In a variety of domains, from robotics to finance, Quality-Diversity algorithms have been used to generate collections of both diverse and high-performing solutions. Multi-Objective Quality-Diversity algorithms have emerged as a promising approach for applying these methods to complex, multi-objective problems. However, existing methods are limited by their search capabilities. For example, Multi-Objective Map-Elites depends on random genetic variations which struggle in high-dimensional search spaces. Despite efforts to enhance search efficiency with gradient-based mutation operators, existing approaches consider updating solutions to improve on each objective separately rather than achieving desired trade-offs. In this work, we address this limitation by introducing Multi-Objective Map-Elites with Preference-Conditioned Policy-Gradient and Crowding Mechanisms: a new Multi-Objective Quality-Diversity algorithm that uses preference-conditioned policy-gradient mutations to efficiently discover promising regions of the objective space and crowding mechanisms to promote a uniform distribution of solutions on the non-dominated front. We evaluate our approach on six robotics locomotion tasks and show that our method outperforms or matches all state-of-the-art Multi-Objective Quality-Diversity methods in all six, including two newly proposed tri-objective tasks. Importantly, our method also achieves a smoother set of trade-offs, as measured by newly-proposed sparsity-based metrics.

To address problems with multiple conflicting objective functions while incorporating preference information from a decision maker (a domain expert), interactive evolutionary multiobjective optimization methods have been proposed and widely adopted. To systematically assess the performance of these methods, it is desirable to have indicators that not only rely on the Pareto dominance relation, where solutions with better objective function values are preferred, but also take the decision maker's preferences into account. In this way, indicator values can meaningfully reflect the decision maker's preferences and the quality of the solutions obtained. In this paper, we propose the first performance indicator that incorporates preference information in the form of two reference points, specifically, an aspiration point, consisting of aspiration levels that represent the decision maker's full satisfaction, and a reservation point, consisting of reservation levels that represent dissatisfaction. The new indicator, termed D-PHI, translates information about reservation and aspiration points into a transformed hypervolume measure that assesses the performance of a solution set. We introduce D-PHI with an extended set of guidelines for preference-based performance evaluation, discuss its theoretical properties, and demonstrate its behavior through empirical examples. Moreover, we define a complementary indicator (CI) based on a so-called achievement scalarizing function. CI offers additional insight into whether aspiration and reservation levels have been achieved, and its usefulness is demonstrated through examples.

The mathematical runtime analysis of evolutionary algorithms traditionally regards the time an algorithm needs to find a solution of a certain quality when initialized with a random population. In practical applications it may be possible to guess solutions that are better than random ones. We start a mathematical runtime analysis for such situations. We observe that different algorithms profit to a very different degree from a better initialization. We also show that the optimal parameterization of an algorithm can depend strongly on the quality of the initial solutions. To overcome this difficulty, self-adjusting and randomized heavy-tailed parameter choices can be profitable. Finally, we observe a larger gap between the performance of the best evolutionary algorithm we found and the corresponding black-box complexity. This could suggest that evolutionary algorithms better exploiting good initial solutions are still to be found. These first findings stem from analyzing the performance of the \((1+1)\) evolutionary algorithm and the static, self-adjusting, and heavy-tailed \((1+(\lambda,\lambda))\) genetic algorithms on the OneMax benchmark. We are optimistic that the question of how to profit from good initial solutions is interesting beyond these first examples.

The NeuroEvolution of Augmenting Topologies (NEAT) algorithm has received considerable recognition in the field of neuroevolution. Its effectiveness is derived from initiating with simple networks and incrementally evolving both their topologies and weights. Although its capability across various challenges is evident, the algorithm's computational efficiency remains an impediment, limiting its scalability potential. To address these limitations, this paper introduces TensorNEAT, a GPU-accelerated library that applies tensorization to the NEAT algorithm. Tensorization reformulates NEAT's diverse network topologies and operations into uniformly shaped tensors, enabling efficient parallel execution across entire populations. TensorNEAT is built upon JAX, leveraging automatic function vectorization and hardware acceleration to significantly enhance computational efficiency. In addition to NEAT, the library supports variants such as CPPN and HyperNEAT, and integrates with benchmark environments like Gym, Brax, and gymnax. Experimental evaluations across various robotic control environments in Brax demonstrate that TensorNEAT delivers up to 500x speedups compared to existing implementations, such as NEAT-Python. The source code for TensorNEAT is publicly available at: https://github.com/EMI-Group/tensorneat .

Dynamic multimodal optimization problems (DMMOPs) demand algorithms capable of swiftly locating and tracking multiple optimal solutions over time. The primary challenge lies in controlling the population diversity to facilitate effective exploration, all within the limitation of computational resources between consecutive environmental changes. In this paper, we study the utilization of density information derived from both current and historical populations to enhance exploration. First, for each active sub-population, we construct a density landscape based on the distribution of concurrently active sub-populations, and establish dominance relationships between candidate solutions in the sub-population based on density and fitness values, directing this sub-population towards exploring low-density promising areas. Then, for each converged sub-population, we construct a density landscape based on the distribution of sub-populations that have historically become extinct, guiding the restart of this sub-population in low-density unexploited areas. Finally, we develop a comprehensive framework of density-assisted evolutionary algorithm (DAEA), which encompasses density-assisted search and restart, also combined with initialization. Moreover, we employ prediction and memory strategies to enhance the performance of DAEA in dynamic environments. Notably, the algorithm relies on an external monitor to detect environmental changes and trigger the dynamic response strategy. DAEA is tested on the CEC’2022 dynamic multimodal optimization benchmark suite, and is compared against several state-of-the-art dynamic multimodal optimization algorithms. The experimental results demonstrate the competitiveness of DAEA in handling DMMOPs. Additionally, experimental results from the berth allocation problem further confirm the applicability of DAEA to real-world dynamic multimodal optimization tasks.

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.

Lengthy evaluation times are common in many optimization problems such as direct policy search tasks, especially when they involve conducting evaluations in the physical world, for example, in robotics applications. Often when evaluating solution over a fixed time period, it becomes clear that the objective value will not increase with additional computation time (e.g., when a two-wheeled robot continuously spins on the spot). In such cases, it makes sense to stop the evaluation early to save computation time. However, most approaches to stop the evaluation are problem specific and need to be specifically designed for the task at hand. Therefore, we propose an early stopping method for direct policy search. The proposed method only looks at the objective value at each timestep and requires no problem-specific knowledge. We test the introduced stopping criterion in five direct policy search environments drawn from games, robotics, and classic control domains and show that it can save up to \(75\%\) of the computation time. We also compare it with problem-specific stopping criteria and show that it performs comparably, while being more generally applicable.

As interest in machine learning and its applications becomes more widespread, how to choose the best models and hyper-parameter settings becomes more important. This problem is known to be challenging for human experts, and consequently, a growing number of methods have been proposed for solving it, giving rise to the area of automated machine learning (AutoML). Many of the most popular AutoML methods are based on Bayesian optimization, which makes only weak assumptions about how modifying hyper-parameters effects the loss of a model. This is a safe assumption that yields robust methods, as the AutoML loss landscapes that relate hyper-parameter settings to loss are poorly understood. We build on recent work on the study of one-dimensional slices of algorithm configuration landscapes by introducing new methods that test n -dimensional landscapes for statistical deviations from uni-modality and convexity, and we use them to show that a diverse set of AutoML loss landscapes are highly structured. We introduce a method for assessing the significance of hyper-parameter partial derivatives, which reveals that most (but not all) AutoML loss landscapes only have a small number of hyper-parameters that interact strongly. To further assess hyper-parameter interactions, we introduce a simplistic optimization procedure that assumes each hyper-parameter can be optimized independently, a single time in sequence, and we show that it obtains configurations that are statistically tied with optimal in all of the n -dimensional AutoML loss landscapes that we studied. Our results suggest many possible new directions for substantially improving the state of the art in AutoML.

Benchmarking and performance analysis play an important role in understanding the behaviour of iterative optimization heuristics (IOHs) such as local search algorithms, genetic and evolutionary algorithms, Bayesian optimization algorithms, etc. This task, however, involves manual setup, execution, and analysis of the experiment on an individual basis, which is laborious and can be mitigated by a generic and well-designed platform. For this purpose, we propose IOHanalyzer, a new user-friendly tool for the analysis, comparison, and visualization of performance data of IOHs. Implemented in R and C++ , IOHanalyzer is fully open source. It is available on CRAN and GitHub. IOHanalyzer provides detailed statistics about fixed-target running times and about fixed-budget performance of the benchmarked algorithms with a real-valued codomain, single-objective optimization tasks. Performance aggregation over several benchmark problems is possible, for example in the form of empirical cumulative distribution functions. Key advantages of IOHanalyzer over other performance analysis packages are its highly interactive design, which allows users to specify the performance measures, ranges, and granularity that are most useful for their experiments, and the possibility to analyze not only performance traces, but also the evolution of dynamic state parameters. IOHanalyzer can directly process performance data from the main benchmarking platforms, including the COCO platform, Nevergrad, the SOS platform, and IOHexperimenter. An R programming interface is provided for users preferring to have a finer control over the implemented functionalities.

Real-world problems typically require the simultaneous optimization of multiple, often conflicting objectives. Many of thesemulti-objective optimization problemsare characterized by wide ranges of uncertainties in their decision variables or objective functions. To cope with such uncertainties,stochasticandrobust optimizationtechniques are widely studied aiming to distinguish candidate solutions with uncertain objectives specified by confidence intervals, probability distributions, sampled data, or uncertainty sets. In this scope, this article first introduces a novel empirical approach for the comparison of candidate solutions with uncertain objectives that can follow arbitrary distributions. The comparison is performed through accurate and efficient calculations of the probability that one solution dominates the other in terms of each uncertain objective. Second, such an operator can be flexibly used and combined with many existing multi-objective optimization frameworks and techniques by just substituting their standard comparison operator, thus easily enabling the Pareto front optimization of problems with multiple uncertain objectives. Third, a new benchmark for evaluating uncertainty-aware optimization techniques is introduced by incorporating different types of uncertainties into a well-known benchmark for multi-objective optimization problems. Fourth, the new comparison operator and benchmark suite are integrated into an existing multi-objective optimization framework that features a selection of multi-objective optimization problems and algorithms. Fifth, the efficiency in terms of performance and execution time of the proposed comparison operator is evaluated on the introduced uncertainty benchmark. Finally, statistical tests are applied giving evidence of the superiority of the new comparison operator in terms of\epsilon-dominance and attainment surfaces in comparison to previously proposed approaches.