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. This explicit use of probabilistic models in optimization offers some significant advantages over other types of metaheuristics. This paper discusses these advantages and outlines many of the different types of EDAs. In addition, some of the most powerful efficiency enhancement techniques applied to EDAs are discussed and some of the key theoretical results relevant to EDAs are outlined.

An introduction and survey of estimation of distribution algorithms

Linking Entropy to Estimation of Distribution Algorithms.- Entropy-based Convergence Measurement in Discrete Estimation of Distribution Algorithms.- Real-coded Bayesian Optimization Algorithm.- The CMA Evolution Strategy: A Comparing Review.- Estimation of Distribution Programming: EDA-based Approach to Program Generation.- Multi-objective Optimization with the Naive ID A.- A Parallel Island Model for Estimation of Distribution Algorithms.- GA-EDA: A New Hybrid Cooperative Search Evolutionary Algorithm.- Bayesian Classifiers in Optimization: An EDA-like Approach.- Feature Ranking Using an EDA-based Wrapper Approach.- Learning Linguistic Fuzzy Rules by Using Estimation of Distribution Algorithms as the Search Engine in the COR Methodology.- Estimation of Distribution Algorithm with 2-opt Local Search for the Quadratic Assignment Problem.

Towards a new evolutionary computation : advances in the estimation of distribution algorithms

Towards a New Evolutionary Computation

Calibrating the parameters of an evolutionary algorithm (EA) is a laborious task. The highly stochastic nature of an EA typically leads to a high variance of the measurements. The standard statistical method to reduce variance is measurement replication, i.e., averaging over several test runs with identical parameter settings. The computational cost of measurement replication scales with the variance and is often too high to allow for results of statistical significance. In this paper we study an alternative: the REVAC method for Relevance Estimation and Value Calibration, and we investigate how different levels of measurement replication influence the cost and quality of its calibration results. Two sets ofof experiments are reported: calibrating a genetic algorithm on standard benchmark problems, and calibrating a complex simulation in evolutionary agent-based economics. We find that measurement replication is not essential to REVAC, which emerges as a strong and efficient alternative to existing statistical methods.

Efficient relevance estimation and value calibration of evolutionary algorithm parameters

We investigate the spatial distribution and the global frequency of agents who can either cooperate or defect. The agent interaction is described by a deterministic, non-iterated prisoner's dilemma game, further each agent only locally interacts with his neighbors. Based on a detailed analysis of the local payoff structures we derive critical conditions for the invasion or the spatial coexistence of cooperators and defectors. These results are concluded in a phase diagram that allows us to identify five regimes, each characterized by a distinct spatiotemporal dynamics and a corresponding final spatial structure. In addition to the complete invasion of defectors, we find coexistence regimes with either a majority of cooperators in large spatial domains, or a minority of cooperators organized in small non-stationary domains or in small clusters. The analysis further allowed a verification of computer simulation results by Nowak and May (1993). Eventually, we present simulation results of a true 5-person game on a lattice. This modification leads to non-uniform spatial interactions that may even enhance the effect of cooperation.

Evolution of cooperation in a spatial prisoner's dilemma

Estimation of Distribution Algorithms (EDA) have been proposed as an extension of genetic algorithms. In this paper we explain the relationship of EDA to algorithms developed in statistics, artificial intelligence, and statistical physics. The major design issues are discussed within a general interdisciplinary framework. It is shown that maximum entropy approximations play a crucial role. All proposed algorithms try to minimize the Kullback-Leibler divergence KLD between the unknown distribution p(x) and a class q(x) of approximations. However, the Kullback-Leibler divergence is not symmetric. Approximations which suppose that the function to be optimized is additively decomposed (ADF) minimize KLD(q||p), the methods which learn the approximate model from data minimize KLD(p||q). This minimization is identical to maximizing the log-likelihood. In the paper three classes of algorithms are discussed. FDA uses the ADF to compute an approximate factorization of the unknown distribution. The factors are marginal distributions, whose values are computed from samples. The second class is represented by the Bethe-Kikuchi approach which has recently been rediscovered in statistical physics. Here the values of the marginals are computed from a difficult constrained minimization problem. The third class learns the factorization from the data. We analyze our learning algorithm LFDA in detail. It is shown that learning is faced with two problems: first, to detect the important dependencies be- tween the variables, and second, to create an acyclic Bayesian network of bounded clique size.

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The Estimation of Distributions and the Minimum Relative Entropy Principle

We perform a rigorous stochastic analysis of both deterministic and stochastic cellular automata. The theory uses a mesoscopic view, i.e. it works with probabilities instead of individual configurations used in micro-simulations. An exact stochastic analysis can be done using the theory of Markov processes. But this analysis is restricted to small problems only. For larger problems we compute the distribution using a factorization into marginals. These marginals are then approximated by the given marginals of low order with iterative proportional fitting using the maximum entropy principle. This method has been developed in probabilistic logic. Our method leads to a set of difference equations which have to be iterated numerically. We use the exact methods as well as our approximations to investigate the popular nonlinear voter model (NLVM). We show that the "phase transitions" regarded in recent papers are artifacts of the mean-field approximation. They do not show up in the real automata. There exist many mathematical peculiarities of the NLVM which raise doubts concerning the suitability of the model. As an alternative we propose the Exponential Voter Model which depends on a single parameter only, the inverse "temperature" β. Our proposed method to perform a stochastic analysis is not restricted to cellular automata, but can be applied to more general discrete stochastic systems.

Stochastic analysis of cellular automata with application to the voter model

The success of evolutionary algorithms, in particular Factorized Distribution Algorithms (FDA), for many pattern recognition tasks heavily depends on our ability to reduce the number of function evaluations.

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A Maximum Entropy Approach to Sampling in EDA – The Single Connected Case

We consider the finite homogeneous Markov chain induced by a class of one-dimensional asynchronous cellular automata—automata that are allowed to change only one cell per iteration. Furthermore, we confine to totalistic automata, where transitions depend only on the number of 1s in the neighborhood of the current cell. We consider three different cases: (i) size of neighborhood equals length of the automaton; (ii) size of neighborhood two, length of automaton arbitrary; and (iii) size of neighborhood three, length of automaton arbitrary. For each case, the associated Markov chain proves to be ergodic. We derive simple-form stationary distributions, in case (i) by lumping states with respect to the number of 1s in the automaton, and in cases (ii) and (iii) by considering the number of 0–1 borders within the automaton configuration. For the three-neighborhood automaton, we analyze also the Markov chain at the boundary of the parameter domain, and the symmetry of the entropy. Finally, we show that if the local transition rule is exponential, the stationary probability is the Boltzmann distribution of the Ising model.

Markov Chain Analysis for One-Dimensional Asynchronous Cellular Automata

We assume that the function to be optimized is additively decomposed (ADF). Then the interaction graph $G_{ADF}$ can be used to compute exact or approximate factorizations. For many practical problems only approximate factorizations lead to efficient optimization algorithms. The relation between the approximation used by the FDA algorithm and the minimum relative entropy principle is discussed. A new algorithm is presented, derived from the Bethe-Kikuchi approach in statistical physics. It minimizes the relative entropy to a Boltzmann distribution with fixed $eta$. We shortly compare different factorizations and algorithms within the FDA software. We use 2-d Ising spin glass problems and Kaufman's n-k function as examples.

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Robin Höns

Papers

The Estimation of Distributions and the Minimum Relative Entropy Principle

Stochastic analysis of cellular automata with application to the voter model

A Maximum Entropy Approach to Sampling in EDA – The Single Connected Case

Markov Chain Analysis for One-Dimensional Asynchronous Cellular Automata

The Factorized Distribution Algorithm and the Minimum Relative Entropy Principle