Evolutionary multiobjective optimization (EMO) has made significant strides over the past two decades. However, as problem scales and complexities increase, traditional EMO algorithms face substantial performance limitations due to insufficient parallelism and scalability. While most work has focused on algorithm design to address these challenges, little attention has been given to hardware acceleration, thereby leaving a clear gap between EMO algorithms and advanced computing devices, such as GPUs. To bridge the gap, we propose to parallelize EMO algorithms on GPUs via the tensorization methodology. By employing tensorization, the data structures and operations of EMO algorithms are transformed into concise tensor representations, which seamlessly enables automatic utilization of GPU computing. We demonstrate the effectiveness of our approach by applying it to three representative EMO algorithms: NSGA-III, MOEA/D, and HypE. To comprehensively assess our methodology, we introduce a multiobjective robot control benchmark using a GPU-accelerated physics engine. Our experiments show that the tensorized EMO algorithms achieve speedups of up to 1113x compared to their CPU-based counterparts, while maintaining solution quality and effectively scaling population sizes to hundreds of thousands. Furthermore, the tensorized EMO algorithms efficiently tackle complex multiobjective robot control tasks, producing high-quality solutions with diverse behaviors. Source codes are available at https://github.com/EMI-Group/evomo.
Representation learning is a pivotal area in the field of machine learning, focusing on the development of methods to automatically discover the representations or features needed for a given task from raw data. Unlike traditional feature engineering, which requires manual crafting of features, representation learning aims to learn features that are more useful and relevant for tasks such as classification, prediction, and clustering. We introduce Principal Orthogonal Latent Components Analysis Network (POLCA Net), an approach to mimic and extend PCA and LDA capabilities to non-linear domains. POLCA Net combines an autoencoder framework with a set of specialized loss functions to achieve effective dimensionality reduction, orthogonality, variance-based feature sorting, high-fidelity reconstructions, and additionally, when used with classification labels, a latent representation well suited for linear classifiers and low dimensional visualization of class distribution as well.
Multiobjective optimization problems (MOPs) are prevalent in machine learning, with applications in multi-task learning, learning under fairness or robustness constraints, etc. Instead of reducing multiple objective functions into a scalar objective, MOPs aim to optimize for the so-called Pareto optimality or Pareto set learning, which involves optimizing more than one objective function simultaneously, over models with thousands / millions of parameters. Existing benchmark libraries for MOPs mainly focus on evolutionary algorithms, most of which are zeroth-order / meta-heuristic methods that do not effectively utilize higher-order information from objectives and cannot scale to large-scale models with thousands / millions of parameters. In light of the above gap, this paper introduces LibMOON, the first multiobjective optimization library that supports state-of-the-art gradient-based methods, provides a fair benchmark, and is open-sourced for the community.
Machine learning has driven an exponential increase in computational demand, leading to massive data centers that consume significant amounts of energy and contribute to climate change. This makes sustainable data center control a priority. In this paper, we introduce SustainDC, a set of Python environments for benchmarking multi-agent reinforcement learning (MARL) algorithms for data centers (DC). SustainDC supports custom DC configurations and tasks such as workload scheduling, cooling optimization, and auxiliary battery management, with multiple agents managing these operations while accounting for the effects of each other. We evaluate various MARL algorithms on SustainDC, showing their performance across diverse DC designs, locations, weather conditions, grid carbon intensity, and workload requirements. Our results highlight significant opportunities for improvement of data center operations using MARL algorithms. Given the increasing use of DC due to AI, SustainDC provides a crucial platform for the development and benchmarking of advanced algorithms essential for achieving sustainable computing and addressing other heterogeneous real-world challenges.
Evolutionary multiobjective optimization has witnessed remarkable progress during the past decades. However, existing algorithms often encounter computational challenges in large-scale scenarios, primarily attributed to the absence of hardware acceleration. In response, we introduce a Tensorized Reference Vector Guided Evolutionary Algorithm (TensorRVEA) for harnessing the advancements of GPU acceleration. In TensorRVEA, the key data structures and operators are fully transformed into tensor forms for leveraging GPU-based parallel computing. In numerical benchmark tests involving large-scale populations and problem dimensions, TensorRVEA consistently demonstrates high computational performance, achieving up to over 1000$\times$ speedups. Then, we applied TensorRVEA to the domain of multiobjective neuroevolution for addressing complex challenges in robotic control tasks. Furthermore, we assessed TensorRVEA's extensibility by altering several tensorized reproduction operators. Experimental results demonstrate promising scalability and robustness of TensorRVEA. Source codes are available at \url{https://github.com/EMI-Group/tensorrvea}.
Manual parameter tuning of cyber-physical systems is a common practice, but it is labor-intensive. Bayesian Optimization (BO) offers an automated alternative, yet its black-box nature reduces trust and limits human-BO collaborative system tuning. Experts struggle to interpret BO recommendations due to the lack of explanations. This paper addresses the post-hoc BO explainability problem for cyber-physical systems. We introduce TNTRules (Tune-No-Tune Rules), a novel algorithm that provides both global and local explanations for BO recommendations. TNTRules generates actionable rules and visual graphs, identifying optimal solution bounds and ranges, as well as potential alternative solutions. Unlike existing explainable AI (XAI) methods, TNTRules is tailored specifically for BO, by encoding uncertainty via a variance pruning technique and hierarchical agglomerative clustering. A multi-objective optimization approach allows maximizing explanation quality. We evaluate TNTRules using established XAI metrics (Correctness, Completeness, and Compactness) and compare it against adapted baseline methods. The results demonstrate that TNTRules generates high-fidelity, compact, and complete explanations, significantly outperforming three baselines on 5 multi-objective testing functions and 2 hyperparameter tuning problems.
Transfer optimization enables data-efficient optimization of a target task by leveraging experiential priors from related source tasks. This is especially useful in multiobjective optimization settings where a set of trade-off solutions is sought under tight evaluation budgets. In this paper, we introduce a novel concept of \textit{inverse transfer} in multiobjective optimization. Inverse transfer stands out by employing Bayesian inverse Gaussian process models to map performance vectors in the objective space to population search distributions in task-specific decision space, facilitating knowledge transfer through objective space unification. Building upon this idea, we introduce the first Inverse Transfer Evolutionary Multiobjective Optimizer (invTrEMO). A key highlight of invTrEMO is its ability to harness the common objective functions prevalent in many application areas, even when decision spaces do not precisely align between tasks. This allows invTrEMO to uniquely and effectively utilize information from heterogeneous source tasks as well. Furthermore, invTrEMO yields high-precision inverse models as a significant byproduct, enabling the generation of tailored solutions on-demand based on user preferences. Empirical studies on multi- and many-objective benchmark problems, as well as a practical case study, showcase the faster convergence rate and modelling accuracy of the invTrEMO relative to state-of-the-art evolutionary and Bayesian optimization algorithms. The source code of the invTrEMO is made available at https://github.com/LiuJ-2023/invTrEMO.
Classical evolutionary approaches for multiobjective optimization are quite accurate but incur a lot of queries to the objectives; this can be prohibitive when objectives are expensive oracles. A sample-efficient approach to solving multiobjective optimization is via Gaussian process (GP) surrogates and Bayesian optimization (BO). Multiobjective Bayesian optimization (MOBO) involves the construction of an acquisition function which is optimized to acquire new observation candidates sequentially. This ``inner'' optimization can be hard due to various reasons: acquisition functions being nonconvex, nondifferentiable and/or unavailable in analytical form; batch sampling usually exacerbates these problems and the success of MOBO heavily relies on this inner optimization. This, ultimately, affects their sample efficiency. To overcome these challenges, we propose a Thompson sampling (TS) based approach ($q\texttt{POTS}$). Whereas TS chooses candidates according to the probability that they are optimal, $q\texttt{POTS}$ chooses candidates according to the probability that they are Pareto optimal. Instead of a hard acquisition function optimization, $q\texttt{POTS}~$ solves a cheap multiobjective optimization on the GP posteriors with evolutionary approaches. This way we get the best of both worlds: accuracy of evolutionary approaches and sample-efficiency of MOBO. New candidates are chosen on the posterior GP Pareto frontier according to a maximin distance criterion. $q\texttt{POTS}~$ is endowed with theoretical guarantees, a natural exploration-exploitation trade-off, and superior empirical performance.
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.
Sparsity is a highly desired feature in deep neural networks (DNNs) since it ensures numerical efficiency, improves the interpretability of models (due to the smaller number of relevant features), and robustness. For linear models, it is well known that there exists a \emph{regularization path} connecting the sparsest solution in terms of the $\ell^1$ norm, i.e., zero weights and the non-regularized solution. Very recently, there was a first attempt to extend the concept of regularization paths to DNNs by means of treating the empirical loss and sparsity ($\ell^1$ norm) as two conflicting criteria and solving the resulting multiobjective optimization problem for low-dimensional DNN. However, due to the non-smoothness of the $\ell^1$ norm and the high number of parameters, this approach is not very efficient from a computational perspective for high-dimensional DNNs. To overcome this limitation, we present an algorithm that allows for the approximation of the entire Pareto front for the above-mentioned objectives in a very efficient manner for high-dimensional DNNs with millions of parameters. We present numerical examples using both deterministic and stochastic gradients. We furthermore demonstrate that knowledge of the regularization path allows for a well-generalizing network parametrization. To the best of our knowledge, this is the first algorithm to compute the regularization path for non-convex multiobjective optimization problems (MOPs) with millions of degrees of freedom.
Improving the fairness of machine learning models is a nuanced task that requires decision makers to reason about multiple, conflicting criteria. The majority of fair machine learning methods transform the error-fairness trade-off into a single objective problem with a parameter controlling the relative importance of error versus fairness. We propose instead to directly optimize the error-fairness tradeoff by using multi-objective optimization. We present a flexible framework for defining the fair machine learning task as a weighted classification problem with multiple cost functions. This framework is agnostic to the underlying prediction model as well as the metrics. We use multiobjective optimization to define the sample weights used in model training for a given machine learner, and adapt the weights to optimize multiple metrics of fairness and accuracy across a set of tasks. To reduce the number of optimized parameters, and to constrain their complexity with respect to population subgroups, we propose a novel meta-model approach that learns to map protected attributes to sample weights, rather than optimizing those weights directly. On a set of real-world problems, this approach outperforms current state-of-the-art methods by finding solution sets with preferable error/fairness trade-offs.
In order to deploy machine learning in a real-world self-driving laboratory where data acquisition is costly and there are multiple competing design criteria, systems need to be able to intelligently sample while balancing performance trade-offs and constraints. For these reasons, we present an active learning process based on multiobjective black-box optimization with continuously updated machine learning models. This workflow is built on open-source technologies for real-time data streaming and modular multiobjective optimization software development. We demonstrate a proof of concept for this workflow through the autonomous operation of a continuous-flow chemistry laboratory, which identifies ideal manufacturing conditions for the electrolyte 2,2,2-trifluoroethyl methyl carbonate.
The problem of approximating the Pareto front of a multiobjective optimization problem can be reformulated as the problem of finding a set that maximizes the hypervolume indicator. This paper establishes the analytical expression of the Hessian matrix of the mapping from a (fixed size) collection of $n$ points in the $d$-dimensional decision space (or $m$ dimensional objective space) to the scalar hypervolume indicator value. To define the Hessian matrix, the input set is vectorized, and the matrix is derived by analytical differentiation of the mapping from a vectorized set to the hypervolume indicator. The Hessian matrix plays a crucial role in second-order methods, such as the Newton-Raphson optimization method, and it can be used for the verification of local optimal sets. So far, the full analytical expression was only established and analyzed for the relatively simple bi-objective case. This paper will derive the full expression for arbitrary dimensions ($m\geq2$ objective functions). For the practically important three-dimensional case, we also provide an asymptotically efficient algorithm with time complexity in $O(n\log n)$ for the exact computation of the Hessian Matrix' non-zero entries. We establish a sharp bound of $12m-6$ for the number of non-zero entries. Also, for the general $m$-dimensional case, a compact recursive analytical expression is established, and its algorithmic implementation is discussed. Also, for the general case, some sparsity results can be established; these results are implied by the recursive expression. To validate and illustrate the analytically derived algorithms and results, we provide a few numerical examples using Python and Mathematica implementations. Open-source implementations of the algorithms and testing data are made available as a supplement to this paper.
The ongoing advancements in network architecture design have led to remarkable achievements in deep learning across various challenging computer vision tasks. Meanwhile, the development of neural architecture search (NAS) has provided promising approaches to automating the design of network architectures for lower prediction error. Recently, the emerging application scenarios of deep learning have raised higher demands for network architectures considering multiple design criteria: number of parameters/floating-point operations, and inference latency, among others. From an optimization point of view, the NAS tasks involving multiple design criteria are intrinsically multiobjective optimization problems; hence, it is reasonable to adopt evolutionary multiobjective optimization (EMO) algorithms for tackling them. Nonetheless, there is still a clear gap confining the related research along this pathway: on the one hand, there is a lack of a general problem formulation of NAS tasks from an optimization point of view; on the other hand, there are challenges in conducting benchmark assessments of EMO algorithms on NAS tasks. To bridge the gap: (i) we formulate NAS tasks into general multi-objective optimization problems and analyze the complex characteristics from an optimization point of view; (ii) we present an end-to-end pipeline, dubbed $\texttt{EvoXBench}$, to generate benchmark test problems for EMO algorithms to run efficiently -- without the requirement of GPUs or Pytorch/Tensorflow; (iii) we instantiate two test suites comprehensively covering two datasets, seven search spaces, and three hardware devices, involving up to eight objectives. Based on the above, we validate the proposed test suites using six representative EMO algorithms and provide some empirical analyses. The code of $\texttt{EvoXBench}$ is available from $\href{https://github.com/EMI-Group/EvoXBench}{\rm{here}}$.
In the field of evolutionary multiobjective optimization, the decision maker (DM) concerns conflicting objectives. In the real-world applications, there usually exist more than one DM and each DM concerns parts of these objectives. Multiparty multiobjective optimization problems (MPMOPs) are proposed to depict the MOP with multiple decision makers involved, where each party concerns about certain some objectives of all. However, in the evolutionary computation field, there is not much attention paid on MPMOPs. This paper constructs a series of MPMOPs based on distance minimization problems (DMPs), whose Pareto optimal solutions can be vividly visualized. To address MPMOPs, the new proposed algorithm OptMPNDS3 uses the multiparty initializing method to initialize the population and takes JADE2 operator to generate the offsprings. OptMPNDS3 is compared with OptAll, OptMPNDS and OptMPNDS2 on the problem suite. The result shows that OptMPNDS3 is strongly comparable to other algorithms
The deployment of the sensor nodes (SNs) always plays a decisive role in the system performance of wireless sensor networks (WSNs). In this work, we propose an optimal deployment method for practical heterogeneous WSNs which gives a deep insight into the trade-off between the reliability and deployment cost. Specifically, this work aims to provide the optimal deployment of SNs to maximize the coverage degree and connection degree, and meanwhile minimize the overall deployment cost. In addition, this work fully considers the heterogeneity of SNs (i.e. differentiated sensing range and deployment cost) and three-dimensional (3-D) deployment scenarios. This is a multi-objective optimization problem, non-convex, multimodal and NP-hard. To solve it, we develop a novel swarm-based multi-objective optimization algorithm, known as the competitive multi-objective marine predators algorithm (CMOMPA) whose performance is verified by comprehensive comparative experiments with ten other stateof-the-art multi-objective optimization algorithms. The computational results demonstrate that CMOMPA is superior to others in terms of convergence and accuracy and shows excellent performance on multimodal multiobjective optimization problems. Sufficient simulations are also conducted to evaluate the effectiveness of the CMOMPA based optimal SNs deployment method. The results show that the optimized deployment can balance the trade-off among deployment cost, sensing reliability and network reliability. The source code is available on https://github.com/iNet-WZU/CMOMPA.
In this paper, we propose a new end-to-end methodology to optimize the energy performance as well as comfort and air quality in large buildings without any renovation work. We introduce a metamodel based on recurrent neural networks and trained to predict the behavior of a general class of buildings using a database sampled from a simulation program. This metamodel is then deployed in different frameworks and its parameters are calibrated using the specific data of two real buildings. Parameters are estimated by comparing the predictions of the metamodel with real data obtained from sensors using the CMA-ES algorithm, a derivative free optimization procedure. Then, energy consumptions are optimized while maintaining a target thermal comfort and air quality, using the NSGA-II multi-objective optimization procedure. The numerical experiments illustrate how this metamodel ensures a significant gain in energy efficiency, up to almost 10%, while being computationally much more appealing than numerical models and flexible enough to be adapted to several types of buildings.
The field of machine learning for drug discovery is witnessing an explosion of novel methods. These methods are often benchmarked on simple physicochemical properties such as solubility or general druglikeness, which can be readily computed. However, these properties are poor representatives of objective functions in drug design, mainly because they do not depend on the candidate's interaction with the target. By contrast, molecular docking is a widely successful method in drug discovery to estimate binding affinities. However, docking simulations require a significant amount of domain knowledge to set up correctly which hampers adoption. To this end, we present DOCKSTRING, a bundle for meaningful and robust comparison of ML models consisting of three components: (1) an open-source Python package for straightforward computation of docking scores; (2) an extensive dataset of docking scores and poses of more than 260K ligands for 58 medically-relevant targets; and (3) a set of pharmaceutically-relevant benchmark tasks including regression, virtual screening, and de novo design. The Python package implements a robust ligand and target preparation protocol that allows non-experts to obtain meaningful docking scores. Our dataset is the first to include docking poses, as well as the first of its size that is a full matrix, thus facilitating experiments in multiobjective optimization and transfer learning. Overall, our results indicate that docking scores are a more appropriate evaluation objective than simple physicochemical properties, yielding more realistic benchmark tasks and molecular candidates.
Traceless Genetic Programming (TGP) is a Genetic Programming (GP) variant that is used in cases where the focus is rather the output of the program than the program itself. The main difference between TGP and other GP techniques is that TGP does not explicitly store the evolved computer programs. Two genetic operators are used in conjunction with TGP: crossover and insertion. In this paper, we shall focus on how to apply TGP for solving multi-objective optimization problems which are quite unusual for GP. Each TGP individual stores the output of a computer program (tree) representing a point in the search space. Numerical experiments show that TGP is able to solve very fast and very well the considered test problems.
Deep reinforcement learning (DRL) has recently shown its success in tackling complex combinatorial optimization problems. When these problems are extended to multiobjective ones, it becomes difficult for the existing DRL approaches to flexibly and efficiently deal with multiple subproblems determined by weight decomposition of objectives. This paper proposes a concise meta-learning-based DRL approach. It first trains a meta-model by meta-learning. The meta-model is fine-tuned with a few update steps to derive submodels for the corresponding subproblems. The Pareto front is then built accordingly. Compared with other learning-based methods, our method can greatly shorten the training time of multiple submodels. Due to the rapid and excellent adaptability of the meta-model, more submodels can be derived so as to increase the quality and diversity of the found solutions. The computational experiments on multiobjective traveling salesman problems and multiobjective vehicle routing problem with time windows demonstrate the superiority of our method over most of learning-based and iteration-based approaches.