Playing Mastermind With Constant-Size Memory

We analyze the classic board game of Mastermind with $n$ holes and a constant number of colors. A result of Chv\'atal (Combinatorica 3 (1983), 325-329) states that the codebreaker can find the secret code with $\Theta(n / \log n)$ questions. We show that this bound remains valid if the codebreaker may only store a constant number of guesses and answers. In addition to an intrinsic interest in this question, our result also disproves a conjecture of Droste, Jansen, and Wegener (Theory of Computing Systems 39 (2006), 525-544) on the memory-restricted black-box complexity of the OneMax function class.Comment: 23 page

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

OneMax in Black-Box Models with Several Restrictions

Black-box complexity studies lower bounds for the efficiency of general-purpose black-box optimization algorithms such as evolutionary algorithms and other search heuristics. Different models exist, each one being designed to analyze a different aspect of typical heuristics such as the memory size or the variation operators in use. While most of the previous works focus on one particular such aspect, we consider in this work how the combination of several algorithmic restrictions influence the black-box complexity. Our testbed are so-called OneMax functions, a classical set of test functions that is intimately related to classic coin-weighing problems and to the board game Mastermind. We analyze in particular the combined memory-restricted ranking-based black-box complexity of OneMax for different memory sizes. While its isolated memory-restricted as well as its ranking-based black-box complexity for bit strings of length $n$ is only of order $n/\log n$, the combined model does not allow for algorithms being faster than linear in $n$, as can be seen by standard information-theoretic considerations. We show that this linear bound is indeed asymptotically tight. Similar results are obtained for other memory- and offspring-sizes. Our results also apply to the (Monte Carlo) complexity of OneMax in the recently introduced elitist model, in which only the best-so-far solution can be kept in the memory. Finally, we also provide improved lower bounds for the complexity of OneMax in the regarded models. Our result enlivens the quest for natural evolutionary algorithms optimizing OneMax in $o(n \log n)$ iterations.Comment: This is the full version of a paper accepted to GECCO 201

Toward a complexity theory for randomized search heuristics : black-box models

Randomized search heuristics are a broadly used class of general-purpose algorithms. Analyzing them via classical methods of theoretical computer science is a growing field. While several strong runtime bounds exist, a powerful complexity theory for such algorithms is yet to be developed. We contribute to this goal in several aspects. In a first step, we analyze existing black-box complexity models. Our results indicate that these models are not restrictive enough. This remains true if we restrict the memory of the algorithms under consideration. These results motivate us to enrich the existing notions of black-box complexity by the additional restriction that not actual objective values, but only the relative quality of the previously evaluated solutions may be taken into account by the algorithms. Many heuristics belong to this class of algorithms. We show that our ranking-based model gives more realistic complexity estimates for some problems, while for others the low complexities of the previous models still hold. Surprisingly, our results have an interesting game-theoretic aspect as well.We show that analyzing the black-box complexity of the OneMaxn function class—a class often regarded to analyze how heuristics progress in easy parts of the search space—is the same as analyzing optimal winning strategies for the generalized Mastermind game with 2 colors and length-n codewords. This connection was seemingly overlooked so far in the search heuristics community.Randomisierte Suchheuristiken sind vielseitig einsetzbare Algorithmen, die aufgrund ihrer hohen Flexibilität nicht nur im industriellen Kontext weit verbreitet sind. Trotz zahlreicher erfolgreicher Anwendungsbeispiele steckt die Laufzeitanalyse solcher Heuristiken noch in ihren Kinderschuhen. Insbesondere fehlt es uns an einem guten Verständnis, in welchen Situationen problemunabhängige Heuristiken in kurzer Laufzeit gute Lösungen liefern können. Eine Komplexitätstheorie ähnlich wie es sie in der klassischen Algorithmik gibt, wäre wünschenswert. Mit dieser Arbeit tragen wir zur Entwicklung einer solchen Komplexitätstheorie für Suchheuristiken bei. Wir zeigen anhand verschiedener Beispiele, dass existierende Modelle die Schwierigkeit eines Problems nicht immer zufriedenstellend erfassen. Wir schlagen daher ein weiteres Modell vor. In unserem Ranking-Based Black-Box Model lernen die Algorithmen keine exakten Funktionswerte, sondern bloß die Rangordnung der bislang angefragten Suchpunkte. Dieses Modell gibt für manche Probleme eine bessere Einschätzung der Schwierigkeit. Wir zeigen jedoch auch, dass auch im neuen Modell Probleme existieren, deren Komplexität als zu gering einzuschätzen ist. Unsere Ergebnisse haben auch einen spieltheoretischen Aspekt. Optimale Gewinnstrategien für den Rater im Mastermindspiel (auch SuperHirn) mit n Positionen entsprechen genau optimalen Algorithmen zur Maximierung von OneMaxn-Funktionen. Dieser Zusammenhang wurde scheinbar bislang übersehen. Diese Arbeit ist in englischer Sprache verfasst

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