℓ and exponential in d while the (1+1) EA is efficient only for valleys of small length. Moreover, we show that both SSWM and Metropolis can also efficiently optimize a rugged function consisting of consecutive valleys
Simulated Annealing is a Polynomial-Time Approximation Scheme for the Minimum Spanning Tree Problem
We prove that Simulated Annealing with an appropriate cooling schedule
computes arbitrarily tight constant-factor approximations to the minimum
spanning tree problem in polynomial time. This result was conjectured by
Wegener (2005). More precisely, denoting by , and
the number of vertices and edges as well as the maximum and minimum edge weight
of the MST instance, we prove that simulated annealing with initial temperature
and multiplicative cooling schedule with factor ,
where , with probability at least computes in
time a spanning tree with weight
at most times the optimum weight, where . Consequently, for any
, we can choose in such a way that a
-approximation is found in time
with
probability at least . In the special case of so-called
-separated weights, this algorithm computes an optimal solution
(again in time ), which is a significant speed-up over Wegener's runtime
guarantee of .Comment: 19 pages. Full version of a paper appearing at GECCO 202
First Steps Towards a Runtime Comparison of Natural and Artificial Evolution
Evolutionary algorithms (EAs) form a popular optimisation paradigm inspired
by natural evolution. In recent years the field of evolutionary computation has
developed a rigorous analytical theory to analyse their runtime on many
illustrative problems. Here we apply this theory to a simple model of natural
evolution. In the Strong Selection Weak Mutation (SSWM) evolutionary regime the
time between occurrence of new mutations is much longer than the time it takes
for a new beneficial mutation to take over the population. In this situation,
the population only contains copies of one genotype and evolution can be
modelled as a (1+1)-type process where the probability of accepting a new
genotype (improvements or worsenings) depends on the change in fitness.
We present an initial runtime analysis of SSWM, quantifying its performance
for various parameters and investigating differences to the (1+1)EA. We show
that SSWM can have a moderate advantage over the (1+1)EA at crossing fitness
valleys and study an example where SSWM outperforms the (1+1)EA by taking
advantage of information on the fitness gradient
Towards a Runtime Comparison of Natural and Artificial Evolution
Evolutionary algorithms (EAs) form a popular optimisation paradigm inspired by natural evolution. In recent years the field of evolutionary computation has developed a rigorous analytical theory to analyse the runtimes of EAs on many illustrative problems. Here we apply this theory to a simple model of natural evolution. In the Strong Selection Weak Mutation (SSWM) evolutionary regime the time between occurrences of new mutations is much longer than the time it takes for a mutated genotype to take over the population. In this situation, the population only contains copies of one genotype and evolution can be modelled as a stochastic process evolving one genotype by means of mutation and selection between the resident and the mutated genotype. The probability of accepting the mutated genotype then depends on the change in fitness. We study this process, SSWM, from an algorithmic perspective, quantifying its expected optimisation time for various parameters and investigating differences to a similar evolutionary algorithm, the well-known (1+1) EA. We show that SSWM can have a moderate advantage over the (1+1) EA at crossing fitness valleys and study an example where SSWM outperforms the (1+1) EA by taking advantage of information on the fitness gradient
Characterising fitness landscapes with fitness-probability cloud and its applications to algorithm configuration
Metaheuristics are approximation optimisation techniques widely applied to solve complex optimisation problems. Despite a large number of developed metaheuristic algorithms, a limited amount of work has been done to understand on which kinds of problems the proposed algorithm will perform well or poorly and why. A useful solution to this dilemma is to use fitness landscape analysis to gain an in-depth understanding of which algorithms, or algorithm variants are best suited for solving which kinds of problem instances, even to dynamically determine the best algorithm configuration during different stages of a search algorithm.
This thesis for the first time bridges the gap between fitness landscape analysis and algorithm configuration, i.e., finding the best suited configuration of a given algorithm for solving a particular problem instance. Studies in this thesis contribute to the following:
a. Developing a novel and effective approach to characterise fitness landscapes and measure problem difficulty with respect to algorithms.
b. Incorporating fitness landscape analysis in building a generic (problem-independent) approach, which can perform automatic algorithm configuration on a per-instance base, and in designing novel and effective algorithm configurations.
c. Incorporating fitness landscape analysis in establishing a generic framework for designing adaptive heuristic algorithms
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Extending the class of order-k delineable problems for the gene expression messy genetic algorithm
This paper revisits the recently introduced gene expression messy genetic algorithm (GEMGA) and offers some modifications to the extend the class of order-k delineable problems (class of problems that can be solved using a bounded order of relations) in GEMGA. The fundamental components that control the delineability of relations are reviewed in the light of the recently proposed SEARCH framework. Modified class and relation comparison statistics of GEMGA are proposed. The sample complexity of this improved version of GEMGA is shown to be subquadratic. Theoretical conclusions are also substantiated by experimental results for large, multimodal order-k delineable problems with respect to class average comparison statistic. We also present results for the recently constructed Goldberg-Lewei test functions
On the Runtime Analysis of Selection Hyper-heuristics for Pseudo-Boolean Optimisation
Rather than manually deciding on a suitable algorithm configuration for a given optimisation problem, hyper-heuristics are high-level search algorithms which evolve the heuristic to be applied. While there are numerous reported successful applications of hyper-heuristics to
combinatorial optimisation problems, it is not yet fully understood how well they perform and on which problem classes they are effective. Selection hyper-heuristics (SHHs) employ smart methodologies to select from a pre-defined set of low-level heuristics which to apply
in the next decision step. This thesis extends and improves upon the existing foundational
understanding of the behaviour and performance of SHHs, providing insights into how and when they can be successfully applied by analysing the time complexity of SHHs on a variety of unimodal and multimodal problem classes.
Through a rigorous theoretical analysis, we show that while four commonly applied simple SHHs from the literature do not learn to select the most promising low-level heuristics, generalising them such that application of the chosen heuristic occurs over a longer period
of time allows for vastly improved performance. Furthermore, we prove that extending the size of the set of low-level heuristics can improve the performance of the generalised SHHs, outperforming SHHs with smaller sets of low-level heuristics. We show that allowing
the SHH to automatically adapt the length of the learning period may further improve the performance and outperform non-adaptive variants. SHHs selecting between two move-acceptance operators are also analysed on two classes of multimodal benchmark functions. An analysis of the performance of simple SHHs on these functions provides insights into the effectiveness of the presented methodologies for escaping from local optima
Analysis of combinatorial search spaces for a class of NP-hard problems, An
2011 Spring.Includes bibliographical references.Given a finite but very large set of states X and a real-valued objective function ƒ defined on X, combinatorial optimization refers to the problem of finding elements of X that maximize (or minimize) ƒ. Many combinatorial search algorithms employ some perturbation operator to hill-climb in the search space. Such perturbative local search algorithms are state of the art for many classes of NP-hard combinatorial optimization problems such as maximum k-satisfiability, scheduling, and problems of graph theory. In this thesis we analyze combinatorial search spaces by expanding the objective function into a (sparse) series of basis functions. While most analyses of the distribution of function values in the search space must rely on empirical sampling, the basis function expansion allows us to directly study the distribution of function values across regions of states for combinatorial problems without the need for sampling. We concentrate on objective functions that can be expressed as bounded pseudo-Boolean functions which are NP-hard to solve in general. We use the basis expansion to construct a polynomial-time algorithm for exactly computing constant-degree moments of the objective function ƒ over arbitrarily large regions of the search space. On functions with restricted codomains, these moments are related to the true distribution by a system of linear equations. Given low moments supplied by our algorithm, we construct bounds of the true distribution of ƒ over regions of the space using a linear programming approach. A straightforward relaxation allows us to efficiently approximate the distribution and hence quickly estimate the count of states in a given region that have certain values under the objective function. The analysis is also useful for characterizing properties of specific combinatorial problems. For instance, by connecting search space analysis to the theory of inapproximability, we prove that the bound specified by Grover's maximum principle for the Max-Ek-Lin-2 problem is sharp. Moreover, we use the framework to prove certain configurations are forbidden in regions of the Max-3-Sat search space, supplying the first theoretical confirmation of empirical results by others. Finally, we show that theoretical results can be used to drive the design of algorithms in a principled manner by using the search space analysis developed in this thesis in algorithmic applications. First, information obtained from our moment retrieving algorithm can be used to direct a hill-climbing search across plateaus in the Max-k-Sat search space. Second, the analysis can be used to control the mutation rate on a (1+1) evolutionary algorithm on bounded pseudo-Boolean functions so that the offspring of each search point is maximized in expectation. For these applications, knowledge of the search space structure supplied by the analysis translates to significant gains in the performance of search