Complexity Theory for Discrete Black-Box Optimization Heuristics

A predominant topic in the theory of evolutionary algorithms and, more generally, theory of randomized black-box optimization techniques is running time analysis. Running time analysis aims at understanding the performance of a given heuristic on a given problem by bounding the number of function evaluations that are needed by the heuristic to identify a solution of a desired quality. As in general algorithms theory, this running time perspective is most useful when it is complemented by a meaningful complexity theory that studies the limits of algorithmic solutions. In the context of discrete black-box optimization, several black-box complexity models have been developed to analyze the best possible performance that a black-box optimization algorithm can achieve on a given problem. The models differ in the classes of algorithms to which these lower bounds apply. This way, black-box complexity contributes to a better understanding of how certain algorithmic choices (such as the amount of memory used by a heuristic, its selective pressure, or properties of the strategies that it uses to create new solution candidates) influences performance. In this chapter we review the different black-box complexity models that have been proposed in the literature, survey the bounds that have been obtained for these models, and discuss how the interplay of running time analysis and black-box complexity can inspire new algorithmic solutions to well-researched problems in evolutionary computation. We also discuss in this chapter several interesting open questions for future work.Comment: This survey article is to appear (in a slightly modified form) in the book "Theory of Randomized Search Heuristics in Discrete Search Spaces", which will be published by Springer in 2018. The book is edited by Benjamin Doerr and Frank Neumann. Missing numbers of pointers to other chapters of this book will be added as soon as possibl

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 $(1+(\lambda,\lambda))$~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 $\lambda$ 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

Unbiased Black-Box Complexities of Jump Functions

We analyze the unbiased black-box complexity of jump functions with small, medium, and large sizes of the fitness plateau surrounding the optimal solution. Among other results, we show that when the jump size is $(1/2 - \varepsilon)n$, that is, only a small constant fraction of the fitness values is visible, then the unbiased black-box complexities for arities $3$ and higher are of the same order as those for the simple \textsc{OneMax} function. Even for the extreme jump function, in which all but the two fitness values $n/2$ and $n$ are blanked out, polynomial-time mutation-based (i.e., unary unbiased) black-box optimization algorithms exist. This is quite surprising given that for the extreme jump function almost the whole search space (all but a $\Theta(n^{-1/2})$ fraction) is a plateau of constant fitness. To prove these results, we introduce new tools for the analysis of unbiased black-box complexities, for example, selecting the new parent individual not by comparing the fitnesses of the competing search points, but also by taking into account the (empirical) expected fitnesses of their offspring.Comment: This paper is based on results presented in the conference versions [GECCO 2011] and [GECCO 2014

Parameterized Complexity Analysis of Randomized Search Heuristics

This chapter compiles a number of results that apply the theory of parameterized algorithmics to the running-time analysis of randomized search heuristics such as evolutionary algorithms. The parameterized approach articulates the running time of algorithms solving combinatorial problems in finer detail than traditional approaches from classical complexity theory. We outline the main results and proof techniques for a collection of randomized search heuristics tasked to solve NP-hard combinatorial optimization problems such as finding a minimum vertex cover in a graph, finding a maximum leaf spanning tree in a graph, and the traveling salesperson problem.Comment: This is a preliminary version of a chapter in the book "Theory of Evolutionary Computation: Recent Developments in Discrete Optimization", edited by Benjamin Doerr and Frank Neumann, published by Springe

A Fitness Function Elimination Theory For Blackbox Optimization And Problem Class Learning

The modern view of optimization is that optimization algorithms are not designed in a vacuum, but can make use of information regarding the broad class of objective functions from which a problem instance is drawn. Using this knowledge, we want to design optimization algorithms that execute quickly (efficiency), solve the objective function with minimal samples (performance), and are applicable over a wide range of problems (abstraction). However, we present a new theory for blackbox optimization from which, we conclude that of these three desired characteristics, only two can be maximized by any algorithm. We put forward an alternate view of optimization where we use knowledge about the problem class and samples from the problem instance to identify which problem instances from the class are being solved. From this Elimination of Fitness Functions approach, an idealized optimization algorithm that minimizes sample counts over any problem class, given complete knowledge about the class, is designed. This theory allows us to learn more about the difficulty of various problems, and we are able to use it to develop problem complexity bounds. We present general methods to model this algorithm over a particular problem class and gain efficiency at the cost of specifically targeting that class. This is demonstrated over the Generalized Leading-Ones problem and a generalization called LO∗∗ , and efficient algorithms with optimal performance are derived and analyzed. We also iii tighten existing bounds for LO∗∗∗. Additionally, we present a probabilistic framework based on our Elimination of Fitness Functions approach that clarifies how one can ideally learn about the problem class we face from the objective functions. This problem learning increases the performance of an optimization algorithm at the cost of abstraction. In the context of this theory, we re-examine the blackbox framework as an algorithm design framework and suggest several improvements to existing methods, including incorporating problem learning, not being restricted to blackbox framework and building parametrized algorithms. We feel that this theory and our recommendations will help a practitioner make substantially better use of all that is available in typical practical optimization algorithm design scenarios