Multiplicative Drift Analysis

Evolutionary algorithms are randomized search heuristics that are often described as robust general purpose problem solvers. It is known, however, that the performance of an evolutionary algorithm may be very sensitive to the setting of some of its parameters. A different perspective is to investigate changes in the expected optimization time due to small changes in the fitness landscape. A class of fitness functions where the expected optimization time of the (1+1) evolutionary algorithm is of the same magnitude for almost all of its members is the set of linear fitness functions. Using linear functions as a starting point, a model of a fitness landscape is devised that incorporates important properties of linear functions. Unexpectedly, the expected optimization time of the (1+1) evolutionary algorithm is clearly larger for this fitness model than on linear functions.

Analysis of speedups in parallel evolutionary algorithms and [formula omitted] EAs for combinatorial optimization

Evolutionary algorithms are popular heuristics for solving various combinatorial problems as they are easy to apply and often produce good results. Island models parallelize evolution by using different populations, called islands, which are connected by a graph structure as communication topology. Each island periodically communicates copies of good solutions to neighboring islands in a process called migration. We consider the speedup gained by island models in terms of the parallel running time for problems from combinatorial optimization: sorting (as maximization of sortedness), shortest paths, and Eulerian cycles. Different search operators are considered. The results show in which settings and up to what degree evolutionary algorithms can be parallelized efficiently. Along the way, we also investigate how island models deal with plateaus. In particular, we show that natural settings lead to exponential vs. logarithmic speedups, depending on the frequency of migration.KeywordsEvolutionary AlgorithmMutation OperatorLinear SpeedupShort Path ProblemSearch PointThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Drift analysis is one of the state-of-the-art techniques for the runtime analysis of randomized search heuristics (RSHs) such as evolutionary algorithms (EAs), simulated annealing,etc. The vast majority of existing drift theorems yield bounds on the expected value of the hitting time for a target state, for example the set of optimal solutions, without making additional statements on the distribution of this time. We address this lack by providing a general drift theorem that includes bounds on the upper and lower tail of the hitting time distribution. The new tail bounds are applied to prove very precise sharp-concentration results on the running time of a simple EA on standard benchmark problems, including the class of general linear functions. On all these problems, the probability of deviating by anr-factor in lower-order terms of the expected time decreases exponentially withr. The usefulness of the theorem outside the theory of RSHs is demonstrated by deriving tail bounds on the number of cycles in random permutations. All these results handle a position-dependent (variable) drift that was not covered by previous drift theorems with tail bounds. Finally, user-friendly specializations of the general drift theorem are given.

We consider the expected runtime of non-elitist evolutionary algorithms (EAs), when they are applied to a family of fitness functions with a plateau of second-best fitness in a Hamming ball of radius r around a unique global optimum. On one hand, using the level-based theorems, we obtain polynomial upper bounds on the expected runtime for some modes of non-elitist EA based on unbiased mutation and the bitwise mutation in particular. On the other hand, we show that the EA with fitness proportionate selection is inefficient if the bitwise mutation is used with the standard settings of mutation probability.

Many-objective optimization focuses on solving optimization problems with four or more objectives. Effort has been made mainly on studying continuous problems, with interesting results and for which several optimizers have been proposed. Nevertheless, combinatorial problems have not received as much attention, making this an open research area. An important result on continuous problems states that the problem does not necessarily becomes more difficult while more objectives are considered, but, does this result hold for combinatorial problems? This investigation takes this subject on by studying a many-objective combinatorial problem, particularly, the pickup and delivery problem (PDP), which is an important combinatorial optimization problem in the transportation industry and consists in finding a collection of routes with minimum cost. Traditionally, cost has been associated with the number of routes and the total travel distance, however, some other objectives emerge in many applications, for example, travel time, workload imbalance, and uncollected profit. If we consider all these objectives equally important, PDP can be tackled as a many-objective problem. This study is concerned with the study of the performance of three multi-objective evolutionary algorithms on the PDP varying the number of objectives, in order to analyze the change of PDP’s difficulty when the number of objectives is increased. Results show that the problem becomes more difficult to solve while more objectives are considered.

Toward SLA-constrained service composition: An approach based on a fuzzy linguistic preference model and an evolutionary algorithm

Evolutionary algorithms are randomized search heuristics whose general variants have been successfully applied in black box optimization. In this scenario the function f to be optimized is not known in advance and knowledge on f can be obtained only by sampling search points a revealing the value of f(a). In order to analyze the behavior of different variants of evolutionary algorithms on certain functions f, the expected runtime until some optimal search point is sampled and the success probability, i.e., the probability that an optimal search point is among the first sampled points, are of particular interest. Here a simple method for the analysis is discussed and applied to several functions. For specific situations more involved techniques are necessary. Two such results are presented. First, it is shown that the most simple evolutionary algorithm optimizes each pseudo-boolean linear function in an expected time of O(n log n). Second, an example is shown where crossover decreases the expected runtime from superpolynomial to polynomial.

First-hitting times under drift

Estimation of distribution algorithms (EDAs) solve an optimization problem heuristically by finding a probability distribution focused around its optima. Starting with the uniform distribution, points are sampled with respect to this distribution and the distribution is changed according to the function values of the sampled points. Although there are many successful experiments suggesting the usefulness of EDAs, there are only few rigorous theoretical results apart from convergence results without time bounds. Here we present first rigorous runtime analyses of a simple EDA, the compact genetic algorithm (cGA), for linear pseudo-Boolean functions on n variables. We prove a general lower bound for all functions and a general upper bound for all linear functions. Simple test functions show that not all linear functions are optimized in the same runtime by the cGA.

We study unbiased (1 + 1) evolutionary algorithms on linear functions with an unknown number n of bits with non-zero weight. Static algorithms achieve an optimal runtime of O(n(ln n)2+e), however, it remained unclear whether more dynamic parameter policies could yield better runtime guarantees. We consider two setups: one where the mutation rate follows a fixed schedule, and one where it may be adapted depending on the history of the run. For the first setup, we give a schedule that achieves a runtime of (1±o(1))βn ln n, where β ≈ 3.552, which is an asymptotic improvement over the runtime of the static setup. Moreover, we show that no schedule admits a better runtime guarantee and that the optimal schedule is essentially unique. For the second setup, we show that the runtime can be further improved to (1 ± o(1))en ln n, which matches the performance of algorithms that know n in advance. Finally, we study the related model of initial segment uncertainty with static position-dependent mutation rates, and derive asymptotically optimal lower bounds. This answers a question by Doerr, Doerr, and Kotzing.

One important goal of black-box complexity theory is the development of complexity models allowing to derive meaningful lower bounds for whole classes of randomized search heuristics. Complementing classical runtime analysis, black-box models help us to understand how algorithmic choices such as the population size, the variation operators, or the selection rules influence the optimization time. One example for such a result is the $$\varOmega (n \log n)$$ lower bound for unary unbiased algorithms on functions with a unique global optimum (Lehre and Witt in Algorithmica 64:623–642, 2012), which tells us that higher arity operators or biased sampling strategies are needed when trying to beat this bound. In lack of analyzing techniques, such non-trivial lower bounds are very rare in the existing literature on black-box optimization and therefore remain to be one of the main challenges in black-box complexity theory. With this paper we contribute to our technical toolbox for lower bound computations by proposing a new type of information-theoretic argument. We regard the permutation- and bit-invariant version of LeadingOnes and prove that its $$(1+1)$$ elitist black-box complexity is $$\varOmega (n^2)$$ , a bound that is matched by $$(1+1)$$ -type evolutionary algorithms. The $$(1+1)$$ elitist complexity of LeadingOnes is thus considerably larger than its unrestricted one, which is known to be of order $$n\log \log n$$ (Afshani et al. in Lecture notes in computer science, vol 8066, pp 1–11. Springer, New York, 2013). The $$\varOmega (n^2)$$ lower bound does not rely on the fact that elitist black-box algorithms are not allowed to make use of absolute fitness values. In contrast, we show that even if absolute fitness values are revealed to the otherwise elitist algorithm, it cannot significantly profit from this additional information. Our result thus shows that for LeadingOnes the memory-restriction, together with the selection requirement, has a substantial impact on the best possible performance.

For many years, Evolutionary Algorithms (EAs) have been applied to improve Neural Networks (NNs) architectures. They have been used for solving different problems, such as training the networks (adjusting the weights), designing network topology, optimizing global parameters, and selecting features. Here, we provide a systematic brief survey about applications of the EAs on the specific domain of the recurrent NNs named Reservoir Computing (RC). At the beginning of the 2000s, the RC paradigm appeared as a good option for employing recurrent NNs without dealing with the inconveniences of the training algorithms. RC models use a nonlinear dynamic system, with fixed recurrent neural network named the reservoir, and learning process is restricted to adjusting a linear parametric function. However, an RC model has several hyper-parameters, therefore EAs are helpful tools to figure out optimal RC architectures. We provide an overview of the results on the area, discuss novel advances, and we present our vision regarding the new trends and still open questions.

We study the stochastic versions of a broad class of combinatorial problems where the weights of the elements in the input dataset are uncertain. The class of problems that we study includes shortest paths, minimum weight spanning trees, and minimum weight matchings over probabilistic graphs, and other combinatorial problems like knapsack. We observe that the expected value is inadequate in capturing different types of risk averse or risk-prone behaviors, and instead we consider a more general objective which is to maximize the expected utility of the solution for some given utility function, rather than the expected weight (expected weight becomes a special case). We show that we can obtain a polynomial time approximation algorithm with additive error ϵ for any ϵ >; 0, if there is a pseudopolynomial time algorithm for the exact version of the problem (This is true for the problems mentioned above) and the maximum value of the utility function is bounded by a constant. Our result generalizes several prior results on stochastic shortest path, stochastic spanning tree, and stochastic knapsack. Our algorithm for utility maximization makes use of the separability of exponential utility and a technique to decompose a general utility function into exponential utility functions, which may be useful in other stochastic optimization problems.

Recently, ant colony optimization (ACO) algorithms have proven to be efficient in uncertain environments, such as noisy or dynamically changing fitness functions. Most of these analyses have focused on combinatorial problems such as path finding. We rigorously analyze an ACO algorithm optimizing linear pseudo-Boolean functions under additive posterior noise. We study noise distributions whose tails decay exponentially fast, including the classical case of additive Gaussian noise. Without noise, the classical [Formula: see text] EA outperforms any ACO algorithm, with smaller [Formula: see text] being better; however, in the case of large noise, the [Formula: see text] EA fails, even for high values of [Formula: see text] (which are known to help against small noise). In this article, we show that ACO is able to deal with arbitrarily large noise in a graceful manner; that is, as long as the evaporation factor [Formula: see text] is small enough, dependent on the variance [Formula: see text] of the noise and the dimension n of the search space, optimization will be successful. We also briefly consider the case of prior noise and prove that ACO can also efficiently optimize linear functions under this noise model.