Ke Shang

Ke Shang, Zhiyun Xiao, Yuxuan Liu et al.

Dynamic multi-objective optimization with a changing number of objectives has recently attracted increasing attention due to its relevance to real-world problems whose evaluation criteria may evolve over time. However, existing benchmark test suites for this problem setting suffer from a fundamental limitation: when the number of objectives changes, the objective functions themselves also change implicitly. This makes it difficult to isolate and evaluate an algorithm's capability to handle dynamics in the number of objectives alone. In this paper, we analyze this issue in detail and show that several theoretical properties claimed in prior studies rely on an assumption that is violated by commonly used test suites. To address this problem, we propose a scalable benchmark test suite in which the objective functions are fixed throughout the optimization process, while the number of active objectives changes over time. Our benchmark is constructed by defining a maximum-objective problem and dynamically selecting subsets of objectives. To avoid degeneracy issues in classical DTLZ and WFG problems, we adopt Minus-DTLZ and Minus-WFG formulations, in which all objectives are mutually conflicting. Extensive benchmark studies using representative algorithms from the literature demonstrate the usefulness and flexibility of the proposed test suite.

Nan Gao, Jia Li, Huaibo Huang et al.

Blind face restoration (BFR) is a highly challenging problem due to the uncertainty of data degradation patterns. Current BFR methods have realized certain restored productions but with inherent neural degradations that limit real-world generalization in complicated scenarios. In this paper, we propose a plug-and-play framework InfoBFR to tackle neural degradations, e.g., prior bias, topological distortion, textural distortion, and artifact residues, which achieves high-generalization face restoration in diverse wild and heterogeneous scenes. Specifically, based on the results from pre-trained BFR models, InfoBFR considers information compression using manifold information bottleneck (MIB) and information compensation with efficient diffusion LoRA to conduct information optimization. InfoBFR effectively synthesizes high-fidelity faces without attribute and identity distortions. Comprehensive experimental results demonstrate the superiority of InfoBFR over state-of-the-art GAN-based and diffusion-based BFR methods, with around 70ms consumption, 16M trainable parameters, and nearly 85% BFR-boosting. It is promising that InfoBFR will be the first plug-and-play restorer universally employed by diverse BFR models to conquer neural degradations.

Nan Gao, Jia Li, Huaibo Huang et al.

Blind face restoration (BFR) is a highly challenging problem due to the uncertainty of degradation patterns. Current methods have low generalization across photorealistic and heterogeneous domains. In this paper, we propose a Diffusion-Information-Diffusion (DID) framework to tackle diffusion manifold hallucination correction (DiffMAC), which achieves high-generalization face restoration in diverse degraded scenes and heterogeneous domains. Specifically, the first diffusion stage aligns the restored face with spatial feature embedding of the low-quality face based on AdaIN, which synthesizes degradation-removal results but with uncontrollable artifacts for some hard cases. Based on Stage I, Stage II considers information compression using manifold information bottleneck (MIB) and finetunes the first diffusion model to improve facial fidelity. DiffMAC effectively fights against blind degradation patterns and synthesizes high-quality faces with attribute and identity consistencies. Experimental results demonstrate the superiority of DiffMAC over state-of-the-art methods, with a high degree of generalization in real-world and heterogeneous settings. The source code and models will be public.

Tianye Shu, Ke Shang, Cheng Gong et al.

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.

Ke Shang, Tianye Shu, Hisao Ishibuchi

Hypervolume contribution is an important concept in evolutionary multi-objective optimization (EMO). It involves in hypervolume-based EMO algorithms and hypervolume subset selection algorithms. Its main drawback is that it is computationally expensive in high-dimensional spaces, which limits its applicability to many-objective optimization. Recently, an R2 indicator variant (i.e., $R_2^{\text{HVC}}$ indicator) is proposed to approximate the hypervolume contribution. The $R_2^{\text{HVC}}$ indicator uses line segments along a number of direction vectors for hypervolume contribution approximation. It has been shown that different direction vector sets lead to different approximation quality. In this paper, we propose \textit{Learning to Approximate (LtA)}, a direction vector set generation method for the $R_2^{\text{HVC}}$ indicator. The direction vector set is automatically learned from training data. The learned direction vector set can then be used in the $R_2^{\text{HVC}}$ indicator to improve its approximation quality. The usefulness of the proposed LtA method is examined by comparing it with other commonly-used direction vector set generation methods for the $R_2^{\text{HVC}}$ indicator. Experimental results suggest the superiority of LtA over the other methods for generating high quality direction vector sets.

Ke Shang, Tianye Shu, Hisao Ishibuchi et al.

In the evolutionary multi-objective optimization (EMO) field, the standard practice is to present the final population of an EMO algorithm as the output. However, it has been shown that the final population often includes solutions which are dominated by other solutions generated and discarded in previous generations. Recently, a new EMO framework has been proposed to solve this issue by storing all the non-dominated solutions generated during the evolution in an archive and selecting a subset of solutions from the archive as the output. The key component in this framework is the subset selection from the archive which usually stores a large number of candidate solutions. However, most studies on subset selection focus on small candidate solution sets for environmental selection. There is no benchmark test suite for large-scale subset selection. This paper aims to fill this research gap by proposing a benchmark test suite for subset selection from large candidate solution sets, and comparing some representative methods using the proposed test suite. The proposed test suite together with the benchmarking studies provides a baseline for researchers to understand, use, compare, and develop subset selection methods in the EMO field.

Weiyu Chen, Hisao Ishibuchi, Ke Shang

Subset selection is an important component in evolutionary multiobjective optimization (EMO) algorithms. Clustering, as a classic method to group similar data points together, has been used for subset selection in some fields. However, clustering-based methods have not been evaluated in the context of subset selection from solution sets obtained by EMO algorithms. In this paper, we first review some classic clustering algorithms. We also point out that another popular subset selection method, i.e., inverted generational distance (IGD)-based subset selection, can be viewed as clustering. Then, we perform a comprehensive experimental study to evaluate the performance of various clustering algorithms in different scenarios. Experimental results are analyzed in detail, and some suggestions about the use of clustering algorithms for subset selection are derived. Additionally, we demonstrate that decision maker's preference can be introduced to clustering-based subset selection.

Ke Shang, Hisao Ishibuchi, Weiyu Chen et al.

Hypervolume is widely used in the evolutionary multi-objective optimization (EMO) field to evaluate the quality of a solution set. For a solution set with $μ$ solutions on a Pareto front, a larger hypervolume means a better solution set. Investigating the distribution of the solution set with the largest hypervolume is an important topic in EMO, which is the so-called hypervolume optimal $μ$-distribution. Theoretical results have shown that the $μ$ solutions are uniformly distributed on a linear Pareto front in two dimensions. However, the $μ$ solutions are not always uniformly distributed on a single-line Pareto front in three dimensions. They are only uniform when the single-line Pareto front has one constant objective. In this paper, we further investigate the hypervolume optimal $μ$-distribution in three dimensions. We consider the line- and plane-based Pareto fronts. For the line-based Pareto fronts, we extend the single-line Pareto front to two-line and three-line Pareto fronts, where each line has one constant objective. For the plane-based Pareto fronts, the linear triangular and inverted triangular Pareto fronts are considered. First, we show that the $μ$ solutions are not always uniformly distributed on the line-based Pareto fronts. The uniformity depends on how the lines are combined. Then, we show that a uniform solution set on the plane-based Pareto front is not always optimal for hypervolume maximization. It is locally optimal with respect to a $(μ+1)$ selection scheme. Our results can help researchers in the community to better understand and utilize the hypervolume indicator.

Weiyu Chen, Hisao Ishibuchi, Ke Shang

Subset selection is an interesting and important topic in the field of evolutionary multi-objective optimization (EMO). Especially, in an EMO algorithm with an unbounded external archive, subset selection is an essential post-processing procedure to select a pre-specified number of solutions as the final result. In this paper, we discuss the efficiency of greedy subset selection for the hypervolume, IGD and IGD+ indicators. Greedy algorithms usually efficiently handle subset selection. However, when a large number of solutions are given (e.g., subset selection from tens of thousands of solutions in an unbounded external archive), they often become time-consuming. Our idea is to use the submodular property, which is known for the hypervolume indicator, to improve their efficiency. First, we prove that the IGD and IGD+ indicators are also submodular. Next, based on the submodular property, we propose an efficient greedy inclusion algorithm for each indicator. Then, we demonstrate through computational experiments that the proposed algorithms are much faster than the standard greedy subset selection algorithms.

Hisao Ishibuchi, Lie Meng Pang, Ke Shang

It is assumed in the evolutionary multi-objective optimization (EMO) community that a final solution is selected by a decision maker from a non-dominated solution set obtained by an EMO algorithm. The number of solutions to be presented to the decision maker can be totally different. In some cases, the decision maker may want to examine only a few representative solutions from which a final solution is selected. In other cases, a large number of non-dominated solutions may be needed to visualize the Pareto front. In this paper, we suggest the use of a general EMO framework with three solution sets to handle various situations with respect to the required number of solutions. The three solution sets are the main population of an EMO algorithm, an external archive to store promising solutions, and a final solution set which is presented to the decision maker. The final solution set is selected from the archive. Thus the population size and the archive size can be arbitrarily specified as long as the archive size is not smaller than the required number of solutions. The final population is not necessarily to be a good solution set since it is not presented to the decision maker. Through computational experiments, we show the advantages of this framework over the standard final population and final archive frameworks. We also discuss how to select a final solution set and how to explain the reason for the selection, which is the first attempt towards an explainable EMO framework.

Lie Meng Pang, Hisao Ishibuchi, Ke Shang

The development of efficient and effective evolutionary multi-objective optimization (EMO) algorithms has been an active research topic in the evolutionary computation community. Over the years, many EMO algorithms have been proposed. The existing EMO algorithms are mainly developed based on the final population framework. In the final population framework, the final population of an EMO algorithm is presented to the decision maker. Thus, it is required that the final population produced by an EMO algorithm is a good solution set. Recently, the use of solution selection framework was suggested for the design of EMO algorithms. This framework has an unbounded external archive to store all the examined solutions. A pre-specified number of solutions are selected from the archive as the final solutions presented to the decision maker. When the solution selection framework is used, EMO algorithms can be designed in a more flexible manner since the final population is not necessarily to be a good solution set. In this paper, we examine the design of MOEA/D under these two frameworks. We use an offline genetic algorithm-based hyper-heuristic method to find the optimal configuration of MOEA/D in each framework. The DTLZ and WFG test suites and their minus versions are used in our experiments. The experimental results suggest the possibility that a more flexible, robust and high-performance MOEA/D algorithm can be obtained when the solution selection framework is used.

Lie Meng Pang, Hisao Ishibuchi, Ke Shang

In the evolutionary multi-objective optimization (EMO) community, it is usually assumed that the final population is presented to the decision maker as the result of the execution of an EMO algorithm. Recently, an unbounded external archive was used to evaluate the performance of EMO algorithms in some studies where a pre-specified number of solutions are selected from all the examined non-dominated solutions. In this framework, which is referred to as the solution selection framework, the final population does not have to be a good solution set. Thus, the solution selection framework offers higher flexibility to the design of EMO algorithms than the final population framework. In this paper, we examine the design of MOEA/D under these two frameworks. First, we show that the performance of MOEA/D is improved by linearly changing the reference point specification during its execution through computational experiments with various combinations of initial and final specifications. Robust and high performance of the solution selection framework is observed. Then, we examine the use of a genetic algorithm-based offline hyper-heuristic method to find the best configuration of MOEA/D in each framework. Finally, we further discuss solution selection after the execution of an EMO algorithm in the solution selection framework.

Weiyu Chen, Hisao Ishibuhci, Ke Shang

Subset selection is a popular topic in recent years and a number of subset selection methods have been proposed. Among those methods, hypervolume subset selection is widely used. Greedy hypervolume subset selection algorithms can achieve good approximations to the optimal subset. However, when the candidate set is large (e.g., an unbounded external archive with a large number of solutions), the algorithm is very time-consuming. In this paper, we propose a new lazy greedy algorithm exploiting the submodular property of the hypervolume indicator. The core idea is to avoid unnecessary hypervolume contribution calculation when finding the solution with the largest contribution. Experimental results show that the proposed algorithm is hundreds of times faster than the original greedy inclusion algorithm and several times faster than the fastest known greedy inclusion algorithm on many test problems.

Hisao Ishibuchi, Lie Meng Pang, Ke Shang

In general, a multi-objective optimization problem does not have a single optimal solution but a set of Pareto optimal solutions, which forms the Pareto front in the objective space. Various evolutionary algorithms have been proposed to approximate the Pareto front using a pre-specified number of solutions. Hundreds of solutions are obtained by their single run. The selection of a single final solution from the obtained solutions is assumed to be done by a human decision maker. However, in many cases, the decision maker does not want to examine hundreds of solutions. Thus, it is needed to select a small subset of the obtained solutions. In this paper, we discuss subset selection from a viewpoint of the final decision making. First we briefly explain existing subset selection studies. Next we formulate an expected loss function for subset selection. We also show that the formulated function is the same as the IGD plus indicator. Then we report experimental results where the proposed approach is compared with other indicator-based subset selection methods.

Weiyu Chen, Hisao Ishibuchi, Ke Shang

Recently, the discretization of decision and objective spaces has been discussed in the literature. In some studies, it is shown that the decision space discretization improves the performance of evolutionary multi-objective optimization (EMO) algorithms on continuous multi-objective test problems. In other studies, it is shown that the objective space discretization improves the performance on combinatorial multi-objective problems. However, the effect of the simultaneous discretization of both spaces has not been examined in the literature. In this paper, we examine the effects of the decision space discretization, objective space discretization and simultaneous discretization on the performance of NSGA-II through computational experiments on the DTLZ and WFG problems. Using various settings about the number of decision variables and the number of objectives, our experiments are performed on four types of problems: standard problems, large-scale problems, many-objective problems, and large-scale many-objective problems. We show that the decision space discretization has a positive effect for large-scale problems and the objective space discretization has a positive effect for many-objective problems. We also show the discretization of both spaces is useful for large-scale many-objective problems.