Improved Runtime Bounds for the Univariate Marginal Distribution Algorithm via Anti-Concentration

Unlike traditional evolutionary algorithms which produce offspring via genetic operators, Estimation of Distribution Algorithms (EDAs) sample solutions from probabilistic models which are learned from selected individuals. It is hoped that EDAs may improve optimisation performance on epistatic fitness landscapes by learning variable interactions. However, hardly any rigorous results are available to support claims about the performance of EDAs, even for fitness functions without epistasis. The expected runtime of the Univariate Marginal Distribution Algorithm (UMDA) on OneMax was recently shown to be in $\mathcal{O}\left(n\lambda\log \lambda\right)$ by Dang and Lehre (GECCO 2015). Later, Krejca and Witt (FOGA 2017) proved the lower bound $\Omega\left(\lambda\sqrt{n}+n\log n\right)$ via an involved drift analysis. We prove a $\mathcal{O}\left(n\lambda\right)$ bound, given some restrictions on the population size. This implies the tight bound $\Theta\left(n\log n\right)$ when $\lambda=\mathcal{O}\left(\log n\right)$, matching the runtime of classical EAs. Our analysis uses the level-based theorem and anti-concentration properties of the Poisson-Binomial distribution. We expect that these generic methods will facilitate further analysis of EDAs.Comment: 19 pages, 1 figur

Monetary policy and stability during six periods in US economic history: 1959–2008: a novel, nonlinear monetary policy rule

We investigate the monetary policy of the Federal Reserve Board during six periods in US economic history 1959–2008. In particular, we examine the Fed’s response to changes in three guiding variables: inflation, π, unemployment, U, and industrial production, y, during periods with low and high economic stability. We identify separate responses for the Fed’s change in interest rate depending upon (i) the current rate, FF, and the guiding variables’ level below or above their average values and (ii) recent movements in inflation and unemployment. The change in rate, FF, can then be calculated. We identify policies that both increased and decreased economic stability

Investigating possible causal relations among physical, chemical and biological variables across regions in the Gulf of Maine

We examine potential causal relations between ecosystem variables in four regions of the Gulf of Maine under two major assumptions: (i) a causal cyclic variable will precede, or lead, its effect variable; e.g., a peak (through) in the causal variable will come before a peak (through) in the effect variable. (ii) If physical variables determine regional ecosystem properties, then independent clusters of observations of physical, biological and interaction variables from the same stations will show similar patterns. We use the leading–lagging-strength method to establish leading strength and potential causality, and we use principal component analysis, to establish if regions differ in their ecological characteristics. We found that several relationships for physical and chemical variables were significant, and consistent with ‘‘common knowledge’’ of causal relations. In contrast, relationships that included biological variables differed among regions. In spite of these findings, we found that physical and chemical characteristics of near shore and pelagic regions of the Gulf of Maine translate into unique biological assemblages and unique physical–biologi- cal interaction

A Parameterized Complexity Analysis of Bi-level Optimisation with Evolutionary Algorithms

Bi-level optimisation problems have gained increasing interest in the field of combinatorial optimisation in recent years. With this paper, we start the runtime analysis of evolutionary algorithms for bi-level optimisation problems. We examine two NP-hard problems, the generalised minimum spanning tree problem (GMST), and the generalised travelling salesman problem (GTSP) in the context of parameterised complexity. For the generalised minimum spanning tree problem, we analyse the two approaches presented by Hu and Raidl (2012) with respect to the number of clusters that distinguish each other by the chosen representation of possible solutions. Our results show that a (1+1) EA working with the spanning nodes representation is not a fixed-parameter evolutionary algorithm for the problem, whereas the global structure representation enables to solve the problem in fixed-parameter time. We present hard instances for each approach and show that the two approaches are highly complementary by proving that they solve each other's hard instances very efficiently. For the generalised travelling salesman problem, we analyse the problem with respect to the number of clusters in the problem instance. Our results show that a (1+1) EA working with the global structure representation is a fixed-parameter evolutionary algorithm for the problem

Level-Based Analysis of Genetic Algorithms and Other Search Processes

The fitness-level technique is a simple and old way to derive upper bounds for the expected runtime of simple elitist evolutionary algorithms (EAs). Recently, the technique has been adapted to deduce the runtime of algorithms with non-elitist populations and unary variation operators [2,8]. In this paper, we show that the restriction to unary variation operators can be removed. This gives rise to a much more general analytical tool which is applicable to a wide range of search processes. As introductory examples, we provide simple runtime analyses of many variants of the Genetic Algorithm on well-known benchmark functions, such as OneMax, LeadingOnes, and the sorting problem

Self-adaptation of mutation rates in non-elitist populations

The runtime of evolutionary algorithms (EAs) depends critically on their parameter settings, which are often problem-specific. Automated schemes for parameter tuning have been developed to alleviate the high costs of manual parameter tuning. Experimental results indicate that self-adaptation, where parameter settings are encoded in the genomes of individuals, can be effective in continuous optimisation. However, results in discrete optimisation have been less conclusive. Furthermore, a rigorous runtime analysis that explains how self adaptation can lead to asymptotic speedups has been missing. This paper provides the first such analysis for discrete, population-based EAs. We apply level-based analysis to show how a self-adaptive EA is capable of fine-tuning its mutation rate, leading to exponential speedups over EAs using fixed mutation rates

Self-adaptation of mutation rates in non-elitist populations

The runtime of evolutionary algorithms (EAs) depends critically on their parameter settings, which are often problem-specific. Automated schemes for parameter tuning have been developed to alleviate the high costs of manual parameter tuning. Experimental results indicate that self-adaptation, where parameter settings are encoded in the genomes of individuals, can be effective in continuous optimisation. However, results in discrete optimisation have been less conclusive. Furthermore, a rigorous runtime analysis that explains how self adaptation can lead to asymptotic speedups has been missing. This paper provides the first such analysis for discrete, population-based EAs. We apply level-based analysis to show how a self-adaptive EA is capable of fine-tuning its mutation rate, leading to exponential speedups over EAs using fixed mutation rates

Escaping Local Optima Using Crossover with Emergent Diversity

Population diversity is essential for avoiding premature convergence in Genetic Algorithms and for the effective use of crossover. Yet the dynamics of how diversity emerges in populations are not well understood. We use rigorous run time analysis to gain insight into population dynamics and Genetic Algorithm performance for the (μ+1) Genetic Algorithm and the Jump test function. We show that the interplay of crossover followed by mutation may serve as a catalyst leading to a sudden burst of diversity. This leads to significant improvements of the expected optimisation time compared to mutation-only algorithms like the (1+1) Evolutionary Algorithm. Moreover, increasing the mutation rate by an arbitrarily small constant factor can facilitate the generation of diversity, leading to even larger speedups. Experiments were conducted to complement our theoretical findings and further highlight the benefits of crossover on the function class