This paper shows how to evolve numerically the maximum entropy probability distributions for a given set of constraints, which is a variational calculus problem. An evolutionary algorithm can obtain approximations to some well-known analytical results, but is even more flexible and can find distributions for which a closed formula cannot be readily stated. The numerical approach handles distributions over finite intervals. We show that there are two ways of conducting the procedure: by direct optimization of the Lagrangian of the constrained problem, or by optimizing the entropy among the subset of distributions which fulfill the constraints. An incremental evolutionary strategy easily obtains the uniform, the exponential, the Gaussian, the log-normal, the Laplace, among other distributions, once the constrained problem is solved with any of the two methods. Solutions for mixed ("chimera") distributions can be also found. We explain why many of the distributions are symmetrical and continuous, but some are not.
The recent successes of deep learning and deep reinforcement learning have firmly established their statuses as state-of-the-art artificial learning techniques. However, longstanding drawbacks of these approaches, such as their poor sample efficiencies and limited generalization capabilities point to a need for re-thinking the way such systems are designed and deployed. In this paper, we emphasize how the use of these learning systems, in conjunction with a specific variation of evolutionary algorithms could lead to the emergence of unique characteristics such as the automated acquisition of a variety of desirable behaviors and useful sets of behavior priors. This could pave the way for learning to occur in a generalized and continual manner, with minimal interactions with the environment. We discuss the anticipated improvements from such neuro-evolutionary frameworks, along with the associated challenges, as well as its potential for application to a number of research areas.
Existing multi-strategy adaptive differential evolution (DE) commonly involves trials of multiple strategies and then rewards better-performing ones with more resources. However, the trials of an exploitative or explorative strategy may result in over-exploitation or over-exploration. To improve the performance, this paper proposes a new strategy adaptation method, named explicit adaptation scheme (Ea scheme), which separates multiple strategies and employs them on-demand. It is done by dividing the evolution process into several Selective-candidate with Similarity Selection (SCSS) generations and adaptive generations. In the SCSS generations, the exploitation and exploration needs are learnt by utilizing a balanced strategy. To meet these needs, in adaptive generations, two other strategies, exploitative or explorative is adaptively used. Experimental studies on benchmark functions demonstrate the effectiveness of Ea scheme when compared with its variants and other adaptation methods. Furthermore, performance comparisons with state-of-the-art evolutionary algorithms and swarm intelligence-based algorithms show that EaDE is very competitive.
Most decision tree induction algorithms are based on a greedy top-down recursive partitioning strategy for tree growth. In this paper, we propose several methods for induction of decision trees and their ensembles based on evolutionary algorithms. The main difference of our approach is using real-valued vector representation of decision tree that allows to use a large number of different optimization algorithms, as well as optimize the whole tree or ensemble for avoiding local optima. Differential evolution and evolution strategies were chosen as optimization algorithms, as they have good results in reinforcement learning problems. We test the predictive performance of this methods using several public UCI data sets, and the proposed methods show better quality than classical methods.
Neural networks (NN) have been recently applied together with evolutionary algorithms (EAs) to solve dynamic optimization problems. The applied NN estimates the position of the next optimum based on the previous time best solutions. After detecting a change, the predicted solution can be employed to move the EA's population to a promising region of the solution space in order to accelerate convergence and improve accuracy in tracking the optimum. While previous works show improvement of the results, they neglect the overhead created by NN. In this work, we reflect the time spent on training NN in the optimization time and compare the results with a baseline EA. We explore if by considering the generated overhead, NN is still able to improve the results, and under which condition is able to do so. The main difficulties to train the NN are: 1) to get enough samples to generalize predictions for new data, and 2) to obtain reliable samples. As NN needs to collect data at each time step, if the time horizon is short, we will not be able to collect enough samples to train the NN. To alleviate this, we propose to consider more individuals on each change to speed up sample collection in shorter time steps. In environments with a high frequency of changes, the solutions produced by EA are likely to be far from the real optimum. Using unreliable train data for the NN will, in consequence, produce unreliable predictions. Also, as the time spent for NN stays fixed regardless of the frequency, a higher frequency of change will mean a higher produced overhead by the NN in proportion to the EA. In general, after considering the generated overhead, we conclude that NN is not suitable in environments with a high frequency of changes and/or short time horizons. However, it can be promising for the low frequency of changes, and especially for the environments that changes have a pattern.
Fitness landscapes are a useful concept to study the dynamics of meta-heuristics. In the last two decades, they have been applied with success to estimate the optimization power of several types of evolutionary algorithms, including genetic algorithms and genetic programming. However, so far they have never been used to study the performance of machine learning algorithms on unseen data, and they have never been applied to neuroevolution. This paper aims at filling both these gaps, applying for the first time fitness landscapes to neuroevolution and using them to infer useful information about the predictive ability of the method. More specifically, we use a grammar-based approach to generate convolutional neural networks, and we study the dynamics of three different mutations to evolve them. To characterize fitness landscapes, we study autocorrelation and entropic measure of ruggedness. The results show that these measures are appropriate for estimating both the optimization power and the generalization ability of the considered neuroevolution configurations.
Known as two cornerstones of problem solving by search, exploitation and exploration are extensively discussed for implementation and application of evolutionary algorithms (EAs). However, only a few researches focus on evaluation and theoretical estimation of exploitation and exploration. Considering that exploitation and exploration are two issues regarding global search and local search, this paper proposes to evaluate them via the success probability and the one-step improvement rate computed in different domains of integration. Then, case studies are performed by analyzing performances of (1+1) random univariate search and (1+1) evolutionary programming on the sphere function and the cheating problem. By rigorous theoretical analysis, we demonstrate that both exploitation and exploration of the investigated elitist EAs degenerate exponentially with the problem dimension $n$. Meanwhile, it is also shown that maximization of exploitation and exploration can be achieved by setting an appropriate value for the standard deviation $\sigma$ of Gaussian mutation, which is positively related to the distance from the present solution to the center of the promising region.
A wave energy converter (WEC) similar to the CETO system developed by Carnegie Clean Energy is considered for design optimisation. This WEC is able to absorb power from heave, surge and pitch motion modes, making the optimisation problem nontrivial. The WEC dynamics is simulated using the spectral-domain model taking into account hydrodynamic forces, viscous drag, and power take-off forces. The design parameters for optimisation include the buoy radius, buoy height, tether inclination angles, and control variables (damping and stiffness). The WEC design is optimised for the wave climate at Albany test site in Western Australia considering unidirectional irregular waves. Two objective functions are considered: (i) maximisation of the annual average power output, and (ii) minimisation of the levelised cost of energy (LCoE) for a given sea site. The LCoE calculation is approximated as a ratio of the produced energy to the significant mass of the system that includes the mass of the buoy and anchor system. Six different heuristic optimisation methods are applied in order to evaluate and compare the performance of the best known evolutionary algorithms, a swarm intelligence technique and a numerical optimisation approach. The results demonstrate that if we are interested in maximising energy production without taking into account the cost of manufacturing such a system, the buoy should be built as large as possible (20 m radius and 30 m height). However, if we want the system that produces cheap energy, then the radius of the buoy should be approximately 11-14~m while the height should be as low as possible. These results coincide with the overall design that Carnegie Clean Energy has selected for its CETO 6 multi-moored unit. However, it should be noted that this study is not informed by them, so this can be seen as an independent validation of the design choices.
Randomized search heuristics such as evolutionary algorithms are frequently applied to dynamic combinatorial optimization problems. Within this paper, we present a dynamic model of the classic Weighted Vertex Cover problem and analyze the runtime performances of the well-studied algorithms Randomized Local Search and (1+1) EA adapted to it, to contribute to the theoretical understanding of evolutionary computing for problems with dynamic changes. In our investigations, we use an edge-based representation based on the dual form of the Linear Programming formulation for the problem and study the expected runtime that the adapted algorithms require to maintain a 2-approximate solution when the given weighted graph is modified by an edge-editing or weight-editing operation. Considering the weights on the vertices may be exponentially large with respect to the size of the graph, the step size adaption strategy is incorporated, with or without the 1/5-th rule that is employed to control the increasing/decreasing rate of the step size. Our results show that three of the four algorithms presented in the paper can recompute 2-approximate solutions for the studied dynamic changes in polynomial expected runtime, but the (1+1) EA with 1/5-th Rule requires pseudo-polynomial expected runtime.
The placement scheme of virtual machines (VMs) to physical servers (PSs) is crucial to lowering operational cost for cloud providers. Evolutionary algorithms (EAs) have been performed promising-solving on virtual machine placement (VMP) problems in the past. However, as growing demand for cloud services, the existing EAs fail to implement in large-scale virtual machine placement (LVMP) problem due to the high time complexity and poor scalability. Recently, the multi-factorial optimization (MFO) technology has surfaced as a new search paradigm in evolutionary computing. It offers the ability to evolve multiple optimization tasks simultaneously during the evolutionary process. This paper aims to apply the MFO technology to the LVMP problem in heterogeneous environment. Firstly, we formulate a deployment cost based VMP problem in the form of the MFO problem. Then, a multi-factorial evolutionary algorithm (MFEA) embedded with greedy-based allocation operator is developed to address the established MFO problem. After that, a re-migration and merge operator is designed to offer the integrated solution of the LVMP problem from the solutions of MFO problem. To assess the effectiveness of our proposed method, the simulation experiments are carried on large-scale and extra large-scale VMs test data sets. The results show that compared with various heuristic methods, our method could shorten optimization time significantly and offer a competitive placement solution for the LVMP problem in heterogeneous environment.
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.
Multi-objective optimisation of damper placement in dynamically symmetric adjacent buildings is considered with identical viscoelastic dampers used for vibration control. First, exhaustive search is used to describe the solution space in terms of various quantities of interest such as maximum top floor displacement, maximum floor acceleration, base shear, and interstorey drift. With the help of examples, it is pointed out that the Pareto fronts in these problems contain a very small number of solutions. The effectiveness of two commonly used multi-objective evolutionary algorithms, viz., NSGA-II and MOPSO, is evaluated for a specific example.
We propose a projection pursuit (PP) algorithm based on Gaussian mixture models (GMMs). The negentropy obtained from a multivariate density estimated by GMMs is adopted as the PP index to be maximised. For a fixed dimension of the projection subspace, the GMM-based density estimation is projected onto that subspace, where an approximation of the negentropy for Gaussian mixtures is computed. Then, Genetic Algorithms (GAs) are used to find the optimal, orthogonal projection basis by maximising the former approximation. We show that this semi-parametric approach to PP is flexible and allows highly informative structures to be detected, by projecting multivariate datasets onto a subspace, where the data can be feasibly visualised. The performance of the proposed approach is shown on both artificial and real datasets.
Working with complex, high-level MOEA meta-models such as Multiobjec-tive Optimization Hierarchic Genetic Strategy (MO-mHGS) with multi-deme support usually requires dedicated implementation and configuration for each internal (single-deme) algorithm variant. If we generalize meta-model, we can simplify whole simulation process and bind any internal algorithm (we denote it as a driver), without providing redundant meta-model implementations. This idea has become a fundamental of Evogil platform. Our aim was to allow construct-ing custom hybrid models or combine existing solutions in runtime simulation environment. We define hybrid solution as a composition of a meta-model and a driver (or multiple drivers). Meta-model uses drivers to perform evolutionary calculations and process their results. Moreover, Evogil provides set of ready-made solutions divided into two groups (multi-deme meta-models and single-deme drivers), as well as processing tools (quality metrics, statistics and plotting scripts), simulation management and results persistence layer.
Although Deep Neural Networks have seen great success in recent years through various changes in overall architectures and optimization strategies, their fundamental underlying design remains largely unchanged. Computational neuroscience on the other hand provides more biologically realistic models of neural processing mechanisms, but they are still high level abstractions of the actual experimentally observed behaviour. Here a model is proposed that bridges Neuroscience, Machine Learning and Evolutionary Algorithms to evolve individual soma and synaptic compartment models of neurons in a scalable manner. Instead of attempting to manually derive models for all the observed complexity and diversity in neural processing, we propose an Evolvable Neural Unit (ENU) that can approximate the function of each individual neuron and synapse. We demonstrate that this type of unit can be evolved to mimic Integrate-And-Fire neurons and synaptic Spike-Timing-Dependent Plasticity. Additionally, by constructing a new type of neural network where each synapse and neuron is such an evolvable neural unit, we show it is possible to evolve an agent capable of learning to solve a T-maze environment task. This network independently discovers spiking dynamics and reinforcement type learning rules, opening up a new path towards biologically inspired artificial intelligence.
Reinforcement learning, evolutionary algorithms and imitation learning are three principal methods to deal with continuous control tasks. Reinforcement learning is sample efficient, yet sensitive to hyper-parameters setting and needs efficient exploration; Evolutionary algorithms are stable, but with low sample efficiency; Imitation learning is both sample efficient and stable, however it requires the guidance of expert data. In this paper, we propose Recruitment-imitation Mechanism (RIM) for evolutionary reinforcement learning, a scalable framework that combines advantages of the three methods mentioned above. The core of this framework is a dual-actors and single critic reinforcement learning agent. This agent can recruit high-fitness actors from the population of evolutionary algorithms, which instructs itself to learn from experience replay buffer. At the same time, low-fitness actors in the evolutionary population can imitate behavior patterns of the reinforcement learning agent and improve their adaptability. Reinforcement and imitation learners in this framework can be replaced with any off-policy actor-critic reinforcement learner or data-driven imitation learner. We evaluate RIM on a series of benchmarks for continuous control tasks in Mujoco. The experimental results show that RIM outperforms prior evolutionary or reinforcement learning methods. The performance of RIM's components is significantly better than components of previous evolutionary reinforcement learning algorithm, and the recruitment using soft update enables reinforcement learning agent to learn faster than that using hard update.
Biological and cultural inspired optimization algorithms are nowadays part of the basic toolkit of a great many research domains. By mimicking processes in nature and animal societies, these general-purpose search algorithms promise to deliver optimal or near-optimal solutions using hardly any information on the optimization problems they are set to tackle. Here we study the performances of a cultural-inspired algorithm -- the imitative learning search -- as well as of asexual and sexual variants of evolutionary algorithms in finding the global maxima of NK-fitness landscapes. The main performance measure is the total number of agent updates required by the algorithms to find those global maxima and the baseline performance, which establishes the effectiveness of the cooperative algorithms, is set by the blind search in which the agents explore the problem space (binary strings) by flipping bits at random. We find that even for smooth landscapes that exhibit a single maximum, the evolutionary algorithms do not perform much better than the blind search due to the stochastic effects of the genetic roulette. The imitative learning is immune to this effect thanks to the deterministic choice of the fittest string in the population, which is used as a model for imitation. The tradeoff is that for rugged landscapes the imitative learning search is more prone to be trapped in local maxima than the evolutionary algorithms. In fact, in the case of rugged landscapes with a mild density of local maxima, the blind search either beats or matches the cooperative algorithms regardless of whether the task is to find the global maximum or to find the fittest state within a given runtime.
Evolutionary algorithms, inspired by natural evolution, aim to optimize difficult objective functions without computing derivatives. Here we detail the relationship between population genetics and evolutionary optimization and formulate a new evolutionary algorithm. Optimization of a continuous objective function is analogous to searching for high fitness phenotypes on a fitness landscape. We summarize how natural selection moves a population along the non-euclidean gradient that is induced by the population on the fitness landscape (the natural gradient). Under normal approximations common in quantitative genetics, we show how selection is related to Newton's method in optimization. We find that intermediate selection is most informative of the fitness landscape. We describe the generation of new phenotypes and introduce an operator that recombines the whole population to generate variants that preserve normal statistics. Finally, we introduce a proof-of-principle algorithm that combines natural selection, our recombination operator, and an adaptive method to increase selection. Our algorithm is similar to covariance matrix adaptation and natural evolutionary strategies in optimization, and has similar performance. The algorithm is extremely simple in implementation with no matrix inversion or factorization, does not require storing a covariance matrix, and may form the basis of more general model-based optimization algorithms with natural gradient updates.
An underlying assumption of Stackelberg Games (SGs) is perfect rationality of the players. However, in real-life situations (which are often modeled by SGs) the followers (terrorists, thieves, poachers or smugglers) -- as humans in general -- may act not in a perfectly rational way, as their decisions may be affected by biases of various kinds which bound rationality of their decisions. One of the popular models of bounded rationality (BR) is Anchoring Theory (AT) which claims that humans have a tendency to flatten probabilities of available options, i.e. they perceive a distribution of these probabilities as being closer to the uniform distribution than it really is. This paper proposes an efficient formulation of AT in sequential extensive-form SGs (named ATSG), suitable for Mixed-Integer Linear Program (MILP) solution methods. ATSG is implemented in three MILP/LP-based state-of-the-art methods for solving sequential SGs and two recently introduced non-MILP approaches: one relying on Monte Carlo sampling (O2UCT) and the other one (EASG) employing Evolutionary Algorithms. Experimental evaluation indicates that both non-MILP heuristic approaches scale better in time than MILP solutions while providing optimal or close-to-optimal solutions. Except for competitive time scalability, an additional asset of non-MILP methods is flexibility of potential BR formulations they are able to incorporate. While MILP approaches accept BR formulations with linear constraints only, no restrictions on the BR form are imposed in either of the two non-MILP methods.
Estimation of Distribution Algorithms (EDAs) are one branch of Evolutionary Algorithms (EAs) in the broad sense that they evolve a probabilistic model instead of a population. Many existing algorithms fall into this category. Analogous to genetic drift in EAs, EDAs also encounter the phenomenon that updates of the probabilistic model not justified by the fitness move the sampling frequencies to the boundary values. This can result in a considerable performance loss. This paper proves the first sharp estimates of the boundary hitting time of the sampling frequency of a neutral bit for several univariate EDAs. For the UMDA that selects $\mu$ best individuals from $\lambda$ offspring each generation, we prove that the expected first iteration when the frequency of the neutral bit leaves the middle range $[\tfrac 14, \tfrac 34]$ and the expected first time it is absorbed in 0 or 1 are both $\Theta(\mu)$. The corresponding hitting times are $\Theta(K^2)$ for the cGA with hypothetical population size $K$. This paper further proves that for PBIL with parameters $\mu$, $\lambda$, and $\rho$, in an expected number of $\Theta(\mu/\rho^2)$ iterations the sampling frequency of a neutral bit leaves the interval $[\Theta(\rho/\mu),1-\Theta(\rho/\mu)]$ and then always the same value is sampled for this bit, that is, the frequency approaches the corresponding boundary value with maximum speed. For the lower bounds implicit in these statements, we also show exponential tail bounds. If a bit is not neutral, but neutral or has a preference for ones, then the lower bounds on the times to reach a low frequency value still hold. An analogous statement holds for bits that are neutral or prefer the value zero.