Using Prior Knowledge and Learning from Experience in Estimation of Distribution Algorithms

Estimation of distribution algorithms (EDAs) are stochastic optimization techniques that explore the space of potential solutions by building and sampling explicit probabilistic models of promising candidate solutions. One of the primary advantages of EDAs over many other stochastic optimization techniques is that after each run they leave behind a sequence of probabilistic models describing useful decompositions of the problem. This sequence of models can be seen as a roadmap of how the EDA solves the problem. While this roadmap holds a great deal of information about the problem, until recently this information has largely been ignored. My thesis is that it is possible to exploit this information to speed up problem solving in EDAs in a principled way. The main contribution of this dissertation will be to show that there are multiple ways to exploit this problem-specific knowledge. Most importantly, it can be done in a principled way such that these methods lead to substantial speedups without requiring parameter tuning or hand-inspection of models

Estimation of Distribution Algorithms and Minimum Relative Entropy

In the field of optimization using probabilistic models of the search space, this thesis identifies and elaborates several advancements in which the principles of maximum entropy and minimum relative entropy from information theory are used to estimate a probability distribution. The probability distribution within the search space is represented by a graphical model (factorization, Bayesian network or junction tree). An estimation of distribution algorithm (EDA) is an evolutionary optimization algorithm which uses a graphical model to sample a population within the search space and then estimates a new graphical model from the selected individuals of the population. - So far, the Factorized Distribution Algorithm (FDA) builds a factorization or Bayesian network from a given additive structure of the objective function to be optimized using a greedy algorithm which only considers a subset of the variable dependencies. Important connections can be lost by this method. This thesis presents a heuristic subfunction merge algorithm which is able to consider all dependencies between the variables (as long as the marginal distributions of the model do not become too large). On a 2-D grid structure, this algorithm builds a pentavariate factorization which allows to solve the deceptive grid benchmark problem with a much smaller population size than the conventional factorization. Especially for small population sizes, calculating large marginal distributions from smaller ones using Maximum Entropy and iterative proportional fitting leads to a further improvement. - The second topic is the generalization of graphical models to loopy structures. Using the Bethe-Kikuchi approximation, the loopy graphical model (region graph) can learn the Boltzmann distribution of an objective function by a generalized belief propagation algorithm (GBP). It minimizes the free energy, a notion adopted from statistical physics which is equivalent to the relative entropy to the Boltzmann distribution. Previous attempts to combine the Kikuchi approximation with EDA have relied on an expensive Gibbs sampling procedure for generating a population from this loopy probabilistic model. In this thesis a combination with a factorization is presented which allows more efficient sampling. The free energy is generalized to incorporate the inverse temperature ß. The factorization building algorithm mentioned above can be employed here, too. The dynamics of GBP is investigated, and the method is applied on Ising spin glass ground state search. Small instances (7 x 7) are solved without difficulty. Larger instances (10 x 10 and 15 x 15) do not converge to the true optimum with large ß, but sampling from the factorization can find the optimum with about 1000-10000 sampling attempts, depending on the instance. If GBP does not converge, it can be replaced by a concave-convex procedure which guarantees convergence. - Third, if no probabilistic structure is given for the objective function, a Bayesian network can be learned to capture the dependencies in the population. The relative entropy between the population-induced distribution and the Bayesian network distribution is equivalent to the log-likelihood of the model. The log-likelihood has been generalized to the BIC/MDL score which reduces overfitting by punishing complicated structure of the Bayesian network. A previous information theoretic analysis of BIC/MDL in the context of EDA is continued, and empiric evidence is given that the method is able to learn the correct structure of an objective function, given a sufficiently large population. - Finally, a way to reduce the search space of EDA is presented by combining it with a local search heuristics. The Kernighan Lin hillclimber, known originally for the traveling salesman problem and graph bipartitioning, is generalized to arbitrary binary problems. It can be applied in a stand-alone manner, as an iterative 1+1 search algorithm, or combined with EDA. On the MAXSAT problem it performs in a similar scale to the specialized SAT solver Walksat. An analysis of the Kernighan Lin local optima indicates that the combination with an EDA is favorable. The thesis shows how evolutionary optimization can be improved using interdisciplinary results from information theory, statistics, probability calculus and statistical physics. The principles of information theory for estimating probability distributions are applicable in many areas. EDAs are a good application because an improved estimation affects directly the optimization success.Estimation of Distribution Algorithms und Minimierung der relativen Entropie Im Bereich der Optimierung mit probabilistischen Modellen des Suchraums werden einige Fortschritte identifiziert und herausgearbeitet, in denen die Prinzipien der maximalen Entropie und der minimalen relativen Entropie aus der Informationstheorie verwendet werden, um eine Wahrscheinlichkeitsverteilung zu schätzen. Die Wahrscheinlichkeitsverteilung im Suchraum wird durch ein graphisches Modell beschrieben (Faktorisierung, Bayessches Netz oder Verbindungsbaum). Ein Estimation of Distribution Algorithm (EDA) ist ein evolutionärer Optimierungsalgorithmus, der mit Hilfe eines graphischen Modells eine Population im Suchraum erzeugt und dann anhand der selektierten Individuen dieser Population ein neues graphisches Modell erzeugt. - Bislang baut der Factorized Distribution Algorithm (FDA) eine Faktorisierung oder ein Bayessches Netz aus einer gegebenen additiven Struktur der Zielfunktion durch einen Greedy-Algorithmus, der nur einen Teil der Verbindungen zwischen den Variablen berücksichtigt. Wichtige verbindungen können durch diese Methode verloren gehen. Diese Arbeit stellt einen heuristischen Subfunktionenverschmelzungsalgorithmus vor, der in der Lage ist, alle Abhängigkeiten zwischen den Variablen zu berücksichtigen (wofern die Randverteilungen des Modells nicht zu groß werden). Auf einem 2D-Gitter erzeugt dieser Algorithmus eine pentavariate Faktorisierung, die es ermöglicht, das Deceptive-Grid-Testproblem mit viel kleinerer Populationsgröße zu lösen als mit der konventionellen Faktorisierung. Insbesondere für kleine Populationsgrößen kann das Ergebnis noch verbessert werden, wenn große Randverteilungen aus kleineren vermittels des Prinzips der maximalen Entropie und des Iterative Proportional Fitting- Algorithmus berechnet werden. - Das zweite Thema ist die Verallgemeinerung graphischer Modelle zu zirkulären Strukturen. Mit der Bethe-Kikuchi-Approximation kann das zirkuläre graphische Modell (der Regionen-Graph) die Boltzmannverteilung einer Zielfunktion durch einen generalisierten Belief Propagation-Algorithmus (GBP) lernen. Er minimiert die freie Energie, eine Größe aus der statistischen Physik, die äquivalent zur relativen Entropie zur Boltzmannverteilung ist. Frühere Versuche, die Kikuchi-Approximation mit EDA zu verbinden, benutzen einen aufwendigen Gibbs-Sampling-Algorithmus, um eine Population aus dem zirkulären Wahrscheinlichkeitsmodell zu erzeugen. In dieser Arbeit wird eine Verbindung mit Faktorisierungen vorgestellt, die effizienteres Sampling erlaubt. Die freie Energie wird um die inverse Temperatur ß erweitert. Der oben erwähnte Algorithmus zur Erzeugung einer Faktorisierung kann auch hier angewendet werden. Die Dynamik von GBP wird untersucht und auf Ising-Modelle angewendet. Kleine Probleme (7 x 7) werden ohne Schwierigkeit gelöst. Größere Probleme (10 x 10 und 15 x 15) konvergieren mit großem ß nicht mehr zum wahren Optimum, aber durch Sampling von der Faktorisierung kann das Optimum bei einer Samplegröße von 1000 bis 10000, je nach Probleminstanz, gefunden werden. Wenn GBP nicht konvergiert, kann es durch eine Konkav-Konvex-Prozedur ersetzt werden, die Konvergenz garantiert. - Drittens kann, wenn für die Zielfunktion keine Struktur gegeben ist, ein Bayessches Netz gelernt werden, um die Abhängigkeiten in der Population zu erfassen. Die relative Entropie zwischen der Populationsverteilung und der Verteilung durch das Bayessche Netz ist äquivalent zur Log-Likelihood des Modells. Diese wurde erweitert zum BIC/MDL-Kriterium, das Überanpassung lindert, indem komplizierte Strukturen bestraft werden. Eine vorangegangene informationstheoretische Analyse von BIC/MDL im EDA-Bereich wird erweitert, und empirisch wird belegt, daß die Methode die korrekte Struktur einer Zielfunktion bei genügend großer Population lernen kann. - Schließlich wird vorgestellt, wie durch eine lokale Suchheuristik der Suchraum von EDA reduziert werden kann. Der Kernighan-Lin-Hillclimber, der ursprünglich für das Problem des Handlungsreisenden und Graphen-Bipartitionierung konzipiert ist, wird für beliebige binäre Probleme erweitert. Er kann allein angewandt werden, als iteratives 1+1-Suchverfahren, oder in Kombination mit EDA. Er löst das MAXSAT-Problem in ähnlicher Größenordnung wie der spezialisierte Hillclimber Walksat. Eine Analyse der lokalen Optima von Kernighan-Lin zeigt, daß die Kombination mit EDA vorteilhaft ist. Die Arbeit zeigt, wie evolutionäre Optimierung verbessert werden kann, indem interdisziplinäre Ergebnisse aus Informationstheorie, Statistik, Wahrscheinlichkeitsrechnung und statistischer Physik eingebracht werden. Die Prinzipien der Informationstheorie zur Schätzung von Wahrscheinlichkeitsverteilungen lassen sich in vielen Bereichen anwenden. EDAs sind eine gute Anwendung, denn eine verbesserte Schätzung beeinflußt direkt den Optimierungserfolg

The Use of Automated Search in Deriving Software Testing Strategies

Testing a software artefact using every one of its possible inputs would normally cost too much, and take too long, compared to the benefits of detecting faults in the software. Instead, a testing strategy is used to select a small subset of the inputs with which to test the software. The criterion used to select this subset affects the likelihood that faults in the software will be detected. For some testing strategies, the criterion may result in subsets that are very efficient at detecting faults, but implementing the strategy -- deriving a 'concrete strategy' specific to the software artefact -- is so difficult that it is not cost-effective to use that strategy in practice. In this thesis, we propose the use of metaheuristic search to derive concrete testing strategies in a cost-effective manner. We demonstrate a search-based algorithm that derives concrete strategies for 'statistical testing', a testing strategy that has a good fault-detecting ability in theory, but which is costly to implement in practice. The cost-effectiveness of the search-based approach is enhanced by the rigorous empirical determination of an efficient algorithm configuration and associated parameter settings, and by the exploitation of low-cost commodity GPU cards to reduce the time taken by the algorithm. The use of a flexible grammar-based representation for the test inputs ensures the applicability of the algorithm to a wide range of software

A Field Guide to Genetic Programming

xiv, 233 p. : il. ; 23 cm.Libro ElectrónicoA Field Guide to Genetic Programming (ISBN 978-1-4092-0073-4) is an introduction to genetic programming (GP). GP is a systematic, domain-independent method for getting computers to solve problems automatically starting from a high-level statement of what needs to be done. Using ideas from natural evolution, GP starts from an ooze of random computer programs, and progressively refines them through processes of mutation and sexual recombination, until solutions emerge. All this without the user having to know or specify the form or structure of solutions in advance. GP has generated a plethora of human-competitive results and applications, including novel scientific discoveries and patentable inventions. The authorsIntroduction -- Representation, initialisation and operators in Tree-based GP -- Getting ready to run genetic programming -- Example genetic programming run -- Alternative initialisations and operators in Tree-based GP -- Modular, grammatical and developmental Tree-based GP -- Linear and graph genetic programming -- Probalistic genetic programming -- Multi-objective genetic programming -- Fast and distributed genetic programming -- GP theory and its applications -- Applications -- Troubleshooting GP -- Conclusions.Contents xi 1 Introduction 1.1 Genetic Programming in a Nutshell 1.2 Getting Started 1.3 Prerequisites 1.4 Overview of this Field Guide I Basics 2 Representation, Initialisation and GP 2.1 Representation 2.2 Initialising the Population 2.3 Selection 2.4 Recombination and Mutation Operators in Tree-based 3 Getting Ready to Run Genetic Programming 19 3.1 Step 1: Terminal Set 19 3.2 Step 2: Function Set 20 3.2.1 Closure 21 3.2.2 Sufficiency 23 3.2.3 Evolving Structures other than Programs 23 3.3 Step 3: Fitness Function 24 3.4 Step 4: GP Parameters 26 3.5 Step 5: Termination and solution designation 27 4 Example Genetic Programming Run 4.1 Preparatory Steps 29 4.2 Step-by-Step Sample Run 31 4.2.1 Initialisation 31 4.2.2 Fitness Evaluation Selection, Crossover and Mutation Termination and Solution Designation Advanced Genetic Programming 5 Alternative Initialisations and Operators in 5.1 Constructing the Initial Population 5.1.1 Uniform Initialisation 5.1.2 Initialisation may Affect Bloat 5.1.3 Seeding 5.2 GP Mutation 5.2.1 Is Mutation Necessary? 5.2.2 Mutation Cookbook 5.3 GP Crossover 5.4 Other Techniques 32 5.5 Tree-based GP 39 6 Modular, Grammatical and Developmental Tree-based GP 47 6.1 Evolving Modular and Hierarchical Structures 47 6.1.1 Automatically Defined Functions 48 6.1.2 Program Architecture and Architecture-Altering 50 6.2 Constraining Structures 51 6.2.1 Enforcing Particular Structures 52 6.2.2 Strongly Typed GP 52 6.2.3 Grammar-based Constraints 53 6.2.4 Constraints and Bias 55 6.3 Developmental Genetic Programming 57 6.4 Strongly Typed Autoconstructive GP with PushGP 59 7 Linear and Graph Genetic Programming 61 7.1 Linear Genetic Programming 61 7.1.1 Motivations 61 7.1.2 Linear GP Representations 62 7.1.3 Linear GP Operators 64 7.2 Graph-Based Genetic Programming 65 7.2.1 Parallel Distributed GP (PDGP) 65 7.2.2 PADO 67 7.2.3 Cartesian GP 67 7.2.4 Evolving Parallel Programs using Indirect Encodings 68 8 Probabilistic Genetic Programming 8.1 Estimation of Distribution Algorithms 69 8.2 Pure EDA GP 71 8.3 Mixing Grammars and Probabilities 74 9 Multi-objective Genetic Programming 75 9.1 Combining Multiple Objectives into a Scalar Fitness Function 75 9.2 Keeping the Objectives Separate 76 9.2.1 Multi-objective Bloat and Complexity Control 77 9.2.2 Other Objectives 78 9.2.3 Non-Pareto Criteria 80 9.3 Multiple Objectives via Dynamic and Staged Fitness Functions 80 9.4 Multi-objective Optimisation via Operator Bias 81 10 Fast and Distributed Genetic Programming 83 10.1 Reducing Fitness Evaluations/Increasing their Effectiveness 83 10.2 Reducing Cost of Fitness with Caches 86 10.3 Parallel and Distributed GP are Not Equivalent 88 10.4 Running GP on Parallel Hardware 89 10.4.1 Master–slave GP 89 10.4.2 GP Running on GPUs 90 10.4.3 GP on FPGAs 92 10.4.4 Sub-machine-code GP 93 10.5 Geographically Distributed GP 93 11 GP Theory and its Applications 97 11.1 Mathematical Models 98 11.2 Search Spaces 99 11.3 Bloat 101 11.3.1 Bloat in Theory 101 11.3.2 Bloat Control in Practice 104 III Practical Genetic Programming 12 Applications 12.1 Where GP has Done Well 12.2 Curve Fitting, Data Modelling and Symbolic Regression 12.3 Human Competitive Results – the Humies 12.4 Image and Signal Processing 12.5 Financial Trading, Time Series, and Economic Modelling 12.6 Industrial Process Control 12.7 Medicine, Biology and Bioinformatics 12.8 GP to Create Searchers and Solvers – Hyper-heuristics xiii 12.9 Entertainment and Computer Games 127 12.10The Arts 127 12.11Compression 128 13 Troubleshooting GP 13.1 Is there a Bug in the Code? 13.2 Can you Trust your Results? 13.3 There are No Silver Bullets 13.4 Small Changes can have Big Effects 13.5 Big Changes can have No Effect 13.6 Study your Populations 13.7 Encourage Diversity 13.8 Embrace Approximation 13.9 Control Bloat 13.10 Checkpoint Results 13.11 Report Well 13.12 Convince your Customers 14 Conclusions Tricks of the Trade A Resources A.1 Key Books A.2 Key Journals A.3 Key International Meetings A.4 GP Implementations A.5 On-Line Resources 145 B TinyGP 151 B.1 Overview of TinyGP 151 B.2 Input Data Files for TinyGP 153 B.3 Source Code 154 B.4 Compiling and Running TinyGP 162 Bibliography 167 Inde

CSM429: Abstract Geometric Crossover for the Permutation Representation

Abstract crossover and abstract mutation are representation-independent operators that are well-defined once a notion of distance over the solution space is defined. They were obtained as generalization of genetic operators for binary strings and real vectors. In this paper we explore how the abstract geometric framework applies to the permutation representation. This representation is challenging for various reasons: because of the inherent difference between permutations and the representations that inspired the abstraction; because the whole notion of geometry over permutation spaces radically departs from traditional geometries and it is almost unexplored mathematical territory; because the many notions of distance available and their subtle interconnections make it hard to see the right distance to use, if any; because the various available interpretations of permutations make ambiguous what a permutation represents, hence, how to treat it; because of the existence of various permutation-like representations that are incorrectly confused with permutations; and finally because of the existence of many mutation and recombination operators and their many variations for the same representation. This article shows that the application of our geometric framework naturally clarifies and unifies an important domain,the permutation representation and the related operators, in which there was little or no hope to find order. In addition the abstract geometric framework is used to improve the design of crossover operators for well-known problems naturally connected with the permutation representation

Optimization of a Quantum Cascade Laser Operating in the Terahertz Frequency Range Using a Multiobjective Evolutionary Algorithm

A quantum cascade (QC) laser is a specific type of semiconductor laser that operates through principles of quantum mechanics. In less than a decade QC lasers are already able to outperform previously designed double heterostructure semiconductor lasers. Because there is a genuine lack of compact and coherent devices which can operate in the far-infrared region the motivation exists for designing a terahertz QC laser. A device operating at this frequency is expected to be more efficient and cost effective than currently existing devices. It has potential applications in the fields of spectroscopy, astronomy, medicine and free-space communication as well as applications to near-space radar and chemical/biological detection. The overarching goal of this research was to find QC laser parameter combinations which can be used to fabricate viable structures. To ensure operation in the THz region the device must conform to the extremely small energy level spacing range from ~10-15 meV. The time and expense of the design and production process is prohibitive, so an alternative to fabrication was necessary. To accomplish this goal a model of a QC laser, developed at Worchester Polytechnic Institute with sponsorship from the Air Force Research Laboratory Sensors Directorate, and the General Multiobjective Parallel Genetic Algorithm (GenMOP), developed at the Air Force Institute of Technology, were integrated to form a computer simulation which stochastically searches for feasible solutions

Risk-Aware Planning for Sensor Data Collection

With the emergence of low-cost unmanned air vehicles, civilian and military organizations are quickly identifying new applications for affordable, large-scale collectives to support and augment human efforts via sensor data collection. In order to be viable, these collectives must be resilient to the risk and uncertainty of operating in real-world environments. Previous work in multi-agent planning has avoided planning for the loss of agents in environments with risk. In contrast, this dissertation presents a problem formulation that includes the risk of losing agents, the effect of those losses on the mission being executed, and provides anticipatory planning algorithms that consider risk. We conduct a thorough analysis of the effects of risk on path-based planning, motivating new solution methods. We then use hierarchical clustering to generate risk-aware plans for a variable number of agents, outperforming traditional planning methods. Next, we provide a mechanism for distributed negotiation of stable plans, utilizing coalitional game theory to provide cost allocation methods that we prove to be fair and stable. Centralized planning with redundancy is then explored, planning for parallel task completion to mitigate risk and provide further increased expected value. Finally, we explore the role of cost uncertainty as additional source of risk, using bi-objective optimization to generate sets of alternative plans. We demonstrate the capability of our algorithms on randomly generated problem instances, showing an improvement over traditional multi-agent planning methods as high as 500% on very large problem instances