Pareto Front Shape-Agnostic Pareto Set Learning in Multi-Objective Optimization

Pareto set learning (PSL) is an emerging approach for acquiring the complete Pareto set of a multi-objective optimization problem. Existing methods primarily rely on the mapping of preference vectors in the objective space to Pareto optimal solutions in the decision space. However, the sampling of preference vectors theoretically requires prior knowledge of the Pareto front shape to ensure high performance of the PSL methods. Designing a sampling strategy of preference vectors is difficult since the Pareto front shape cannot be known in advance. To make Pareto set learning work effectively in any Pareto front shape, we propose a Pareto front shape-agnostic Pareto Set Learning (GPSL) that does not require the prior information about the Pareto front. The fundamental concept behind GPSL is to treat the learning of the Pareto set as a distribution transformation problem. Specifically, GPSL can transform an arbitrary distribution into the Pareto set distribution. We demonstrate that training a neural network by maximizing hypervolume enables the process of distribution transformation. Our proposed method can handle any shape of the Pareto front and learn the Pareto set without requiring prior knowledge. Experimental results show the high performance of our proposed method on diverse test problems compared with recent Pareto set learning algorithms.

Learning Pareto Set for Multi-Objective Continuous Robot Control

For a control problem with multiple conflicting objectives, there exists a set of Pareto-optimal policies called the Pareto set instead of a single optimal policy. When a multi-objective control problem is continuous and complex, traditional multi-objective reinforcement learning (MORL) algorithms search for many Pareto-optimal deep policies to approximate the Pareto set, which is quite resource-consuming. In this paper, we propose a simple and resource-efficient MORL algorithm that learns a continuous representation of the Pareto set in a high-dimensional policy parameter space using a single hypernet. The learned hypernet can directly generate various well-trained policy networks for different user preferences. We compare our method with two state-of-the-art MORL algorithms on seven multi-objective continuous robot control problems. Experimental results show that our method achieves the best overall performance with the least training parameters. An interesting observation is that the Pareto set is well approximated by a curved line or surface in a high-dimensional parameter space. This observation will provide insight for researchers to design new MORL algorithms.

Data-Driven Preference Sampling for Pareto Front Learning

Pareto front learning is a technique that introduces preference vectors in a neural network to approximate the Pareto front. Previous Pareto front learning methods have demonstrated high performance in approximating simple Pareto fronts. These methods often sample preference vectors from a fixed Dirichlet distribution. However, no fixed sampling distribution can be adapted to diverse Pareto fronts. Efficiently sampling preference vectors and accurately estimating the Pareto front is a challenge. To address this challenge, we propose a data-driven preference vector sampling framework for Pareto front learning. We utilize the posterior information of the objective functions to adjust the parameters of the sampling distribution flexibly. In this manner, the proposed method can sample preference vectors from the location of the Pareto front with a high probability. Moreover, we design the distribution of the preference vector as a mixture of Dirichlet distributions to improve the performance of the model in disconnected Pareto fronts. Extensive experiments validate the superiority of the proposed method compared with state-of-the-art algorithms.

Evolutionary Preference Sampling for Pareto Set Learning

Recently, Pareto Set Learning (PSL) has been proposed for learning the entire Pareto set using a neural network. PSL employs preference vectors to scalarize multiple objectives, facilitating the learning of mappings from preference vectors to specific Pareto optimal solutions. Previous PSL methods have shown their effectiveness in solving artificial multi-objective optimization problems (MOPs) with uniform preference vector sampling. The quality of the learned Pareto set is influenced by the sampling strategy of the preference vector, and the sampling of the preference vector needs to be decided based on the Pareto front shape. However, a fixed preference sampling strategy cannot simultaneously adapt the Pareto front of multiple MOPs. To address this limitation, this paper proposes an Evolutionary Preference Sampling (EPS) strategy to efficiently sample preference vectors. Inspired by evolutionary algorithms, we consider preference sampling as an evolutionary process to generate preference vectors for neural network training. We integrate the EPS strategy into five advanced PSL methods. Extensive experiments demonstrate that our proposed method has a faster convergence speed than baseline algorithms on 7 testing problems. Our implementation is available at https://github.com/rG223/EPS.

Improving Critical Node Detection Using Neural Network-based Initialization in a Genetic Algorithm

The Critical Node Problem (CNP) is concerned with identifying the critical nodes in a complex network. These nodes play a significant role in maintaining the connectivity of the network, and removing them can negatively impact network performance. CNP has been studied extensively due to its numerous real-world applications. Among the different versions of CNP, CNP-1a has gained the most popularity. The primary objective of CNP-1a is to minimize the pair-wise connectivity in the remaining network after deleting a limited number of nodes from a network. Due to the NP-hard nature of CNP-1a, many heuristic/metaheuristic algorithms have been proposed to solve this problem. However, most existing algorithms start with a random initialization, leading to a high cost of obtaining an optimal solution. To improve the efficiency of solving CNP-1a, a knowledge-guided genetic algorithm named K2GA has been proposed. Unlike the standard genetic algorithm framework, K2GA has two main components: a pretrained neural network to obtain prior knowledge on possible critical nodes, and a hybrid genetic algorithm with local search for finding an optimal set of critical nodes based on the knowledge given by the trained neural network. The local search process utilizes a cut node-based greedy strategy. The effectiveness of the proposed knowledgeguided genetic algorithm is verified by experiments on 26 realworld instances of complex networks. Experimental results show that K2GA outperforms the state-of-the-art algorithms regarding the best, median, and average objective values, and improves the best upper bounds on the best objective values for eight realworld instances.

Privacy-preserving Continual Federated Clustering via Adaptive Resonance Theory

With the increasing importance of data privacy protection, various privacy-preserving machine learning methods have been proposed. In the clustering domain, various algorithms with a federated learning framework (i.e., federated clustering) have been actively studied and showed high clustering performance while preserving data privacy. However, most of the base clusterers (i.e., clustering algorithms) used in existing federated clustering algorithms need to specify the number of clusters in advance. These algorithms, therefore, are unable to deal with data whose distributions are unknown or continually changing. To tackle this problem, this paper proposes a privacy-preserving continual federated clustering algorithm. In the proposed algorithm, an adaptive resonance theory-based clustering algorithm capable of continual learning is used as a base clusterer. Therefore, the proposed algorithm inherits the ability of continual learning. Experimental results with synthetic and real-world datasets show that the proposed algorithm has superior clustering performance to state-of-the-art federated clustering algorithms while realizing data privacy protection and continual learning ability. The source code is available at \url{https://github.com/Masuyama-lab/FCAC}.

A Parameter-free Adaptive Resonance Theory-based Topological Clustering Algorithm Capable of Continual Learning

In general, a similarity threshold (i.e., a vigilance parameter) for a node learning process in Adaptive Resonance Theory (ART)-based algorithms has a significant impact on clustering performance. In addition, an edge deletion threshold in a topological clustering algorithm plays an important role in adaptively generating well-separated clusters during a self-organizing process. In this paper, we propose a new parameter-free ART-based topological clustering algorithm capable of continual learning by introducing parameter estimation methods. Experimental results with synthetic and real-world datasets show that the proposed algorithm has superior clustering performance to the state-of-the-art clustering algorithms without any parameter pre-specifications.

Effects of Archive Size on Computation Time and Solution Quality for Multi-Objective Optimization

An unbounded external archive has been used to store all nondominated solutions found by an evolutionary multi-objective optimization algorithm in some studies. It has been shown that a selected solution subset from the stored solutions is often better than the final population. However, the use of the unbounded archive is not always realistic. When the number of examined solutions is huge, we must pre-specify the archive size. In this study, we examine the effects of the archive size on three aspects: (i) the quality of the selected final solution set, (ii) the total computation time for the archive maintenance and the final solution set selection, and (iii) the required memory size. Unsurprisingly, the increase of the archive size improves the final solution set quality. Interestingly, the total computation time of a medium-size archive is much larger than that of a small-size archive and a huge-size archive (e.g., an unbounded archive). To decrease the computation time, we examine two ideas: periodical archive update and archiving only in later generations. Compared with updating the archive at every generation, the first idea can obtain almost the same final solution set quality using a much shorter computation time at the cost of a slight increase of the memory size. The second idea drastically decreases the computation time at the cost of a slight deterioration of the final solution set quality. Based on our experimental results, some suggestions are given about how to appropriately choose an archiving strategy and an archive size.

Hybridization of evolutionary algorithm and deep reinforcement learning for multi-objective orienteering optimization

Multi-objective orienteering problems (MO-OPs) are classical multi-objective routing problems and have received a lot of attention in the past decades. This study seeks to solve MO-OPs through a problem-decomposition framework, that is, a MO-OP is decomposed into a multi-objective knapsack problem (MOKP) and a travelling salesman problem (TSP). The MOKP and TSP are then solved by a multi-objective evolutionary algorithm (MOEA) and a deep reinforcement learning (DRL) method, respectively. While the MOEA module is for selecting cities, the DRL module is for planning a Hamiltonian path for these cities. An iterative use of these two modules drives the population towards the Pareto front of MO-OPs. The effectiveness of the proposed method is compared against NSGA-II and NSGA-III on various types of MO-OP instances. Experimental results show that our method exhibits the best performance on almost all the test instances, and has shown strong generalization ability.

Reference Vector Adaptation and Mating Selection Strategy via Adaptive Resonance Theory-based Clustering for Many-objective Optimization

Decomposition-based multiobjective evolutionary algorithms (MOEAs) with clustering-based reference vector adaptation show good optimization performance for many-objective optimization problems (MaOPs). Especially, algorithms that employ a clustering algorithm with a topological structure (i.e., a network composed of nodes and edges) show superior optimization performance to other MOEAs for MaOPs with irregular Pareto optimal fronts (PFs). These algorithms, however, do not effectively utilize information of the topological structure in the search process. Moreover, the clustering algorithms typically used in conventional studies have limited clustering performance, inhibiting the ability to extract useful information for the search process. This paper proposes an adaptive reference vector-guided evolutionary algorithm using an adaptive resonance theory-based clustering with a topological structure. The proposed algorithm utilizes the information of the topological structure not only for reference vector adaptation but also for mating selection. The proposed algorithm is compared with 8 state-of-the-art MOEAs on 78 test problems. Experimental results reveal the outstanding optimization performance of the proposed algorithm over the others on MaOPs with various properties.