GP-GOMEA is among the state-of-the-art for symbolic regression, especially when it comes to finding small and potentially interpretable solutions. A key mechanism employed in any GOMEA variant is the exploitation of linkage, the dependencies between variables, to ensure efficient evolution. In GP-GOMEA, mutual information between node positions in GP trees has so far been used to learn linkage. For this, a fixed expression template is used. This, however, leads to introns for expressions smaller than the full template. As introns have no impact on fitness, their occurrences are not directly linked to selection. Consequently, introns can adversely affect the extent to which mutual information captures dependencies between tree nodes. To overcome this, we propose two new measures for linkage learning, one that explicitly considers introns in mutual information estimates, and one that revisits linkage learning in GP-GOMEA from a grey-box perspective, yielding a measure that needs not to be learned from the population but is derived directly from the template. Across five standard symbolic regression problems, GP-GOMEA achieves substantial improvements using both measures. We also find that the newly learned linkage structure closely reflects the template linkage structure, and that explicitly using the template structure yields the best performance overall.

The Gene-pool Optimal Mixing EA (GOMEA) family of EAs offers a specific means to exploit problem-specific knowledge through linkage learning, i.e., inter-variable dependency detection, expressed using subsets of variables, that should undergo joint variation. Such knowledge can be exploited if faster fitness evaluations are possible when only a few variables are changed in a solution, enabling large speed-ups. The recent-most version of Real-Valued GOMEA (RV-GOMEA) can learn a conditional linkage model during optimization using fitness-based linkage learning, enabling fine-grained dependency exploitation in learning and sampling a Gaussian distribution. However, while the most efficient Gaussian-based EAs, like NES and CMA-ES, employ incremental learning of the Gaussian distribution rather than performing full re-estimation every generation, the recent-most RV-GOMEA version does not employ such incremental learning. In this paper, we therefore study whether incremental distribution estimation can lead to efficiency enhancements of RV-GOMEA. We consider various benchmark problems with varying degrees of overlapping dependencies. We find that, compared to RV-GOMEA and VKD-CMA-ES, the required number of evaluations to reach high-quality solutions can be reduced by a factor of up to 1.5 if population sizes are tuned problem-specifically, while a reduction by a factor of 2–3 can be achieved with generic population-sizing guidelines.

For many real-world optimization problems it is possible to perform partial evaluations, meaning that the impact of changing a few variables on a solution's fitness can be computed very efficiently. It has been shown that such partial evaluations can be excellently leveraged by the Real-Valued Gene-pool Optimal Mixing Evolutionary Algorithm (RV-GOMEA) that uses a linkage model to capture dependencies between problem variables. Recently, conditional linkage models were introduced for RV-GOMEA, expanding its state-of-the-art performance even to problems with overlapping dependencies. However, that work assumed that the dependency structure is known a priori. Fitness-based linkage learning techniques have previously been used to detect dependencies during optimization, but only for non-conditional linkage models. In this work, we combine fitness-based linkage learning and conditional linkage modelling in RV-GOMEA. In addition, we propose a new way to model overlapping dependencies in conditional linkage models to maximize the joint sampling of fully interdependent groups of variables. We compare the resulting novel variant of RV-GOMEA to other variants of RV-GOMEA and VkD-CMA on 12 problems with varying degree of overlapping dependencies. We find that the new RV-GOMEA not only performs best on most problems, also the overhead of learning the conditional linkage models during optimization is often negligible.

Many real-world problems have expensive-to-compute fitness functions and are multi-objective in nature. Surrogate-assisted evolutionary algorithms are often used to tackle such problems. Despite this, literature about analysing the fitness landscapes induced by surrogate models is limited, and even non-existent for multi-objective problems. This study addresses this critical gap by comparing landscapes of the true fitness function with those of surrogate models for multi-objective functions. Moreover, it does so temporally by examining landscape features at different points in time during optimisation, in the vicinity of the population at that point in time. We consider the BBOB bi-objective benchmark functions in our experiments. The results of the fitness landscape analysis reveals significant differences between true and surrogate features at different time points during optimisation. Despite these differences, the true and surrogate landscape features still show high correlations between each other. Furthermore, this study identifies which landscape features are related to search and demonstrates that both surrogate and true landscape features are capable of predicting algorithm performance. These findings indicate that temporal analysis of the landscape features may help to facilitate the design of surrogate switching approaches to improve performance in multi-objective optimisation.

Deploying machine learning models into sensitive domains in our society requires these models to be explainable. Genetic Programming (GP) can offer a way to evolve inherently interpretable expressions. GP-GOMEA is a form of GP that has been found particularly effective at evolving expressions that are accurate yet of limited size and, thus, promote interpretability. Despite this strength, a limitation of GP-GOMEA is template-based. This negatively affects its scalability regarding the arity of operators that can be used, since with increasing operator arity, an increasingly large part of the template tends to go unused. In this paper, we therefore propose two enhancements to GP-GOMEA: (i) semantic subtree inheritance, which performs additional variation steps that consider the semantic context of a subtree, and (ii) greedy child selection, which explicitly considers parts of the template that in standard GP-GOMEA remain unused. We compare different versions of GP-GOMEA regarding search enhancements on a set of continuous and discontinuous regression problems, with varying tree depths and operator sets. Experimental results show that both proposed search enhancements have a generally positive impact on the performance of GP-GOMEA, especially when the set of operators to choose from is large and contains higher-arity operators.

Paragangliomas are rare, primarily slow-growing tumors for which the underlying growth pattern is unknown. Therefore, determining the best care for a patient is hard. Currently, if no significant tumor growth is observed, treatment is often delayed, as treatment itself is not without risk. However, by doing so, the risk of (irreversible) adverse effects due to tumor growth may increase. Being able to predict the growth accurately could assist in determining whether a patient will need treatment during their lifetime and, if so, the timing of this treatment. The aim of this work is to learn the general underlying growth pattern of paragangliomas from multiple tumor growth data sets, in which each data set contains a tumor's volume over time. To do so, we propose a novel approach based on genetic programming to learn a function class, i.e., a parameterized function that can be fit anew for each tumor. We do so in a unique, multi-modal, multi-objective fashion to find multiple potentially interesting function classes in a single run. We evaluate our approach on a synthetic and a real-world data set. By analyzing the resulting function classes, we can effectively explain the general patterns in the data.

In this work, we show that simultaneously training and mixing neural networks is a promising way to conduct Neural Architecture Search (NAS). For hyperparameter optimization, reusing the partially trained weights allows for efficient search, as was previously demonstrated by the Population Based Training (PBT) algorithm. We propose PBT-NAS, an adaptation of PBT to NAS where architectures are improved during training by replacing poorly-performing networks in a population with the result of mixing well-performing ones and inheriting the weights using the shrink-perturb technique. After PBT-NAS terminates, the created networks can be directly used without retraining. PBT-NAS is highly parallelizable and effective: on challenging tasks (image generation and reinforcement learning) PBT-NAS achieves superior performance compared to baselines (random search and mutation-based PBT).

Finding a realistic deformation that transforms one image into another, in case large deformations are required, is considered a key challenge in medical image analysis. Having a proper image registration approach to achieve this could unleash a number of applications requiring information to be transferred between images. Clinical adoption is currently hampered by many existing methods requiring extensive configuration effort before each use, or not being able to (realistically) capture large deformations. A recent multi-objective approach that uses the Multi-Objective Real-Valued Gene-pool Optimal Mixing Evolutionary Algorithm (MO-RV-GOMEA) and a dual-dynamic mesh transformation model has shown promise, exposing the trade-offs inherent to image registration problems and modeling large deformations in 2D. This work builds on this promise and introduces MOREA: the first evolutionary algorithm-based multi-objective approach to deformable registration of 3D images capable of tackling large deformations. MOREA includes a 3D biomechanical mesh model for physical plausibility and is fully GPU-accelerated. We compare MOREA to two state-of-the-art approaches on abdominal CT scans of 4 cervical cancer patients, with the latter two approaches configured for the best results per patient. Without requiring per-patient configuration, MOREA significantly outperforms these approaches on 3 of the 4 patients that represent the most difficult cases.

In a parallel EA one can strictly adhere to the generational clock, and wait for all evaluations in a generation to be done. However, this idle time limits the throughput of the algorithm and wastes computational resources. Alternatively, an EA can be made asynchronous parallel. However, EAs using classic recombination and selection operators (GAs) are known to suffer from an evaluation time bias, which also influences the performance of the approach. Model-Based Evolutionary Algorithms (MBEAs) are more scalable than classic GAs by virtue of capturing the structure of a problem in a model. If this model is learned through linkage learning based on the population, the learned model may also capture biases. Thus, if an asynchronous parallel MBEA is also affected by an evaluation time bias, this could result in learned models to be less suited to solving the problem, reducing performance. Therefore, in this work, we study the impact and presence of evaluation time biases on MBEAs in an asynchronous parallelization setting, and compare this to the biases in GAs. We find that a modern MBEA, GOMEA, is unaffected by evaluation time biases, while the more classical MBEA, ECGA, is affected, much like GAs are.

The aim of Symbolic Regression (SR) is to discover interpretable expressions that accurately describe data. The accuracy of an expression depends on both its structure and coefficients. To keep the structure simple enough to be interpretable, effective coefficient optimisation becomes key. Gradient-based optimisation is clearly effective at training neural networks in Deep Learning (DL), which can essentially be viewed as large, over-parameterised expressions: in this paper, we study how gradient-based optimisation techniques as often used in DL transfer to SR. In particular, we first assess what techniques work well across random SR expressions, independent of any specific SR algorithm. We find that mini-batching and gradient-clipping can be helpful (similar to DL), while second-order optimisers outperform first-order ones (different from DL). Next, we consider whether including gradient-based optimisation in Genetic Programming (GP), a classic SR algorithm, is beneficial. On five real-world datasets, in a generation-based comparison, we find that second-order optimisation outperforms coefficient mutation (or no optimisation). However, in time-based comparisons, performance gaps shrink substantially because the computational expensiveness of second-order optimisation causes GP to perform fewer generations. The interplay of computational costs between the optimisation of structure and coefficients is thus a critical aspect to consider.

Population Based Training (PBT) is an efficient hyperparameter optimization algorithm. PBT is a single-objective algorithm, but many real-world hyperparameter optimization problems involve two or more conflicting objectives. In this work, we therefore introduce a multi-objective version of PBT, MO-PBT. Our experiments on diverse multi-objective hyperparameter optimization problems (Precision/Recall, Accuracy/Fairness, Accuracy/Adversarial Robustness) show that MO-PBT outperforms random search, single-objective PBT, and the state-of-the-art multi-objective hyperparameter optimization algorithm MO-ASHA.

To achieve excellent performance with modern neural networks, having the right network architecture is important. Neural Architecture Search (NAS) concerns the automatic discovery of task-specific network architectures. Modern NAS approaches leverage super-networks whose subnetworks encode candidate neural network architectures. These subnetworks can be trained simultaneously, removing the need to train each network from scratch, thereby increasing the efficiency of NAS.

The Multi-Objective Real-Valued Gene-pool Optimal Mixing Evolutionary Algorithm (MO-RV-GOMEA) has been proven effective and eficient in solving real-world problems. A prime example is optimizing treatment plans for prostate cancer brachytherapy, an internal form of radiation treatment, for which equally important clinical aims from a base protocol are grouped into two objectives and bi-objectively optimized. This use of MO-RV-GOMEA was recently successfully introduced into clinical practice. Brachytherapy can also play an important role in treating cervical cancer. However, using the same approach to optimize treatment plans often does not immediately lead to clinically desirable results. Concordantly, medical experts indicate that they use additional aims beyond the cervix base protocol. Moreover, these aims have different priorities and can be patient-specifically adjusted. For this reason, we propose a novel adaptive objective configuration method to use with MO-RV-GOMEA so that we can accommodate additional aims of this nature. Based on results using only the base protocol, in consultation with medical experts, we configured key additional aims. We show how, for 10 patient cases, the new approach achieves the intended result, properly taking into account the additional aims. Consequently, plans resulting from the new approach are preferred by medical specialists in 8/10 cases.

Model-Based Evolutionary Algorithms (MBEAs) can be highly scalable by virtue of linkage (or variable interaction) learning. This requires, however, that the linkage model can capture the exploitable structure of a problem. Usually, a single type of linkage structure is attempted to be captured using models such as a linkage tree. However, in practice, problems may exhibit multiple linkage structures. This is for instance the case in multi-objective optimization when the objectives have different linkage structures. This cannot be modelled sufficiently well when using linkage models that aim at capturing a single type of linkage structure, deteriorating the advantages brought by MBEAs. Therefore, here, we introduce linkage kernels, whereby a linkage structure is learned for each solution over its local neighborhood. We implement linkage kernels into the MBEA known as GOMEA that was previously found to be highly scalable when solving various problems. We further introduce a novel benchmark function called Best-of-Traps (BoT) that has an adjustable degree of different linkage structures. On both BoT and a worst-case scenario-based variant of the well-known MaxCut problem, we experimentally find a vast performance improvement of linkage-kernel GOMEA over GOMEA with a single linkage tree as well as the MBEA known as DSMGA-II.

Genetic programming (GP) is one of the best approaches today to discover symbolic regression models. To find models that trade off accuracy and complexity, the non-dominated sorting genetic algorithm II (NSGA-II) is widely used. Unfortunately, it has been shown that NSGA-II can be inefficient: in early generations, low-complexity models over-replicate and take over most of the population. Consequently, studies have proposed different approaches to promote diversity. Here, we study the root of this problem, in order to design a superior approach. We find that the over-replication of low complexity-models is due to a lack of evolvability, i.e., the inability to produce offspring with improved accuracy. We therefore extend NSGA-II to track, over time, the evolvability of models of different levels of complexity. With this information, we limit how many models of each complexity level are allowed to survive the generation. We compare this new version of NSGA-II, evoNSGA-II, with the use of seven existing multi-objective GP approaches on ten widely-used data sets, and find that evoNSGA-II is equal or superior to using these approaches in almost all comparisons. Furthermore, our results confirm that evoNSGA-II behaves as intended: models that are more evolvable form the majority of the population. Code: https://github.com/dzhliu/evoNSGA-II

Explainable artificial intelligence (XAI) is an important and rapidly expanding research topic. The goal of XAI is to gain trust in a machine learning (ML) model through clear insights into how the model arrives at its predictions. Genetic programming (GP) is often cited as being uniquely well-suited to contribute to XAI because of its capacity to learn (small) symbolic models that have the potential to be interpreted. Nevertheless, like many ML algorithms, GP typically results in a single best model. However, in practice, the best model in terms of training error may well not be the most suitable one as judged by a domain expert for various reasons, including overfitting, multiple different models existing that have similar accuracy and unwanted errors on particular data points due to typical accuracy measures like mean squared error. Hence, to increase chances that domain experts deem a resulting model plausible, it becomes important to be able to explicitly search for multiple, diverse, high-quality models that trade-off different meanings of accuracy. In this paper, we achieve exactly this with a novel multi-modal multi-tree multi-objective GP approach that extends a modern model-based GP algorithm known as GP-GOMEA that is already effective at searching for small expressions.

Dimensionality reduction (DR) is an important technique for data exploration and knowledge discovery. However, most of the main DR methods are either linear (e.g., PCA), do not provide an explicit mapping between the original data and its lower-dimensional representation (e.g., MDS, t-SNE, isomap), or produce mappings that cannot be easily interpreted (e.g., kernel PCA, neural-based autoencoder). Recently genetic programming (GP) has been used to evolve interpretable DR mappings in the form of symbolic expressions. There exists a number of ways in which GP can be used to this end and no study exists that performs a comparison. In this paper, we fill this gap by comparing existing GP methods as well as devising new ones. We evaluate our methods on several benchmark datasets based on predictive accuracy and on how well the original features can be reconstructed using the lower-dimensional representation only. Finally we qualitatively assess the resulting expressions and their complexity. We find that various GP methods can be competitive with state-of-the-art DR algorithms and that they have the potential to produce interpretable DR mappings.

Domination-based multiobjective (MO) evolutionary algorithms (EAs) are today arguably the most frequently used type of MOEA. These methods, however, stagnate when the majority of the population becomes nondominated, preventing further convergence to the Pareto set. Hypervolume-based MO optimization has shown promising results to overcome this. Direct use of the hypervolume, however, results in no selection pressure for dominated solutions. The recently introduced Sofomore framework overcomes this by solving multiple interleaved single-objective dynamic problems that iteratively improve a single approximation set, based on the uncrowded hypervolume improvement (UHVI). It thereby however loses many advantages of population-based MO optimization, such as handling multimodality. Here, we reformulate the UHVI as a quality measure for approximation sets, called the uncrowded hypervolume (UHV), which can be used to directly solve MO optimization problems with a single-objective optimizer. We use the state-of-the-art gene-pool optimal mixing evolutionary algorithm (GOMEA) that is capable of efficiently exploiting the intrinsically available grey-box properties of this problem. The resulting algorithm, UHV-GOMEA, is compared with Sofomore equipped with GOMEA, and the domination-based MO-GOMEA. In doing so, we investigate in which scenarios either domination-based or hypervolume-based methods are preferred. Finally, we construct a simple hybrid approach that combines MO-GOMEA with UHV-GOMEA and outperforms both.

The Real-Valued Gene-pool Optimal Mixing Evolutionary Algorithm (RV-GOMEA) has previously been successfully used to achieve highly scalable optimization of various real-world problems in a gray-box optimization setting. Deformable Image Registration (DIR) is a multi-objective problem, aimed at finding the most likely non-rigid deformation of a given source image so that it matches a given target image. We specifically consider the case where the deformation model allows for finite-element-type modeling of tissue properties. This optimization problem is non-smooth, necessitating techniques like EAs to get good results. Though the objectives of DIR are non-separable, non-neighboring regions of the deformation grid are conditionally independent. We show that GOMEA allows to exploit such knowledge through the large-scale parallel application of variation steps, where each is only accepted when leading to an improvement, on a Graphics Processing Unit (GPU). On various 2-dimensional DIR problems, we find that this way, similar results can be achieved as when sequential processing is performed, while allowing for substantial speed-ups (up to a factor of 111) for the highest-dimensional problems (i.e., the highest deformation-grid resolution). This work opens the door to the extension of this type of DIR to larger (3-dimensional) deformation grids, and its application to other real-world problems.

We propose a novel surrogate-assisted Evolutionary Algorithm for solving expensive combinatorial optimization problems. We integrate a surrogate model, which is used for fitness value estimation, into a state-of-the-art P3-like variant of the Gene-Pool Optimal Mixing Algorithm (GOMEA) and adapt the resulting algorithm for solving non-binary combinatorial problems. We test the proposed algorithm on an ensemble learning problem. Ensembling several models is a common Machine Learning technique to achieve better performance. We consider ensembles of several models trained on disjoint subsets of a dataset. Finding the best dataset partitioning is naturally a combinatorial non-binary optimization problem. Fitness function evaluations can be extremely expensive if complex models, such as Deep Neural Networks, are used as learners in an ensemble. Therefore, the number of fitness function evaluations is typically limited, necessitating expensive optimization techniques. In our experiments we use five classification datasets from the OpenML-CC18 benchmark and Support-vector Machines as learners in an ensemble. The proposed algorithm demonstrates better performance than alternative approaches, including Bayesian optimization algorithms. It manages to find better solutions using just several thousand fitness function evaluations for an ensemble learning problem with up to 500 variables.

It is known that to achieve efficient scalability of an Evolutionary Algorithm (EA), dependencies (also known as linkage) must be properly taken into account during variation. In a Gray-Box Optimization (GBO) setting, exploiting prior knowledge regarding these dependencies can greatly benefit optimization. We specifically consider the setting where partial evaluations are possible, meaning that the partial modification of a solution can be efficiently evaluated. Such problems are potentially very difficult, for example, non-separable, multimodal, and multiobjective. The Gene-pool Optimal Mixing Evolutionary Algorithm (GOMEA) can effectively exploit partial evaluations, leading to a substantial improvement in performance and scalability. GOMEA was recently shown to be extendable to real-valued optimization through a combination with the real-valued estimation of distribution algorithm AMaLGaM. In this article, we definitively introduce the Real-Valued GOMEA (RV-GOMEA), and introduce a new variant, constructed by combining GOMEA with what is arguably the best-known real-valued EA, the Covariance Matrix Adaptation Evolution Strategies (CMA-ES). Both variants of GOMEA are compared to L-BFGS and the Limited Memory CMA-ES (LM-CMA-ES). We show that both variants of RV-GOMEA achieve excellent performance and scalability in a GBO setting, which can be orders of magnitude better than that of EAs unable to efficiently exploit the GBO setting.

The Gene-pool Optimal Mixing Evolutionary Algorithm (GOMEA) is a model-based EA framework that has been shown to perform well in several domains, including Genetic Programming (GP). Differently from traditional EAs where variation acts blindly, GOMEA learns a model of interdependencies within the genotype, that is, the linkage, to estimate what patterns to propagate. In this article, we study the role of Linkage Learning (LL) performed by GOMEA in Symbolic Regression (SR). We show that the non-uniformity in the distribution of the genotype in GP populations negatively biases LL, and propose a method to correct for this. We also propose approaches to improve LL when ephemeral random constants are used. Furthermore, we adapt a scheme of interleaving runs to alleviate the burden of tuning the population size, a crucial parameter for LL, to SR. We run experiments on 10 real-world datasets, enforcing a strict limitation on solution size, to enable interpretability. We find that the new LL method outperforms the standard one, and that GOMEA outperforms both traditional and semantic GP. We also find that the small solutions evolved by GOMEA are competitive with tuned decision trees, making GOMEA a promising new approach to SR.

Surrogate-assisted evolutionary algorithms have the potential to be of high value for real-world optimization problems when fitness evaluations are expensive, limiting the number of evaluations that can be performed. In this article, we consider the domain of pseudo-Boolean functions in a black-box setting. Moreover, instead of using a surrogate model as an approximation of a fitness function, we propose to precisely learn the coefficients of the Walsh decomposition of a fitness function and use the Walsh decomposition as a surrogate. If the coefficients are learned correctly, then the Walsh decomposition values perfectly match with the fitness function, and, thus, the optimal solution to the problem can be found by optimizing the surrogate without any additional evaluations of the original fitness function. It is known that the Walsh coefficients can be efficiently learned for pseudo-Boolean functions with k -bounded epistasis and known problem structure. We propose to learn dependencies between variables first and, therefore, substantially reduce the number of Walsh coefficients to be calculated. After the accurate Walsh decomposition is obtained, the surrogate model is optimized using GOMEA, which is considered to be a state-of-the-art binary optimization algorithm. We compare the proposed approach with standard GOMEA and two other Walsh decomposition-based algorithms. The benchmark functions in the experiments are well-known trap functions, NK-landscapes, MaxCut, and MAX3SAT problems. The experimental results demonstrate that the proposed approach is scalable at the supposed complexity of O (ℓ log ℓ) function evaluations when the number of subfunctions is O (ℓ) and all subfunctions are k -bounded, outperforming all considered algorithms.

Often, real-world problems are of the gray-box type. It has been shown that the Real-Valued Gene-pool Optimal Mixing Evolutionary Algorithm (RV-GOMEA) is in principle capable of exploiting such a Gray-Box Optimization (GBO) setting using linkage models that capture dependencies between problem variables, resulting in excellent performance and scalability on both benchmark and real-world problems that allow for partial evaluations. However, linkage models proposed for RV-GOMEA so far cannot explicitly capture overlapping dependencies. Consequently, performance degrades if such dependencies exist. In this paper, we therefore introduce various ways of using conditional linkage models in RV-GOMEA. Their use is compared to that of non-conditional models, and to VkD-CMA, whose performance is among the state of the art, on various benchmark problems with overlapping dependencies. We find that RV-GOMEA with conditional linkage models achieves the best scalability on most problems, with conditional models leading to similar or better performance than non-conditional models. We conclude that the introduction of conditional linkage models to RV-GOMEA is an important contribution, as it expands the set of problems for which optimization in a GBO setting results in substantially improved performance and scalability. In future work, conditional linkage models may prove to benefit the optimization of real-world problems.

We introduce a novel surrogate-assisted Genetic Algorithm (GA) for expensive optimization of problems with discrete categorical variables. Specifically, we leverage the strengths of the Gene-pool Optimal Mixing Evolutionary Algorithm (GOMEA), a state-of-the-art GA, and, for the first time, propose to use a convolutional neural network (CNN) as a surrogate model. We propose to train the model on pairwise fitness differences to decrease the number of evaluated solutions that is required to achieve adequate surrogate model training. In providing a proof of principle, we consider relatively standard CNNs, and demonstrate that their capacity is already sufficient to accurately learn fitness landscapes of various well-known benchmark functions. The proposed CS-GOMEA is compared with GOMEA and the widely-used Bayesian-optimization-based expensive optimization frameworks SMAC and Hyperopt, in terms of the number of evaluations that is required to achieve the optimum. In our experiments on binary problems with dimensionalities up to400 variables, CS-GOMEA always found the optimum, whereas SMAC and Hyperopt failed for problem sizes over 16 variables. Moreover, the number of evaluated solutions required by CS-GOMEA to find the optimum was found to scale much better than GOMEA.

In model-based evolutionary algorithms (EAs), the underlying search distribution is adapted to the problem at hand, for example based on dependencies between decision variables. Hill-valley clustering is an adaptive niching method in which a set of solutions is clustered such that each cluster corresponds to a single mode in the fitness landscape. This can be used to adapt the search distribution of an EA to the number of modes, exploring each mode separately. Especially in a black-box setting, where the number of modes is a priori unknown, an adaptive approach is essential for good performance. In this work, we introduce multi-objective hill-valley clustering and combine it with MAMaLGaM, a multi-objective EA, into the multi-objective hill-valley EA (MO-HillVallEA). We empirically show that MO-HillVallEA outperforms MAMaLGaM and other well-known multi-objective optimization algorithms on a set of benchmark functions. Furthermore, and perhaps most important, we show that MO-HillVallEA is capable of obtaining and maintaining multiple approximation sets simultaneously over time.

Semantic Backpropagation (SB) is a recent technique that promotes effective variation in tree-based genetic programming. The basic idea of SB is to provide information on what output is desirable for a specified tree node, by propagating the desired root-node output back to the specified node using inversions of functions encountered along the way. Variation operators then replace the subtree located at the specified node with a tree for which the output is closest to the desired output, by searching in a pre-computed library. In this paper, we propose two contributions to enhance SB specifically for symbolic regression, by incorporating the principles of Keijzer's Linear Scaling (LS). In particular, we show how SB can be used in synergy with the scaled mean squared error, and we show how LS can be adopted within library search. We test our adaptations using the well-known variation operator Random Desired Operator (RDO), comparing to its baseline implementation, and to traditional crossover and mutation. Our experimental results on real-world datasets show that SB enhanced with LS substantially improves the performance of RDO, resulting in overall the best performance among all tested GP algorithms.

The importance and potential of Gray-Box Optimization (GBO) with evolutionary algorithms is becoming increasingly clear lately both for benchmark and real-world problems. We consider the GBO setting where partial evaluations are possible, meaning that sub-functions of the evaluation function are known and can be exploited to improve optimization efficiency. In this paper, we show that the efficiency of GBO can be greatly improved through large-scale parallelism, exploiting the fact that each evaluation function requires the calculation of a number of independent sub-functions. This is especially interesting for real-world problems where often the majority of the computational effort is spent on the evaluation function. Moreover, we show how the best parallelization technique largely depends on factors including the number of sub-functions and their required computation time, revealing that for different parts of the optimization the best parallelization technique should be selected based on these factors. As an illustration, we show how large-scale parallelization can be applied to optimization of high-dose-rate brachytherapy treatment plans for prostate cancer. We find that use of a modern Graphics Processing Unit (GPU) was the most efficient parallelization technique in all realistic scenarios, leading to substantial speed-ups up to a factor of 73.

The Multi-Objective Real-Valued Gene-pool Optimal Mixing Evolutionary Algorithm (MO-RV-GOMEA) has been shown to exhibit excellent performance in solving various bi-objective benchmark and real-world problems. We assess the competence of MO-RV-GOMEA in tackling many-objective problems, which are normally defined as problems with at least four conflicting objectives. Most Pareto dominance-based Multi-Objective Evolutionary Algorithms (MOEAs) typically diminish in performance if the number of objectives is more than three because selection pressure toward the Pareto-optimal front is lost. This is potentially less of an issue for MO-RV-GOMEA because its variation operator creates each offspring solution by iteratively altering a currently existing solution in a few decision variables each time, and changes are only accepted if they result in a Pareto improvement. For most MOEAs, integrating scalarization methods is potentially beneficial in the many-objective context. Here, we investigate the possibility of improving the performance of MO-RV-GOMEA by further guiding improvement checks during solution variation in MO-RV-GOMEA with carefully constructed Tchebycheff scalarizations. Results obtained from experiments performed on a selection of well-known problems from the DTLZ and WFG test suites show that MO-RV-GOMEA is by design already well-suited for many-objective problems. Moreover, by enhancing it with Tchebycheff scalarizations, it outperforms M0EA/D-2TCHMFI, a state-of-the-art decomposition-based MOEA.

Model-based evolutionary algorithms (EAs) adapt an underlying search model to features of the problem at hand, such as the linkage between problem variables. The performance of EAs often deteriorates as multiple modes in the fitness landscape are modelled with a unimodal search model. The number of modes is however often unknown a priori, especially in a black-box setting, which complicates adaptation of the search model. In this work, we focus on models that can adapt to the multi-modality of the fitness landscape. Specifically, we introduce Hill-Valley Clustering, a remarkably simple approach to adaptively cluster the search space in niches, such that a single mode resides in each niche. In each of the located niches, a core search algorithm is initialized to optimize that niche. Combined with an EA and a restart scheme, the resulting Hill-Valley EA (HillVallEA) is compared to current state-of-the-art niching methods on a standard benchmark suite for multi-modal optimization. Numerical results in terms of the detected number of global optima show that, in spite of its simplicity, HillVallEA is competitive within the limited budget of the benchmark suite, and shows superior performance in the long run.

The recently-introduced Gene-pool Optimal Mixing Evolutionary Algorithm (GOMEA) family has been shown to be capable of excellent performance on academic benchmark problems, outperforming other state-of-the-art EAs, especially when efficient partial evaluations are possible. This holds true also for the latest extension, the Multi-Objective Real-Valued GOMEA (MO-RV-GOMEA). In this paper, we apply MO-RV-GOMEA to the real-world multi-objective optimization problem of catheter placement in High-Dose-Rate (HDR) brachytherapy for prostate cancer, a problem that is non-trivial to solve and has high real-world importance and relevance. Due to the underlying geometric structure of the real-valued variables, partial evaluations can be performed, allowing MO-RV-GOMEA to exploit this structure. The performance of MO-RV-GOMEA is tested on three real-world patient cases and compared to a recent state-of-the-art mixed-integer EA that is aimed at a restricted version of the problem. We find that with MO-RV-GOMEA better solutions can be found much faster, making our proposed approach much more realistic to be used in clinical practice, and enabling new insights into both catheter placement for prostate brachytherapy and on objectives used for automated treatment planning. First results indicate that richer problem models are needed to better match real-world clinical preferences.

The recently introduced Gene-pool Optimal Mixing Evolutionary Algorithm for Genetic Programming (GP-GOMEA) has been shown to find much smaller solutions of equally high quality compared to other state-of-the-art GP approaches. This is an interesting aspect as small solutions better enable human interpretation. In this paper, an adaptation of GP-GOMEA to tackle real-world symbolic regression is proposed, in order to find small yet accurate mathematical expressions, and with an application to a problem of clinical interest. For radiotherapy dose reconstruction, a model is sought that captures anatomical patient similarity. This problem is particularly interesting because while features are patient-specific, the variable to regress is a distance, and is defined over patient pairs. We show that on benchmark problems as well as on the application, GP-GOMEA outperforms variants of standard GP. To find even more accurate models, we further consider an evolutionary meta learning approach, where GP-GOMEA is used to construct small, yet effective features for a different machine learning algorithm. Experimental results show how this approach significantly improves the performance of linear regression, support vector machines, and random forest, while providing meaningful and interpretable features.

The recently introduced Gene-pool Optimal Mixing Evolutionary Algorithm (GOMEA) has been shown to be among the state-of-the-art for solving discrete optimization problems. Key to the success of GOMEA is its ability to efficiently exploit the linkage structure of a problem. Here, we introduce the Real-Valued GOMEA (RV-GOMEA), which incorporates several aspects of the real-valued EDA known as AMaLGaM into GOMEA in order to make GOMEA well-suited for real-valued optimization. The key strength of GOMEA to competently exploit linkage structure is effectively preserved in RV-GOMEA, enabling excellent performance on problems that exhibit a linkage structure that is to some degree decomposable. Moreover, the main variation operator of GOMEA enables substantial improvements in performance if the problem allows for partial evaluations, which may be very well possible in many real-world applications. Comparisons of performance with state-of-the-art algorithms such as CMA-ES and AMaLGaM on a set of well-known benchmark problems show that RV-GOMEA achieves comparable, excellent scalability in case of black-box optimization. Moreover, RV-GOMEA achieves unprecedented scalability on problems that allow for partial evaluations, reaching near-optimal solutions for problems with up to millions of real-valued variables within one hour on a normal desktop computer.

The recently introduced Multi-Objective Gene-pool Optimal Mixing Evolutionary Algorithm (MO-GOMEA) exhibits excellent scalability in solving a wide range of challenging discrete multi-objective optimization problems. In this paper, we address scalability issues in solving multi-objective optimization problems with continuous variables by introducing the Multi-Objective Real-Valued GOMEA (MO-RV-GOMEA), which combines MO-GOMEA with aspects of the multi-objective estimation-of-distribution algorithm known as MAMaLGaM. MO-RV-GOMEA exploits linkage structure in optimization problems by performing distribution estimation, adaptation, and sampling as well as solution mixing based on an explicitly-defined linkage model. Such a linkage model can be defined a priori when some problem-specific knowledge is available, or it can be learned from the population. The scalability of MO-RV-GOMEA using different linkage models is compared to the state-of-the-art multi-objective evolutionary algorithms NSGA-II and MAMaLGaM on a wide range of benchmark problems. MO-RV-GOMEA is found to retain the excellent scalability of MO-GOMEA through the successful exploitation of linkage structure, scaling substantially better than NSGA-II and MAMaLGaM. This scalability is even further improved when partial evaluations are possible, achieving strongly sub-linear scalability in terms of the number of evaluations.

Estimation-of-Distribution Algorithms (EDAs) have been applied with quite some success when solving real-valued optimization problems, especially in the case of Black Box Optimization (BBO). Generally the performance of an EDA depends on the match between its driving probability distribution and the landscape of the problem being solved. Because most well-known EDAs, including CMA-ES, NES, and AMaLGaM, use a uni-modal search distribution, they have a high risk of getting trapped in local optima when a problem is multi-modal with a (moderate) number of relatively comparable modes. This risk could potentially be mitigated using niching methods that define multiple regions of interest where separate search distributions govern sub-populations. However, a key question is how to determine a suitable number of niches, especially in BBO. In this paper, we present a novel, adaptive niching approach that determines the niches through hierarchical clustering based on the correlation between the probability densities and fitness values of solutions. We test the performance of a combination of this niching approach with AMaLGaM on both new and well-known niching benchmark problems and find that the new approach properly identifies multiple landscape modes, leading to much better performance on multi-modal problems than with a non-niched, uni-modal EDA.

Brachytherapy is a form of radiotherapy whereby a radiation source is guided near tumors, using devices such as catheter implants. In the present clinical workflow, catheters are first placed inside or close to the tumor based on clinical expertise. Subsequently, software is used to design a plan for the delivery of radiation. Treatment planning is essentially a multi-objective optimization problem, where conflicting objectives represent radiation delivered to tumor cells and healthy cells. However, current clinical software collapses this information into a single-objective, constrained optimization problem. Moreover, catheter positioning is typically not included. As a consequence, it is hard to obtain insight into the true nature of the trade-offs between key planning objectives and the placement of catheters. Such insights are however crucial in understanding how better treatment plans may be constructed. To obtain such insights, we interface with real-world clinical software and derive potential catheter positions for real-world patients. Selecting and configuring catheters requires mixed-integer optimization. For this reason, we extend the recently-proposed Genetic Algorithm for Model-Based mixed-Integer opTimization (GAMBIT) to tackle multi-objective optimization problems. Our results indicate that clinically acceptable plans of high quality may be achievable with less catheters than typically used in current clinical practice.

The Gene-pool Optimal Mixing Evolutionary Algorithm (GOMEA) is a recently introduced model-based EA that has been shown to be capable of outperforming state-of-the-art alternative EAs in terms of scalability when solving discrete optimization problems. One of the key aspects of GOMEA's success is a variation operator that is designed to extensively exploit linkage models by effectively combining partial solutions. Here, we bring the strengths of GOMEA to Genetic Programming (GP), introducing GP-GOMEA. Under the hypothesis of having little problem-specific knowledge, and in an effort to design easy-to-use EAs, GP-GOMEA requires no parameter specification. On a set of well-known benchmark problems we find that GP-GOMEA outperforms standard GP while being on par with more recently introduced, state-of-the-art EAs. We furthermore introduce Input-space Entropy-based Building-block Learning (IEBL), a novel approach to identifying and encapsulating relevant building blocks (subroutines) into new terminals and functions. On problems with an inherent degree of modularity, IEBL can contribute to compact solution representations, providing a large potential for knock-on effects in performance. On the difficult, but highly modular Even Parity problem, GP-GOMEA+IEBL obtains excellent scalability, solving the 14-bit instance in less than 1 hour.

Gradient methods and their value in single-objective, real-valued optimization are well-established. As such, they play a key role in tackling real-world, hard optimization problems such as deformable image registration (DIR). A key question is to which extent gradient techniques can also play a role in a multi-objective approach to DIR. We therefore aim to exploit gradient information within an evolutionary-algorithm-based multi-objective optimization framework for DIR. Although an analytical description of the multi-objective gradient (the set of all Pareto-optimal improving directions) is available, it is nontrivial how to best choose the most appropriate direction per solution because these directions are not necessarily uniformly distributed in objective space. To address this, we employ a Monte-Carlo method to obtain a discrete, spatially-uniformly distributed approximation of the set of Pareto-optimal improving directions. We then apply a diversification technique in which each solution is associated with a unique direction from this set based on its multi- as well as single-objective rank. To assess its utility, we compare a state-of-the-art multi-objective evolutionary algorithm with three different hybrid versions thereof on several benchmark problems and two medical DIR problems. Results show that the diversification strategy successfully leads to unbiased improvement, helping an adaptive hybrid scheme solve all problems, but the evolutionary algorithm remains the most powerful optimization method, providing the best balance between proximity and diversity.

Estimation-of-Distribution Algorithms (EDAs) build and use probabilistic models during optimization in order to automatically discover and use an optimization problems' structure. This is especially useful for black-box optimization where no assumptions are made on the problem being solved, which is characteristic of many cases in solving complex real-world problems. In this paper we consider multi-objective optimization problems with real-valued variables. Although the vast majority of advances in EDA literature concern single-objective optimization, advances have also been made in multi-objective optimization. In this paper we bring together two recent advances, namely incremental Gaussian model building to reduce the required population size and a mixture-based multi-objective framework that has specific methods to better facilitate model-building techniques that span multiple generations. Significantly faster convergence to the optimal Pareto front is achieved on 6 out of 7 artificial benchmark problems from literature. Although results on two of these problems show that building models with higher-order interactions between variables is required, these problems are still artificial. We therefore also consider a more realistic optimization problem in image processing, namely deformable image registration. For this problem too, our results show the need for processing interactions between problem variables, stressing the importance of studying such models.

Recently, foundations rooted in physics have been laid down for the goal of simulating the propagation of a guide wire inside the vasculature. At the heart of the simulation lies the fundamental task of energy minimization. The energy comes from interaction with the vessel wall and the bending of the guide wire. For the simulation to be useful in actual training, obtaining the smallest possible optimization time is key. In this paper, we, therefore, study the influence of using different optimization techniques: a semianalytical approximation algorithm, the conjugate-gradients algorithm, and an evolutionary algorithm (EA), specifically the GLIDE algorithm. Simulation performance has been measured on phantom data. The results show that a substantial reduction in time can be obtained while the error is increased only slightly if conjugate gradients or GLIDE is used.