Rotation symmetric Boolean functions represent an interesting class of Boolean functions as they are relatively rare compared to general Boolean functions. At the same time, the functions in this class can have excellent properties, making them interesting for various practical applications. The usage of metaheuristics to construct rotation symmetric Boolean functions is a direction that has been explored for almost twenty years. Despite that, there are very few results considering evolutionary computation methods. This paper uses several evolutionary algorithms to evolve rotation symmetric Boolean functions with different properties. Despite using generic metaheuristics, we obtain results that are competitive with prior work relying on customized heuristics. Surprisingly, we find that bitstring and floating point encodings work better than the tree encoding. Moreover, evolving highly nonlinear general Boolean functions is easier than rotation symmetric ones.
Training networks consisting of biophysically accurate neuron models could allow for new insights into how brain circuits can organize and solve tasks. We begin by analyzing the extent to which the central algorithm for neural network learning -- stochastic gradient descent through backpropagation (BP) -- can be used to train such networks. We find that properties of biophysically based neural network models needed for accurate modelling such as stiffness, high nonlinearity and long evaluation timeframes relative to spike times makes BP unstable and divergent in a variety of cases. To address these instabilities and inspired by recent work, we investigate the use of "gradient-estimating" evolutionary algorithms (EAs) for training biophysically based neural networks. We find that EAs have several advantages making them desirable over direct BP, including being forward-pass only, robust to noisy and rigid losses, allowing for discrete loss formulations, and potentially facilitating a more global exploration of parameters. We apply our method to train a recurrent network of Morris-Lecar neuron models on a stimulus integration and working memory task, and show how it can succeed in cases where direct BP is inapplicable. To expand on the viability of EAs in general, we apply them to a general neural ODE problem and a stiff neural ODE benchmark and find again that EAs can out-perform direct BP here, especially for the over-parameterized regime. Our findings suggest that biophysical neurons could provide useful benchmarks for testing the limits of BP-adjacent methods, and demonstrate the viability of EAs for training networks with complex components.
The fitness level method is an easy-to-use tool for estimating the hitting time of elitist evolutionary algorithms. Recently, linear lower and upper bounds by fitness levels have been constructed. But these bounds require recursive computation, which makes them difficult to use in practice. We address this shortcoming with a new directed graph (digraph) method that does not require recursive computation and significantly simplifies the calculation of coefficients in the lower bound. In the method, we select a sub-digraph and divide it into fitness levels, then construct an explicit formula for computing the linear lower bound coefficients using transition probabilities restricted to the subdigraph. A major advantage of the new method is the derivation of tight lower bounds on fitness functions with shortcuts, which are difficult to achieve using previous fitness methods. We use three examples (FullyDeceptive, TwoMax1 and Deceptive) to demonstrate that each new lower bound is tight, but previous lower bounds are not. Our work significantly extends the fitness level method from addressing simple fitness functions without shortcuts to more complex functions with shortcuts.
The study of hardest and easiest fitness landscapes is an active area of research. Recently, Kaufmann, Larcher, Lengler and Zou conjectured that for the self-adjusting $(1,\lambda)$-EA, Adversarial Dynamic BinVal (ADBV) is the hardest dynamic monotone function to optimize. We introduce the function Switching Dynamic BinVal (SDBV) which coincides with ADBV whenever the number of remaining zeros in the search point is strictly less than $n/2$, where $n$ denotes the dimension of the search space. We show, using a combinatorial argument, that for the $(1+1)$-EA with any mutation rate $p \in [0,1]$, SDBV is drift-minimizing among the class of dynamic monotone functions. Our construction provides the first explicit example of an instance of the partially-ordered evolutionary algorithm (PO-EA) model with parameterized pessimism introduced by Colin, Doerr and F\'erey, building on work of Jansen. We further show that the $(1+1)$-EA optimizes SDBV in $\Theta(n^{3/2})$ generations. Our simulations demonstrate matching runtimes for both static and self-adjusting $(1,\lambda)$ and $(1+\lambda)$-EA. We further show, using an example of fixed dimension, that drift-minimization does not equal maximal runtime.
The use of Evolutionary Algorithms (EA) for solving Mathematical/Computational Optimization Problems is inspired by the biological processes of Evolution. Few of the primitives involved in the Evolutionary process/paradigm are selection of 'Fit' individuals (from a population sample) for retention, cloning, mutation, discarding, breeding, crossover etc. In the Evolutionary Algorithm abstraction, the individuals are deemed to be solution candidates to an Optimization problem and additional solution(/sets) are built by applying analogies to the above primitives (cloning, mutation etc.) by means of evaluating a 'Fitness' function/criterion. One such algorithm is Differential Evolution (DE) which can be used to compute the minima of functions such as the rastrigin function and rosenbrock function. This work is an attempt to study the result of applying the DE method on these functions with candidate individuals generated on classical Turing modeled computation and comparing the same with those on state of the art Quantum computation.The study benchmarks the convergence of these functions by varying the parameters initialized and reports timing, convergence, and resource utilization results.
Creating and evaluating games manually is an arduous and laborious task. Procedural content generation can aid by creating game artifacts, but usually not an entire game. Evolutionary game design, which combines evolutionary algorithms with automated playtesting, has been used to create novel board games with simple equipment; however, the original approach does not include complex tabletop games with dice, cards, and maps. This work proposes an extension of the approach for tabletop games, evaluating the process by generating variants of Risk, a military strategy game where players must conquer map territories to win. We achieved this using a genetic algorithm to evolve the chosen parameters, as well as a rules-based agent to test the games and a variety of quality criteria to evaluate the new variations generated. Our results show the creation of new variations of the original game with smaller maps, resulting in shorter matches. Also, the variants produce more balanced matches, maintaining the usual drama. We also identified limitations in the process, where, in many cases, where the objective function was correctly pursued, but the generated games were nearly trivial. This work paves the way towards promising research regarding the use of evolutionary game design beyond classic board games.
Evolutionary algorithms (EAs) have achieved remarkable success in tackling complex combinatorial optimization problems. However, EAs often demand carefully-designed operators with the aid of domain expertise to achieve satisfactory performance. In this work, we present the first study on large language models (LLMs) as evolutionary combinatorial optimizers. The main advantage is that it requires minimal domain knowledge and human efforts, as well as no additional training of the model. This approach is referred to as LLM-driven EA (LMEA). Specifically, in each generation of the evolutionary search, LMEA instructs the LLM to select parent solutions from current population, and perform crossover and mutation to generate offspring solutions. Then, LMEA evaluates these new solutions and include them into the population for the next generation. LMEA is equipped with a self-adaptation mechanism that controls the temperature of the LLM. This enables it to balance between exploration and exploitation and prevents the search from getting stuck in local optima. We investigate the power of LMEA on the classical traveling salesman problems (TSPs) widely used in combinatorial optimization research. Notably, the results show that LMEA performs competitively to traditional heuristics in finding high-quality solutions on TSP instances with up to 20 nodes. Additionally, we also study the effectiveness of LLM-driven crossover/mutation and the self-adaptation mechanism in evolutionary search. In summary, our results reveal the great potentials of LLMs as evolutionary optimizers for solving combinatorial problems. We hope our research shall inspire future explorations on LLM-driven EAs for complex optimization challenges.
We introduce EV3, a novel meta-optimization framework designed to efficiently train scalable machine learning models through an intuitive explore-assess-adapt protocol. In each iteration of EV3, we explore various model parameter updates, assess them using pertinent evaluation methods, and then adapt the model based on the optimal updates and previous progress history. EV3 offers substantial flexibility without imposing stringent constraints like differentiability on the key objectives relevant to the tasks of interest, allowing for exploratory updates with intentionally-biased gradients and through a diversity of losses and optimizers. Additionally, the assessment phase provides reliable safety controls to ensure robust generalization, and can dynamically prioritize tasks in scenarios with multiple objectives. With inspiration drawn from evolutionary algorithms, meta-learning, and neural architecture search, we investigate an application of EV3 to knowledge distillation. Our experimental results illustrate EV3's capability to safely explore the modeling landscape, while hinting at its potential applicability across numerous domains due to its inherent flexibility and adaptability. Finally, we provide a JAX implementation of EV3, along with source code for experiments, available at: https://github.com/google-research/google-research/tree/master/ev3.
The schooling behavior of fish can be studied through simulations involving a large number of interacting particles. In such systems, each individual particle is guided by behavior rules, which include aggregation towards a centroid, collision avoidance, and direction alignment. The movement vector of each particle may be expressed as a linear combination of behaviors, with unknown parameters that define a trade-off among several behavioral constraints. A fitness function for collective schooling behavior encompasses all individual particle parameters. For a large number of interacting particles in a complex environment, heuristic methods, such as evolutionary algorithms, are used to optimize the fitness function, ensuring that the resulting decision rule preserves collective behavior. However, these algorithms exhibit slow convergence, making them inefficient in terms of CPU time cost. This paper proposes a CPU-efficient iterative (Cluster, Partition, Refine -- CPR) algorithm for estimating decision rule parameters for a large number of interacting particles. In the first step, we employ the K-Means (unsupervised learning) algorithm to cluster candidate solutions. Then, we partition the search space using Voronoi tessellation over the defined clusters. We assess the quality of each cluster based on the fitness function, with the centroid of their Voronoi cells representing the clusters. Subsequently, we refine the search space by introducing new cells into a number of identified well-fitting Voronoi cells. This process is repeated until convergence. A comparison of the performance of the CPR algorithm with a standard Genetic Algorithm reveals that the former converges faster than the latter. We also demonstrate that the application of the CPR algorithm results in a schooling behavior consistent with empirical observations.
An automated sizing approach for analog circuits using evolutionary algorithms is presented in this paper. A targeted search of the search space has been implemented using a particle generation function and a repair-bounds function that has resulted in faster convergence to the optimal solution. The algorithms are tuned and modified to converge to a better optimal solution with less standard deviation for multiple runs compared to standard versions. Modified versions of the artificial bee colony optimisation algorithm, genetic algorithm, grey wolf optimisation algorithm, and particle swarm optimisation algorithm are tested and compared for the optimal sizing of two operational amplifier topologies. An extensive performance evaluation of all the modified algorithms showed that the modifications have resulted in consistent performance with improved convergence for all the algorithms. The implementation of parallel computation in the algorithms has reduced run time. Among the considered algorithms, the modified artificial bee colony optimisation algorithm gave the most optimal solution with consistent results across multiple runs.
Multiobjective evolutionary algorithms (MOEAs) are major methods for solving multiobjective optimization problems (MOPs). Many MOEAs have been proposed in the past decades, of which the search operators need a carefully handcrafted design with domain knowledge. Recently, some attempts have been made to replace the manually designed operators in MOEAs with learning-based operators (e.g., neural network models). However, much effort is still required for designing and training such models, and the learned operators might not generalize well on new problems. To tackle the above challenges, this work investigates a novel approach that leverages the powerful large language model (LLM) to design MOEA operators. With proper prompt engineering, we successfully let a general LLM serve as a black-box search operator for decomposition-based MOEA (MOEA/D) in a zero-shot manner. In addition, by learning from the LLM behavior, we further design an explicit white-box operator with randomness and propose a new version of decomposition-based MOEA, termed MOEA/D-LO. Experimental studies on different test benchmarks show that our proposed method can achieve competitive performance with widely used MOEAs. It is also promising to see the operator only learned from a few instances can have robust generalization performance on unseen problems with quite different patterns and settings. The results reveal the potential benefits of using pre-trained LLMs in the design of MOEAs.To foster reproducibility and accessibility, the source code is https://github.com/FeiLiu36/LLM4MOEA.
Recently, various Artificial Intelligence (AI) based optimization metaheuristics are proposed and applied for a variety of problems. Cohort Intelligence (CI) algorithm is a socio inspired optimization technique which is successfully applied for solving several unconstrained & constrained real-world problems from the domains such as design, manufacturing, supply chain, healthcare, etc. Generally, real-world problems are constrained in nature. Even though most of the Evolutionary Algorithms (EAs) can efficiently solve unconstrained problems, their performance degenerates when the constraints are involved. In this paper, two novel constraint handling approaches based on modulus and hyperbolic tangent probability distributions are proposed. Constrained CI algorithm with constraint handling approaches based on triangular, modulus and hyperbolic tangent is presented and applied for optimizing advanced manufacturing processes such as Water Jet Machining (WJM), Abrasive Jet Machining (AJM), Ultrasonic Machining (USM) and Grinding process. The solutions obtained using proposed CI algorithm are compared with contemporary algorithms such as Genetic Algorithm, Simulated Annealing, Teaching Learning Based Optimization, etc. The proposed approaches achieved 2%-127% maximization of material removal rate satisfying hard constraints. As compared to the GA, CI with Hyperbolic tangent probability distribution achieved 15%, 2%, 2%, 127%, and 4% improvement in MRR for AJMB, AJMD, WJM, USM, and Grinding processes, respectively contributing to the productivity improvement. The contributions in this paper have opened several avenues for further applicability of the proposed constraint handling approaches for solving complex constrained problems.
Evolutionary Computation (EC) has emerged as a powerful field of Artificial Intelligence, inspired by nature's mechanisms of gradual development. However, EC approaches often face challenges such as stagnation, diversity loss, computational complexity, population initialization, and premature convergence. To overcome these limitations, researchers have integrated learning algorithms with evolutionary techniques. This integration harnesses the valuable data generated by EC algorithms during iterative searches, providing insights into the search space and population dynamics. Similarly, the relationship between evolutionary algorithms and Machine Learning (ML) is reciprocal, as EC methods offer exceptional opportunities for optimizing complex ML tasks characterized by noisy, inaccurate, and dynamic objective functions. These hybrid techniques, known as Evolutionary Machine Learning (EML), have been applied at various stages of the ML process. EC techniques play a vital role in tasks such as data balancing, feature selection, and model training optimization. Moreover, ML tasks often require dynamic optimization, for which Evolutionary Dynamic Optimization (EDO) is valuable. This paper presents the first comprehensive exploration of reciprocal integration between EDO and ML. The study aims to stimulate interest in the evolutionary learning community and inspire innovative contributions in this domain.
Evolutionary algorithms (EAs) are widely used for multi-objective optimization due to their population-based nature. Traditional multi-objective EAs (MOEAs) generate a large set of solutions to approximate the Pareto front, leaving a decision maker (DM) with the task of selecting a preferred solution. However, this process can be inefficient and time-consuming, especially when there are many objectives or the subjective preferences of DM is known. To address this issue, interactive MOEAs (iMOEAs) combine decision making into the optimization process, i.e., update the population with the help of the DM. In contrast to their wide applications, there has existed only two pieces of theoretical works on iMOEAs, which only considered interactive variants of the two simple single-objective algorithms, RLS and (1+1)-EA. This paper provides the first running time analysis (the essential theoretical aspect of EAs) for practical iMOEAs. Specifically, we prove that the expected running time of the well-developed interactive NSGA-II (called R-NSGA-II) for solving the OneMinMax and OneJumpZeroJump problems is $O(n \log n)$ and $O(n^k)$, respectively, which are all asymptotically faster than the traditional NSGA-II. Meanwhile, we present a variant of OneMinMax, and prove that R-NSGA-II can be exponentially slower than NSGA-II. These results provide theoretical justification for the effectiveness of iMOEAs while identifying situations where they may fail. Experiments are also conducted to validate the theoretical results.
Dynamic Optimization Problems (DOPs) are characterized by changes in the fitness landscape that can occur at any time and are common in real world applications. The main issues to be considered include detecting the change in the fitness landscape and reacting in accord. Over the years, several evolutionary algorithms have been proposed to take into account this characteristic during the optimization process. However, the number of available tools or open source codebases for these approaches is limited, making reproducibility and extensive experimentation difficult. To solve this, we developed a component-oriented framework for DOPs called Adjustable Components for Dynamic Problems (AbCD), inspired by similar works in the Multiobjective static domain. Using this framework, we investigate components that were proposed in several popular DOP algorithms. Our experiments show that the performance of these components depends on the problem and the selected components used in a configuration, which differs from the results reported in the literature. Using irace, we demonstrate how this framework can automatically generate DOP algorithm configurations that take into account the characteristics of the problem to be solved. Our results highlight existing problems in the DOP field that need to be addressed in the future development of algorithms and components.
Finding a large set of optima in a multimodal optimization landscape is a challenging task. Classical population-based evolutionary algorithms typically converge only to a single solution. While this can be counteracted by applying niching strategies, the number of optima is nonetheless trivially bounded by the population size. Estimation-of-distribution algorithms (EDAs) are an alternative, maintaining a probabilistic model of the solution space instead of a population. Such a model is able to implicitly represent a solution set far larger than any realistic population size. To support the study of how optimization algorithms handle large sets of optima, we propose the test function EqualBlocksOneMax (EBOM). It has an easy fitness landscape with exponentially many optima. We show that the bivariate EDA mutual-information-maximizing input clustering, without any problem-specific modification, quickly generates a model that behaves very similarly to a theoretically ideal model for EBOM, which samples each of the exponentially many optima with the same maximal probability. We also prove via mathematical means that no univariate model can come close to having this property: If the probability to sample an optimum is at least inverse-polynomial, there is a Hamming ball of logarithmic radius such that, with high probability, each sample is in this ball.
Soft-growing robots are innovative devices that feature plant-inspired growth to navigate environments. Thanks to their embodied intelligence of adapting to their surroundings and the latest innovation in actuation and manufacturing, it is possible to employ them for specific manipulation tasks. The applications of these devices include exploration of delicate/dangerous environments, manipulation of items, or assistance in domestic environments. This work presents a novel approach for design optimization of soft-growing robots, which will be used prior to manufacturing to suggest engineers -- or robot designer enthusiasts -- the optimal dimension of the robot to be built for solving a specific task. I modeled the design process as a multi-objective optimization problem, in which I optimize the kinematic chain of a soft manipulator to reach targets and avoid unnecessary overuse of material and resources. The method exploits the advantages of population-based optimization algorithms, in particular evolutionary algorithms, to transform the problem from multi-objective into a single-objective thanks to an efficient mathematical formulation, the novel rank-partitioning algorithm, and obstacle avoidance integrated within the optimizer operators. I tested the proposed method on different tasks to access its optimality, which showed significant performance in solving the problem. Finally, comparative experiments showed that the proposed method works better than the one existing in the literature in terms of precision, resource consumption, and run time.
Population-based evolutionary algorithms are often considered when approaching computationally expensive black-box optimization problems. They employ a selection mechanism to choose the best solutions from a given population after comparing their objective values, which are then used to generate the next population. This iterative process explores the solution space efficiently, leading to improved solutions over time. However, these algorithms require a large number of evaluations to provide a quality solution, which might be computationally expensive when the evaluation cost is high. In some cases, it is possible to replace the original objective function with a less accurate approximation of lower cost. This introduces a trade-off between the evaluation cost and its accuracy. In this paper, we propose a technique capable of choosing an appropriate approximate function cost during the execution of the optimization algorithm. The proposal finds the minimum evaluation cost at which the solutions are still properly ranked, and consequently, more evaluations can be computed in the same amount of time with minimal accuracy loss. An experimental section on four very different problems reveals that the proposed approach can reach the same objective value in less than half of the time in certain cases.
Feature transformation aims to generate new pattern-discriminative feature space from original features to improve downstream machine learning (ML) task performances. However, the discrete search space for the optimal feature explosively grows on the basis of combinations of features and operations from low-order forms to high-order forms. Existing methods, such as exhaustive search, expansion reduction, evolutionary algorithms, reinforcement learning, and iterative greedy, suffer from large search space. Overly emphasizing efficiency in algorithm design usually sacrifices stability or robustness. To fundamentally fill this gap, we reformulate discrete feature transformation as a continuous space optimization task and develop an embedding-optimization-reconstruction framework. This framework includes four steps: 1) reinforcement-enhanced data preparation, aiming to prepare high-quality transformation-accuracy training data; 2) feature transformation operation sequence embedding, intending to encapsulate the knowledge of prepared training data within a continuous space; 3) gradient-steered optimal embedding search, dedicating to uncover potentially superior embeddings within the learned space; 4) transformation operation sequence reconstruction, striving to reproduce the feature transformation solution to pinpoint the optimal feature space.
With the increasing application of machine learning (ML) algorithms in embedded systems, there is a rising necessity to design low-cost computer arithmetic for these resource-constrained systems. As a result, emerging models of computation, such as approximate and stochastic computing, that leverage the inherent error-resilience of such algorithms are being actively explored for implementing ML inference on resource-constrained systems. Approximate computing (AxC) aims to provide disproportionate gains in the power, performance, and area (PPA) of an application by allowing some level of reduction in its behavioral accuracy (BEHAV). Using approximate operators (AxOs) for computer arithmetic forms one of the more prevalent methods of implementing AxC. AxOs provide the additional scope for finer granularity of optimization, compared to only precision scaling of computer arithmetic. To this end, designing platform-specific and cost-efficient approximate operators forms an important research goal. Recently, multiple works have reported using AI/ML-based approaches for synthesizing novel FPGA-based AxOs. However, most of such works limit usage of AI/ML to designing ML-based surrogate functions used during iterative optimization processes. To this end, we propose a novel data analysis-driven mathematical programming-based approach to synthesizing approximate operators for FPGAs. Specifically, we formulate mixed integer quadratically constrained programs based on the results of correlation analysis of the characterization data and use the solutions to enable a more directed search approach for evolutionary optimization algorithms. Compared to traditional evolutionary algorithms-based optimization, we report up to 21% improvement in the hypervolume, for joint optimization of PPA and BEHAV, in the design of signed 8-bit multipliers.