Rule representations significantly influence the search capabilities and decision boundaries within the search space of Learning Classifier Systems (LCSs). However, it is very difficult to choose an appropriate rule representation for each problem. Additionally, some problems benefit from using different representations for different subspaces within the input space. Thus, an adaptive mechanism is needed to choose an appropriate rule representation for each rule in LCSs. This article introduces a flexible rule representation using a four-parameter beta distribution and integrates it into a fuzzy-style LCS. The four-parameter beta distribution can form various function shapes, and this flexibility enables our LCS to automatically select appropriate representations for different subspaces. Our rule representation can represent crisp/fuzzy decision boundaries in various boundary shapes, such as rectangles and bells, by controlling four parameters, compared to the standard representations such as trapezoidal ones. Leveraging this flexibility, our LCS is designed to adapt the appropriate rule representation for each subspace. Moreover, our LCS has a generalization bias to produce as many crisp rules as possible. Experimental results on real-world classification tasks show that our LCS significantly outperformed LCSs with popular rule representations in test classification accuracy on up to 17 of the 25 datasets tested.

The decision-making process significantly influences the predictions of machine learning models. This is especially important in rule-based systems such as Learning Fuzzy-Classifier Systems (LFCSs) where the selection and application of rules directly determine prediction accuracy and reliability. LFCSs combine evolutionary algorithms with supervised learning to optimize fuzzy classification rules, offering enhanced interpretability and robustness. Despite these advantages, research on improving decision-making mechanisms (i.e., class inference schemes) in LFCSs remains limited. Most LFCSs use voting-based or single-winner-based inference schemes. These schemes rely on classification performance on training data and may not perform well on unseen data, risking overfitting. To address these limitations, this article introduces a novel class inference scheme for LFCSs based on the Dempster–Shafer Theory of Evidence (DS theory). The proposed scheme handles uncertainty well. By using the DS theory, the scheme calculates belief masses (i.e., measures of belief) for each specific class and the “I don’t know” state from each fuzzy rule and infers a class from these belief masses. Unlike the conventional schemes, the proposed scheme also considers the “I don’t know” state that reflects uncertainty, thereby improving the transparency and reliability of LFCSs. Applied to a variant of LFCS (i.e., Fuzzy-UCS), the proposed scheme demonstrates statistically significant improvements in terms of test macro F1 scores across 30 real-world datasets compared to conventional voting-based and single-winner-based fuzzy inference schemes. It forms smoother decision boundaries, provides reliable confidence measures, and enhances the robustness and generalizability of LFCSs in real-world applications.

The performance of surrogate-assisted multiobjective evolutionary algorithms (SAMOEAs) often degrades in high-dimensional problems. Recent studies have shown that decomposition-based approaches are particularly effective in handling high-dimensional search spaces, owing to their problem-simplifying capability. However, existing decomposition-based SAMOEAs are designed to sequentially solve each decomposed subproblem, still unnecessarily consuming function evaluations (FEs) and thus degrading the search efficiency. To address this issue, this paper proposes a novel decomposition-based SAMOEA that employs a multifactorial evolutionary algorithm (MFEA). The proposed algorithm aggregates multiple subproblems randomly and it collectively solves them using a surrogate-assisted MFEA framework. This approach enables the efficient discovery of promising solutions across multiple subproblems in a single FE, enhancing the search efficiency under a limited budget of FEs. Experimental results show that our proposed algorithm outperforms state-of-the-art SAMOEAs on problems with up to 300 dimensions. This suggests that our surrogate-assisted MFEA framework can bring out the further potential of decomposition-based SAMOEAs.

Function approximation is a critical task in various fields. However, existing neural network approaches struggle with locally complex or discontinuous functions due to their reliance on a single global model covering the entire problem space. We propose X-KAN, a novel method that optimizes multiple local Kolmogorov-Arnold Networks (KANs) through an evolutionary rule-based machine learning framework called XCSF. X-KAN combines KAN's high expressiveness with XCSF's adaptive partitioning capability by implementing local KAN models as rule consequents and defining local regions via rule antecedents. Our experimental results on artificial test functions and real-world datasets demonstrate that X-KAN significantly outperforms conventional methods, including XCSF, Multi-Layer Perceptron, and KAN, in terms of approximation accuracy. Notably, X-KAN effectively handles functions with locally complex or discontinuous structures that are challenging for conventional KAN, using a compact set of rules (average 7.2 rules). These results validate the effectiveness of using KAN as a local model in XCSF, which evaluates the rule fitness based on both accuracy and generality. Our X-KAN implementation and an extended version of this paper, including appendices, are available at https://doi.org/10.48550/arXiv.2505.14273.

In surrogate-assisted multiobjective evolutionary al-gorithms (SAMOEAs), approximation and classification models are frequently employed to screen candidate solutions, but there is a tradeoff between both models in terms of the model accuracy and the screening capacity. This tradeoff is highlighted especially when the problem dimension increases, making SAMOEAs difficult to solve high-dimensional problems. This paper proposes a dual surrogate-based SAMOEA for solving high-dimensional expensive multi-objective optimization problems. The proposed algorithm, called DSEAID, is designed to adaptively select either approximation models or classification models dependent on the model accuracy. Compared to existing algorithms which use both approximation and classification models simultaneously, DSEAID possesses a robust framework against the deterioration of the model accuracy. Experimental results on benchmark problems with up to 150 decision variables show that our dual surrogate-based framework is effective in addressing high-dimensional problems. Moreover, we show DSEAID is competitive with state-of-the-art SAMOEAs.

Many real-world applications involve expensive permutation-based combinatorial optimization problems (PCOPs), where the evaluation of solutions becomes a time-consuming process. However, the design of efficient surrogate-based optimizers remains challenging due to the difficulty of constructing surrogate models suitable for the permutation spaces. This paper presents a random forest-assisted local search algorithm for solving expensive PCOPs. Our main motivation for using a random forest model is, in the integer space, to build a surrogate model that learns various important substructures of permutations. In the proposed algorithm, permutation-based solutions are treated as integer vectors in training random forest regression models for the objective function. Then, a population-based local search is performed on the obtained models. Experimental results on benchmark problems with limited budgets of function evaluations show that the proposed algorithms are competitive with state-of-the-art algorithms adapted to expensive PCOPs. This observation suggests that surrogate models trained on the integer representation can be effective for estimating the quality of permutation-based solutions.

This paper proposes an oversampling-guided search framework to improve the search efficiency of multiobjective evolutionary algorithms (MOEAs) for high-dimensional prob-lems. The proposed algorithm uses an oversampling algorithm to generate artificial samples of obtained Pareto solutions, and then it uses those as parent solutions to guide the evolutionary search to good subspaces that have not been sufficiently explored. We in-corporate the proposed algorithm into two popular MOEAs, i.e., NSGA-II and IBEA. Experimental results show that the proposed algorithm significantly improves the performance of NSGA-II and IBEA on problems with up to 200 decision variables. We also discuss a possible direction for further improvement while demonstrating the drawbacks of using oversampling techniques.

Surrogate-assisted multiobjective evolutionary algorithms (MOEAs) have advanced the field of computationally expensive optimization, but their progress is often restricted to low-dimensional problems. This manuscript presents a multiple classifiers-assisted evolutionary algorithm based on decomposition, which is adapted for high-dimensional expensive problems in terms of the following two insights. Compared to approximation-based surrogates, the accuracy of classification-based surrogates is robust for few high-dimensional training samples. Furthermore, multiple local classifiers can hedge the risk of overfitting issues. Accordingly, the proposed algorithm builds multiple classifiers with support vector machines (SVMs) on a decomposition-based multiobjective algorithm, wherein each local classifier is trained for a corresponding scalarization function. Experimental results confirm that the proposed algorithm is competitive to the state-of-the-art algorithms and computationally efficient as well.

Existing works have reported that adaptive differential evolution algorithms, i.e., adaptive DEs, improve the MOEA/D-DE algorithm, but this result is limited to small-scale multi-objective optimization problems. This paper compares four popular adaptive DEs on the MOEA/D-DE framework to evaluate their scalability to the number of decision variables and objectives. Specifically, we employ jDE, JADE, EPSDE, and SaDE in this paper. Our experimental results provide the following novel observations. MOEA/D-DE with JADE derives the best average rank on small-scale problems. However, the performances of MOEA/D-DE with JADE, EPSDE, and SaDE gradually degrade with the increase of the problem scale. In contrast, jDE stably improves the performance of MOEA/D-DE on large-scale problems employed in this paper (i.e., 11 objectives and 100 decision variables). Thus, we find a critical tradeoff among adaptive DEs in terms of the scalability of the MOEA/D-DE framework; a statistical adaption like JADE is suitable for small-scale problems, but a randomization adaptation like jDE is effective with the increase of the problem scale. Our results also suggest that parameter-only adaptation can be suitable for MOEA/D-DE regardless of the problem scale.

The XCS classifier system adaptively controls a rule-generality of a rule-condition through a rule-discovery process. However, there is no proof that the rule-generality can eventually converge to its optimum value even under some ideal assumptions. This paper conducts a convergence analysis of the rule-generality on the rule-discovery process with the ternary alphabet coding. Our analysis provides the first proof that an average rule-generality of rules in a population can converge to its optimum value under some assumptions. This proof can be used to mathematically conclude that the XCS framework has a natural pressure to explore rules toward optimum rules if XCS satisfies our derived conditions. In addition, our theoretical result returns a rough setting-up guideline for the maximum population size, the mutation rate, and the GA threshold, improving the convergence speed of the rule-generality and the XCS performance.

Evolutionary computation has brought great progress to rule-based learning but this progress is often blind to the optimality of the system design. This article theoretically reveals an optimal learning scheme on the most popular evolutionary rule-based learning approach-the accuracy-based classifier system (or XCS). XCS seeks to form accurate, maximally general rules that together classify the state space of a given domain. Previously, setting up the system to perform well has been a “blackart” as no systematic approach to XCS parameter tuning existed. We derive a theoretical approach that mathematically guarantees that XCS identifies the accurate rules, which also returns a theoretically valid XCS parameter setting. Then, we demonstrate our theoretical setting derives the maximum correctness of rule-identification in the fewest iterations possible. We also experimentally show that our theoretical setting enables XCS to easily solve several challenging problems where it had previously struggled.

This paper proposes a competitive-adaptive algorithm tuning framework for meta-heuristic algorithms. Our proposed method, called CAT, is inspired by the Equilibrium Theory in economics, which explains competitors eventually converge to an equilibrium status, e.g. in terms of the price of products. In detail, our proposal runs multiple optimizers with different algorithmic configurations, e.g. mutation variants. Then, the configurations of inferior optimizers are adaptively tuned so that they can derive good solutions that superior ones have derived. This intends to boost the performance even with a limited number of fitness evaluations, by the following technical advantage. The CAT preliminarily validates a search capacity of tuned algorithmic configurations and then constructs an ensemble optimizer by utilizing multiple optimizers. As a case study, this paper applies the CAT to tune the differential evolution algorithms (DEs). Experimental results show that our proposal outperforms the standard DE and an alternative approach i.e. jDE, which adapts hyper-parameters of genetic operators.

This paper proposes a surrogate-assisted MOEA/D using SVM-based surrogates adjusted to subproblems (MOEA/DS3), which intends to achieve the following technical advantages. Firstly, in order to construct a proper surrogate while reducing learning cost to construct surrogates, a surrogate is an SVMclassifier that identifies a specific region of good solutions and thus its learning cost should be lower than a popular alternative approach, i.e., fitness approximation. Secondly, relying on the first advantage, multiple surrogates are constructed and each surrogate, like an expert, is adjusted to each subproblem defined in the MOEA/D framework in order to improve diversity and convergence of the Pareto set. Experimental results show that MOEA/D-S3 outperforms MOEA/D on a number of manyobjective benchmark problems.

This paper proposes a self-adaptation technique of parameter settings used in the XCS learning scheme. Since we adaptively set those settings to their optimum values derived by the recent XCS learning theory, our proposal does not require any trial and error process to find their proper values. Thus, our proposal can always satisfy the optimality of XCS learning scheme, i.e. to distinguish accurate rules from inaccurate rules with the minimum update number of rules. Experimental results on artificial classification problems including overlapping problems show that XCS with our self-adaptation technique significantly outperforms the standard XCS.

This paper proposes Complex-Valued-based Learning Classifier System (CVLCS) that can learn an appropriate policy for the POMDP environments by extending Complex-Valued Reinforcement Learning (CVRL). Concretely, CVLCS explores the optimal policy by not only evolving classifiers but also updating Q-values (i.e., strength) of evolved ones, while CVRL explores the optimal policy by only updating Q-values of the state-action pairs prepared beforehand. To investigate the effectiveness of CVLCS, this paper applies it to various types of the POMDP environments. The experimental results have revealed that (1) CVLCS can derive the good performance which is close to the optimal one and shows such a performance faster than the conventional methods (i.e., Q-Learning as one of CVRL and ZCSM as one of LCS) and (2) CVLCS can stably derive the good performance even in the difficult environments where the conventional methods fail to derive good performance.

Most versions of the XCS Classifier System have been designed to evolve only two rules for each rule discovery invocation, which restricts the search capacity. A difficulty behind generating multiple rules each time is the increase in the probability of deleting immature rules, which conflicts with the requirement that parent rules be sufficiently updated so that fitness represents worth. Thus the aim of this paper is to argue how XCS determines when rules can be deleted safely. The objectives are to certainly identify inaccurate rules and then to maximize how many rules XCS can generate. The proposed method enables adaptation of rule-generation that maximizes the number of generated rules, under the assumption that the reliably inaccurate rules can be replaced with new rules. Experiments show our modification strongly improves the XCS performance on large scale problems, since it can take advantage of multi-point search more efficiently. For example, on the 135-bit multiplexer problem, XCS with our modification requires 1.57 million less training inputs compared with the standard XCS while utilizing the same number of final rules.

This research aims to realize sperm selection support system for assisted reproductive technology. Sperm detection is one of the important component technology. High detection rate is required even the expense of false positive detections so as not to miss promising sperms as much as possible. False positive rate control is key technology for assure high detection rate and adaptive threshold adjustment methods in boosting is proposed. Our experiments evaluate and compare adaptive thresholding method and normal thresholding method as for detection rate and false positive rate of sperm detection.

XCS is the most popular type of Learning Classifier System, but setting optimum parameter values is more of an art than a science. Early theoretical work required the impractical assumption that classifier parameters had fully converged with infinite update times. The aim of this work is to derive a theoretical condition to mathematically guarantee that XCS identifies maximally accurate classifiers, such that subsequent deletion methods can be used optimally, in as few updates as possible. Consequently, our theory provides a universally usable setup guide for three important parameter settings; the learning rate, the accuracy update and the threshold for subsumption deletion. XCS with our best parameter settings solves the 70-bit multiplexer problem with only 21% of instances that the standard XCS setup needs. On a highly class-imbalanced multiplexer problem with inaccurate classifiers having more than 99.99% classification accuracy, our theory enables XCS to identify only 100% accurate classifiers as accurate and thus obtain the optimal performance.

Partial observation can be avoided by extracting both modality specific features and common features from multimodal data. This paper proposes a framework of parameter shared multimodal deep autoencoders which uses complemental multimodal data in order to learn both modality specific and common features. The proposed model shares parameters of networks for each modality, while conventional multimodal deep autoencoder models share top layer neurons of their encoder among the modalities. The parameters of the proposed networks are shared by minimizing the Frobenius norm between those parameters. In experiments, we test the proposal on a classification task with complemental multimodal data. Experiment results show that our framework enables to learn specific and common features of the multimodal data.

A default hierarchy is set of rules containing one or more exceptions to one or more default rules e.g. all dogs are friendly, except my neighbour's. Default hierarchies were the subject of considerable interest in early Learning Classifier Systems research, but they were abandoned due to the considerable difficulty of solving the credit assignment problems they involve. The most popular Learning Classifier System, XCS, and its derivatives do not support default hierarchies because in XCS each rule must be accurate, whereas in a default hierarchy an overgeneral rule may be overridden by a correct rule. In this work we enable XCS to evolve minimal default hierarchies by allowing two conditions in one rule, but evaluating only the accuracy and fitness of the whole as a whole. This simple step avoids the credit assignment issues faced by earlier systems. We call this XCS-DH. Preliminary evaluation of XCS-DH on a number of Boolean functions indicates a strong tendency to exploit the increased expressiveness of its rules. On some functions we observe slower learning and a larger population size, which we attribute to the increased rule expressiveness, which increases the search space. However, we also observe that in a problem that is particularly suitable for XCS-DH representation, and that is sufficient difficult for XCS, XCS-DH's learning rate is faster than XCS's. We take this as confirmation of the potential of learning default hierarchies with XCS-DH.

This paper proposes a novel Learning Classifier System (LCS) which integrates Deep AutoEncoder named DAE to solve high-dimensional problems. In the proposed LCS, DAE starts to compress (encode) an environmental input as a high-dimensional information to an input of LCS as a low-dimensional information and decompresses (decodes) an output of LCS as a low-dimensional information to a system output as a high-dimensional information. Since the compressed inputs are encoded by real value, this paper employs XCSR (i.e., an LCS with real value coding) and combines XCSR with DAE. In order to investigate the effectiveness of the proposed LCS, XCSR with DAE, this paper conducts the preliminary experiment on the benchmark classification problem, i.e., 6-Multiplexer problem. The intensive experiments on the compression from 6 to 5 bits have revealed the following implications: (1) XCSR with DAE performs as well as XCSR even learning from the compressed input data; and (2) XCSR with DAE successfully decodes the compressed rules to extract the rules which are the same as those learned with not compressed input data.

This paper proposes a simple modification of the Cuckoo Search called CS for a dynamic environment. In this paper, we consider a dynamic optimization problem where the global optimum can be cyclically changed depending on time. Our modified CS algorithm holds good candidates in order to effectively explore the search space near those candidates with an intensive local search. Our first experiment tests the prosed method on a set of static optimization problems, which aims at evaluating the potential performance of the proposed method. Then, we apply it to a dynamic optimization problem. Experimental results on the static problems show that the proposed method derives a better performance than the conventional method, which suggest the proposed method potentially has a good capability of finding a good solution. On the dynamic problem, the proposed method also performs well while the conventional method fails to find a better solution.

A learning strategy in Learning Classifier Systems (LCSs) defines how classifiers cover a state-action space in a problem. Previous analyses in classification problems have empirically claimed an adequate learning strategy can be decided depending on the types of noise in the problem. This issue is still arguable from two aspects. First, there lacks comparison of learning strategies in reinforcement learning problems with different types of noise. Second, when we can claim so, a further issue is how should classifiers cover the state-action space in order to improve the stability of LCS performance on as many types of noise as possible? This paper first attempts to empirically conclude these issues on a version of LCSs (i.e., the XCS classifier system). That is, we present a new concept of learning strategy for LCSs, and complement that claim by comparing it with the existing learning strategies on a reinforcement learning problem. Our learning strategy covers all state-action pairs but assigns more classifiers to the highest-return action at each state than other actions. Our results support that claim that existing learning strategies have dependencies on the types of noise in reinforcement learning problems. However, our learning strategy improves the stability of XCS performance compared with the existing strategies on all types of noise employed in this paper.

This paper explores a novel Learning Classifier System (LCS) that can detect shoplifting behavior from the class-imbalanced sequence data of customer-behaviors. The shoplifting behavior detection is related to the sequence labeling as a time-series classification. More importantly, the target problem has a difficulty of the class-imbalanced problem in addition to the time-series classification because the number of data of shoplifters might be much fewer than that of customers who do not shoplift. To tackle this difficult issue, this paper proposes a feature extraction method for the sequence labeling in the class-imbalanced problem and applies it into XCS-SL (LCS for sequence labeling). The proposed LCS (called as XCS-SLFEM (XCS-SL for Feature Extraction of Minority class data)) extracts the features of the minority class from those of the majority one by representing them as the classifiers. The intensive simulation using the customer-behavior dataset that includes the shoplifting behaviors has revealed the following implications: (1) XCS-SLFEM shows the superior performance as compared with XCS-SL in the customer-behavior dataset including the class-imbalanced sequence data; (2) the classification accuracy of XCS-SLFEM increases as the size of memory for the sequence histories increases; and (3) XCS-SLFEM has the potential of predicting shoplifting before the shoplifters do shoplift.

This paper proposes XCS using Attribute Tracking and Feedback (XCS-ATF) that simultaneously extracts both of the generalized and specialized knowledge, and evaluates its effectiveness by investigating how the extracted knowledge contribute to deriving deep/light sleep of aged persons. The data mining of the daily activities of aged person by XCS-ATF has revealed the following implications: (1) XCS-ATF succeeds to extract the knowledge from the dataset including many contradict data; (2) XCS-ATF can extract not only the generalized knowledge as the daily activities that are usually performed with deriving deep/light sleep but also the specialized knowledge as the daily activities (e.g., birthday party) that are not often performed with deriving deep/light sleep, even though the specialized knowledge which does not often occur tends to be deleted as a noise by the general data mining methods; and (3) XCS-ATF can extract the daily activities that provides nine years younger sleep in the healthy aged persons and seven years younger sleep even in dementia persons who are hard to have a deep sleep in comparison with non-dementia persons.

This paper introduces XCS-SL, an extension of XCS for sequence labeling, a form of time-series classification where every input has a class label. Specifically, we consider sequence labeling tasks where on each time step we receive an input/class pair. In sequence labeling the correct class of an input may depend on data received on previous time stamps, so a learner may need to refer to data at previous time stamps. That is, some classification rules (called classifiers' here) must include conditions on previous inputs (a kind of memory). We assume the agent does not know how many conditions on previous inputs are needed to classify the current input, and the number of conditions/memories needed may be different for each input. Hence, using a fixed number of conditions is not a good solution. A novel idea we introduce is classifiers that have a variable-length condition to refer back to data at previous times. The condition can grow and shrink to find a suitable memory size. On a benchmark problem XCS-SL can learn optimal classifiers, and on a real-world sequence labeling task, it derived high classification accuracy and discovered interesting knowledge that shows dependencies between inputs at different times.

We study two existing Learning Classifier Systems (LCSs): XCS, which has a complete map (which covers all actions in each state), and XCSAMm, which has a best action map (which covers only the highest-return action in each state). This allows XCSAM to learn with a smaller population size limit (but larger population size) and to learn faster than XCS on well-behaved tasks. However, many tasks have dif- ficulties like noise and class imbalances. XCS and XCSAM have not been compared on such problems before. This pa- per aims to discover which kind of map is more robust to these difficulties. We apply them to a classification problem (the multiplexer problem) with class imbalance, Gaussian noise or alternating noise (where we return the reward for a different action). We also compare them on real-world data from the UCI repository without adding noise. We analyze how XCSAM focuses on the best action map and introduce a novel deletion mechanism that helps to evolve classifiers towards a best action map. Results show the best action map is more robust (has higher accuracy and sometimes learns faster) in all cases except small amounts of alternat- ing noise.

This paper proposes a novel rule discovery mechanism for the XCS classifier system, which is an extension of the compact genetic algorithm (cGA) to XCS. Our rule discovery mechanism, like cGA, extracts appropriate attributes of classifier conditions through a probability vector and evolves classifiers using the extracted attributes. Unlike cGA, it newly builds the probability vector at every generations (i.e., it keeps no any probability vectors) not so that it requires XCS to have a lot of probability vectors that represent all available attributes, and mutates classifier conditions based on the extracted attributes as attribute feedback. Experimental results show that XCS with our rule discovery mechanism (or XCScGA) can reach optimal performance with fewer rule evaluations and requires smaller population sizes than XCS. Our conclusion is that the proposed rule discovery mechanism promotes a recombination of building blocks, and that our mutation operator works to repair the classifier conditions towards a compact solutions, hence, XCScGA can generate good offspring which represent maximally general, maximally accurate, and compact solutions.

XCS with Adaptive Action Mapping (XCSAM) evolves so- lutions focused on classifiers that advocate the best action in every state. Accordingly, XCSAM usually evolves more compact solutions than XCS which, in contrast, works to- ward solutions representing complete state-action mappings. Experimental results have however shown that, in some prob- lems, XCSAM may produce bigger populations than XCS. In this paper, we extend XCSAM with a novel selection strat- egy to reduce, even further, the size of the solutions XCSAM produces. The proposed strategy selects the parent classi- fiers based both on their fitness values (like XCS) and on the effect they have on the adaptive map. We present experi- mental results showing that XCSAM with the new selection strategy can evolve more compact solutions than XCS which, at the same time, are also maximally general and maximally accurate.