Linear functions play a key role in the runtime analysis of evolutionary algorithms and studies have provided a wide range of new insights and techniques for analyzing evolutionary computation methods. Motivated by studies on separable functions and the optimization behaviour of evolutionary algorithms as well as objective functions from the area of chance constrained optimization, we study the class of objective functions that are weighted sums of two transformed linear functions. Our results show that the (1+1) EA, with a mutation rate depending on the number of overlapping bits of the functions, obtains an optimal solution for these functions in expected time O(n log n), thereby generalizing a well-known result for linear functions to a much wider range of problems.
Almost all adversarial attacks are formulated to add an imperceptible perturbation to an image in order to fool a model. Here, we consider the opposite which is adversarial examples that can fool a human but not a model. A large enough and perceptible perturbation is added to an image such that a model maintains its original decision, whereas a human will most likely make a mistake if forced to decide (or opt not to decide at all). Existing targeted attacks can be reformulated to synthesize such adversarial examples. Our proposed attack, dubbed NKE, is similar in essence to the fooling images, but is more efficient since it uses gradient descent instead of evolutionary algorithms. It also offers a new and unified perspective into the problem of adversarial vulnerability. Experimental results over MNIST and CIFAR-10 datasets show that our attack is quite efficient in fooling deep neural networks. Code is available at https://github.com/aliborji/NKE.
Ensemble learning has gained success in machine learning with major advantages over other learning methods. Bagging is a prominent ensemble learning method that creates subgroups of data, known as bags, that are trained by individual machine learning methods such as decision trees. Random forest is a prominent example of bagging with additional features in the learning process. Evolutionary algorithms have been prominent for optimisation problems and also been used for machine learning. Evolutionary algorithms are gradient-free methods that work with a population of candidate solutions that maintain diversity for creating new solutions. In conventional bagged ensemble learning, the bags are created once and the content, in terms of the training examples, are fixed over the learning process. In our paper, we propose evolutionary bagged ensemble learning, where we utilise evolutionary algorithms to evolve the content of the bags in order to iteratively enhance the ensemble by providing diversity in the bags. The results show that our evolutionary ensemble bagging method outperforms conventional ensemble methods (bagging and random forests) for several benchmark datasets under certain constraints. We find that evolutionary bagging can inherently sustain a diverse set of bags without reduction in performance accuracy.
One of the main problems of evolutionary algorithms is the convergence of the population to local minima. In this paper, we explore techniques that can avoid this problem by encouraging a diverse behavior of the agents through a shared reward system. The rewards are randomly distributed in the environment, and the agents are only rewarded for collecting them first. This leads to an emergence of a novel behavior of the agents. We introduce our approach to the maze problem and compare it to the previously proposed solution, denoted as Novelty Search (Lehman and Stanley, 2011a). We find that our solution leads to an improved performance while being significantly simpler. Building on that, we generalize the problem and apply our approach to a more advanced set of tasks, Atari Games, where we observe a similar performance quality with much less computational power needed.
Multimodal multi-objective problems (MMOPs) commonly arise in real-world problems where distant solutions in decision space correspond to very similar objective values. To obtain all solutions for MMOPs, many multimodal multi-objective evolutionary algorithms (MMEAs) have been proposed. For now, few studies have encompassed most of the recently proposed representative MMEAs and made a comparative comparison. In this study, we first review the related works during the last two decades. Then, we choose 12 state-of-the-art algorithms that utilize different diversity-maintaining techniques and compared their performance on existing test suites. Experimental results indicate the strengths and weaknesses of different techniques on different types of MMOPs, thus providing guidance on how to select/design MMEAs in specific scenarios.
We present a comprehensive global sensitivity analysis of two single-objective and two multi-objective state-of-the-art global optimization evolutionary algorithms as an algorithm configuration problem. That is, we investigate the quality of influence hyperparameters have on the performance of algorithms in terms of their direct effect and interaction effect with other hyperparameters. Using three sensitivity analysis methods, Morris LHS, Morris, and Sobol, to systematically analyze tunable hyperparameters of covariance matrix adaptation evolutionary strategy, differential evolution, non-dominated sorting genetic algorithm III, and multi-objective evolutionary algorithm based on decomposition, the framework reveals the behaviors of hyperparameters to sampling methods and performance metrics. That is, it answers questions like what hyperparameters influence patterns, how they interact, how much they interact, and how much their direct influence is. Consequently, the ranking of hyperparameters suggests their order of tuning, and the pattern of influence reveals the stability of the algorithms.
While the theoretical analysis of evolutionary algorithms (EAs) has made significant progress for pseudo-Boolean optimization problems in the last 25 years, only sporadic theoretical results exist on how EAs solve permutation-based problems. To overcome the lack of permutation-based benchmark problems, we propose a general way to transfer the classic pseudo-Boolean benchmarks into benchmarks defined on sets of permutations. We then conduct a rigorous runtime analysis of the permutation-based $(1+1)$ EA proposed by Scharnow, Tinnefeld, and Wegener (2004) on the analogues of the LeadingOnes and Jump benchmarks. The latter shows that, different from bit-strings, it is not only the Hamming distance that determines how difficult it is to mutate a permutation $\sigma$ into another one $\tau$, but also the precise cycle structure of $\sigma \tau^{-1}$. For this reason, we also regard the more symmetric scramble mutation operator. We observe that it not only leads to simpler proofs, but also reduces the runtime on jump functions with odd jump size by a factor of $\Theta(n)$. Finally, we show that a heavy-tailed version of the scramble operator, as in the bit-string case, leads to a speed-up of order $m^{\Theta(m)}$ on jump functions with jump size $m$. A short empirical analysis confirms these findings, but also reveals that small implementation details like the rate of void mutations can make an important difference.
As we continue to find applications where the currently available noisy devices exhibit an advantage over their classical counterparts, the efficient use of quantum resources is highly desirable. The notion of quantum autoencoders was proposed as a way for the compression of quantum information to reduce resource requirements. Here, we present a strategy to design quantum autoencoders using evolutionary algorithms for transforming quantum information into lower-dimensional representations. We successfully demonstrate the initial applications of the algorithm for compressing different families of quantum states. In particular, we point out that using a restricted gate set in the algorithm allows for efficient simulation of the generated circuits. This approach opens the possibility of using classical logic to find low representations of quantum data, using fewer computational resources.
Co-evolutionary algorithms have a wide range of applications, such as in hardware design, evolution of strategies for board games, and patching software bugs. However, these algorithms are poorly understood and applications are often limited by pathological behaviour, such as loss of gradient, relative over-generalisation, and mediocre objective stasis. It is an open challenge to develop a theory that can predict when co-evolutionary algorithms find solutions efficiently and reliable. This paper provides a first step in developing runtime analysis for population-based competitive co-evolutionary algorithms. We provide a mathematical framework for describing and reasoning about the performance of co-evolutionary processes. An example application of the framework shows a scenario where a simple co-evolutionary algorithm obtains a solution in polynomial expected time. Finally, we describe settings where the co-evolutionary algorithm needs exponential time with overwhelmingly high probability to obtain a solution.
In Evolutionary Robotics, evolutionary algorithms are used to co-optimize morphology and control. However, co-optimizing leads to different challenges: How do you optimize a controller for a body that often changes its number of inputs and outputs? Researchers must then make some choice between centralized or decentralized control. In this article, we study the effects of centralized and decentralized controllers on modular robot performance and morphologies. This is done by implementing one centralized and two decentralized continuous time recurrent neural network controllers, as well as a sine wave controller for a baseline. We found that a decentralized approach that was more independent of morphology size performed significantly better than the other approaches. It also worked well in a larger variety of morphology sizes. In addition, we highlighted the difficulties of implementing centralized control for a changing morphology, and saw that our centralized controller struggled more with early convergence than the other approaches. Our findings indicate that duplicated decentralized networks are beneficial when evolving both the morphology and control of modular robots. Overall, if these findings translate to other robot systems, our results and issues encountered can help future researchers make a choice of control method when co-optimizing morphology and control.
How someone allocates their time is important to their health and well-being. In this paper, we show how evolutionary algorithms can be used to promote health and well-being by optimizing time usage. Based on data from a large population-based child cohort, we design fitness functions to explain health outcomes and introduce constraints for viable time plans. We then investigate the performance of evolutionary algorithms to optimize time use for four individual health outcomes with hypothetical children with different day structures. As the four health outcomes are competing for time allocations, we study how to optimize multiple health outcomes simultaneously in the form of a multi-objective optimization problem. We optimize one-week time-use plans using evolutionary multi-objective algorithms and point out the trade-offs achievable with respect to different health outcomes.
Recent decades have witnessed great advancements in multiobjective evolutionary algorithms (MOEAs) for multiobjective optimization problems (MOPs). However, these progressively improved MOEAs have not necessarily been equipped with scalable and learnable problem-solving strategies for new and grand challenges brought by the scaling-up MOPs with continuously increasing complexity from diverse aspects, mainly including expensive cost of function evaluations, many objectives, large-scale search space, time-varying environments, and multi-task. Under different scenarios, divergent thinking is required in designing new powerful MOEAs for solving them effectively. In this context, research studies on learnable MOEAs with machine learning techniques have received extensive attention in the field of evolutionary computation. This paper begins with a general taxonomy of scaling-up MOPs and learnable MOEAs, followed by an analysis of the challenges that these MOPs pose to traditional MOEAs. Then, we synthetically overview recent advances of learnable MOEAs in solving various scaling-up MOPs, focusing primarily on four attractive directions (i.e., learnable evolutionary discriminators for environmental selection, learnable evolutionary generators for reproduction, learnable evolutionary evaluators for function evaluations, and learnable evolutionary transfer modules for sharing or reusing optimization experience). The insight of learnable MOEAs is offered to readers as a reference to the general track of the efforts in this field.
Genetic algorithms have unique properties which are useful when applied to black box optimization. Using selection, crossover, and mutation operators, candidate solutions may be obtained without the need to calculate a gradient. In this work, we study results obtained from using quantum-enhanced operators within the selection mechanism of a genetic algorithm. Our approach frames the selection process as a minimization of a binary quadratic model with which we encode fitness and distance between members of a population, and we leverage a quantum annealing system to sample low energy solutions for the selection mechanism. We benchmark these quantum-enhanced algorithms against classical algorithms over various black-box objective functions, including the OneMax function, and functions from the IOHProfiler library for black-box optimization. We observe a performance gain in average number of generations to convergence for the quantum-enhanced elitist selection operator in comparison to classical on the OneMax function. We also find that the quantum-enhanced selection operator with non-elitist selection outperform benchmarks on functions with fitness perturbation from the IOHProfiler library. Additionally, we find that in the case of elitist selection, the quantum-enhanced operators outperform classical benchmarks on functions with varying degrees of dummy variables and neutrality.
Hyperparameter optimization constitutes a large part of typical modern machine learning workflows. This arises from the fact that machine learning methods and corresponding preprocessing steps often only yield optimal performance when hyperparameters are properly tuned. But in many applications, we are not only interested in optimizing ML pipelines solely for predictive accuracy; additional metrics or constraints must be considered when determining an optimal configuration, resulting in a multi-objective optimization problem. This is often neglected in practice, due to a lack of knowledge and readily available software implementations for multi-objective hyperparameter optimization. In this work, we introduce the reader to the basics of multi-objective hyperparameter optimization and motivate its usefulness in applied ML. Furthermore, we provide an extensive survey of existing optimization strategies, both from the domain of evolutionary algorithms and Bayesian optimization. We illustrate the utility of MOO in several specific ML applications, considering objectives such as operating conditions, prediction time, sparseness, fairness, interpretability and robustness.
With the ever increasing complexity of specifications, manual sizing for analog circuits recently became very challenging. Especially for innovative, large-scale circuits designs, with tens of design variables, operating conditions and conflicting objectives to be optimized, design engineers spend many weeks, running time-consuming simulations, in their attempt at finding the right configuration. Recent years brought machine learning and optimization techniques to the field of analog circuits design, with evolutionary algorithms and Bayesian models showing good results for circuit sizing. In this context, we introduce a design optimization method based on Generalized Differential Evolution 3 (GDE3) and Gaussian Processes (GPs). The proposed method is able to perform sizing for complex circuits with a large number of design variables and many conflicting objectives to be optimized. While state-of-the-art methods reduce multi-objective problems to single-objective optimization and potentially induce a prior bias, we search directly over the multi-objective space using Pareto dominance and ensure that diverse solutions are provided to the designers to choose from. To the best of our knowledge, the proposed method is the first to specifically address the diversity of the solutions, while also focusing on minimizing the number of simulations required to reach feasible configurations. We evaluate the introduced method on two voltage regulators showing different levels of complexity and we highlight that the proposed innovative candidate selection method and survival policy leads to obtaining feasible solutions, with a high degree of diversity, much faster than with GDE3 or Bayesian Optimization-based algorithms.
Directed Evolution (DE), a landmark wet-lab method originated in 1960s, enables discovery of novel protein designs via evolving a population of candidate sequences. Recent advances in biotechnology has made it possible to collect high-throughput data, allowing the use of machine learning to map out a protein's sequence-to-function relation. There is a growing interest in machine learning-assisted DE for accelerating protein optimization. Yet the theoretical understanding of DE, as well as the use of machine learning in DE, remains limited. In this paper, we connect DE with the bandit learning theory and make a first attempt to study regret minimization in DE. We propose a Thompson Sampling-guided Directed Evolution (TS-DE) framework for sequence optimization, where the sequence-to-function mapping is unknown and querying a single value is subject to costly and noisy measurements. TS-DE updates a posterior of the function based on collected measurements. It uses a posterior-sampled function estimate to guide the crossover recombination and mutation steps in DE. In the case of a linear model, we show that TS-DE enjoys a Bayesian regret of order $\tilde O(d^{2}\sqrt{MT})$, where $d$ is feature dimension, $M$ is population size and $T$ is number of rounds. This regret bound is nearly optimal, confirming that bandit learning can provably accelerate DE. It may have implications for more general sequence optimization and evolutionary algorithms.
In evolutionary multiobjective optimization, effectiveness refers to how an evolutionary algorithm performs in terms of converging its solutions into the Pareto front and also diversifying them over the front. This is not an easy job, particularly for optimization problems with more than three objectives, dubbed many-objective optimization problems. In such problems, classic Pareto-based algorithms fail to provide sufficient selection pressure towards the Pareto front, whilst recently developed algorithms, such as decomposition-based ones, may struggle to maintain a set of well-distributed solutions on certain problems (e.g., those with irregular Pareto fronts). Another issue in some many-objective optimizers is rapidly increasing computational requirement with the number of objectives, such as hypervolume-based algorithms and shift-based density estimation (SDE) methods. In this paper, we aim to address this problem and develop an effective and efficient evolutionary algorithm (E3A) that can handle various many-objective problems. In E3A, inspired by SDE, a novel population maintenance method is proposed to select high-quality solutions in the environmental selection procedure. We conduct extensive experiments and show that E3A performs better than 11 state-of-the-art many-objective evolutionary algorithms in quickly finding a set of well-converged and well-diversified solutions.
Evolutionary algorithms (EAs), such as the genetic algorithm (GA), offer an elegant way to handle combinatorial optimization problems (COPs). However, limited by expertise and resources, most users do not have enough capability to implement EAs to solve COPs. An intuitive and promising solution is to outsource evolutionary operations to a cloud server, whilst it suffers from privacy concerns. To this end, this paper proposes a novel computing paradigm, evolution as a service (EaaS), where a cloud server renders evolutionary computation services for users without sacrificing users' privacy. Inspired by the idea of EaaS, this paper designs PEGA, a novel privacy-preserving GA for COPs. Specifically, PEGA enables users outsourcing COPs to the cloud server holding a competitive GA and approximating the optimal solution in a privacy-preserving manner. PEGA features the following characteristics. First, any user without expertise and enough resources can solve her COPs. Second, PEGA does not leak contents of optimization problems, i.e., users' privacy. Third, PEGA has the same capability as the conventional GA to approximate the optimal solution. We implements PEGA falling in a twin-server architecture and evaluates it in the traveling salesman problem (TSP, a widely known COP). Particularly, we utilize encryption cryptography to protect users' privacy and carefully design a suit of secure computing protocols to support evolutionary operators of GA on encrypted data. Privacy analysis demonstrates that PEGA does not disclose the contents of the COP to the cloud server. Experimental evaluation results on four TSP datasets show that PEGA is as effective as the conventional GA in approximating the optimal solution.
Evolutionary algorithms solve problems by simulating the evolution of a population of candidate solutions. We focus on evolving permutations for ordering problems like the traveling salesperson problem (TSP), as well as assignment problems like the quadratic assignment problem (QAP) and largest common subgraph (LCS). We propose cycle mutation, a new mutation operator whose inspiration is the well known cycle crossover operator, and the concept of a permutation cycle. We use fitness landscape analysis to explore the problem characteristics for which cycle mutation works best. As a prerequisite, we develop new permutation distance measures: cycle distance, $k$-cycle distance, and cycle edit distance. The fitness landscape analysis predicts that cycle mutation is better suited for assignment and mapping problems than it is for ordering problems. We experimentally validate these findings showing cycle mutation's strengths on problems like QAP and LCS, and its limitations on problems like the TSP, while also showing that it is less prone to local optima than commonly used alternatives. We integrate cycle mutation into the open-source Chips-n-Salsa library, and the new distance metrics into the open-source JavaPermutationTools library.
In more recent years, there has been increasing research interest in exploiting the use of application specific hardware for solving optimisation problems. Examples of solvers that use specialised hardware are IBM's Quantum System One and D-wave's Quantum Annealer (QA) and Fujitsu's Digital Annealer (DA). These solvers have been developed to optimise problems faster than traditional meta-heuristics implemented on general purpose machines. Previous research has shown that these solvers (can optimise many problems much quicker than exact solvers such as GUROBI and CPLEX. Such conclusions have not been made when comparing hardware solvers with classical evolutionary algorithms. Making a fair comparison between traditional evolutionary algorithms, such as Genetic Algorithm (GA), and the DA (or other similar solvers) is challenging because the later benefits from the use of application specific hardware while evolutionary algorithms are often implemented on general-purpose machines. Moreover, quantum or quantum-inspired solvers are limited to solving problems in a specific format. A common formulation used is Quadratic Unconstrained Binary Optimisation (QUBO). Many optimisation problems are however constrained and have natural representations that are non-binary. Converting such problems to QUBO can lead to more problem difficulty and/or larger search space. The question addressed in this paper is whether quantum or quantum-inspired solvers can optimise QUBO transformations of combinatorial optimisation problems faster than classical evolutionary algorithms applied to the same problems in their natural representations. We show that the DA often present better average objective function values than GA on Travelling Salesman, Quadratic Assignment and Multi-dimensional Knapsack Problem instances.