In Symbolic Regression (SR), achieving a proper balance between accuracy and interpretability remains a key challenge. The Genetic Programming variant of the Gene-pool Optimal Mixing Evolutionary Algorithm (GP-GOMEA) is of particular interest as it achieves state-of-the-art performance using a template that limits the size of expressions. A recently introduced expansion, modular GP-GOMEA, is capable of decomposing expressions using multiple subexpressions, further increasing chances of interpretability. However, modular GP-GOMEA may create larger expressions, increasing the need to balance size and accuracy. A multi-objective variant of GP-GOMEA exists, which can be used, for instance, to optimize for size and accuracy simultaneously, discovering their trade-off. However, even with enhancements that we propose in this paper to improve the performance of multi-objective modular GP-GOMEA, when optimizing for size and accuracy, the single-objective version in which a multi-objective archive is used only for logging, still consistently finds a better average hypervolume. We consequently analyze when a single-objective approach should be preferred. Additionally, we explore an objective that stimulates re-use in multi-objective modular GP-GOMEA.

Medical applications often involve several data modalities, particularly medical images and clinical information, which can be combined to enhance the decision-making process by improving accuracy. Multimodal learning approaches can leverage all available data for increased robustness in the resulting models, consequently outperforming unimodal approaches. Furthermore, AI frameworks must be human-verifiable and interpretable to be deployed in real-world situations, considering legal and privacy aspects. Due to the opaque nature of Deep Learning (DL) methods, interpretability is often limited despite their state-of-the-art performance in many tasks. Genetic Programming (GP) can provide compact and interpretable symbolic expressions for tabular data but is less effective for image analysis. We introduce MultiFIX: a new interpretability-focused pipeline for multimodal learning that leverages the strengths of DL and GP to explicitly engineer features from different data types and combine them to make the final prediction. The MultiFIX pipeline comprises two stages: the training stage, where a DL (black-box) model is trained using different training procedures to extract relevant features from each modality; and the inference stage, where the resulting model is transformed to be interpretable. Image features are explained with attention maps by Grad-CAM, and inherently interpretable symbolic expressions evolved with GP fully replace the tabular feature engineering block, and the fusion of the extracted features to predict the target label. To show the application potential of the presented pipeline, we demonstrate MultiFIX with a Melanoma Risk Assessment dataset. Results show that MultiFIX outperforms unimodal models while offering explanations that can be straightforwardly analysed and are consistent with the expectations.

The publisher regrets that one of the authors is missing in the PDF and web version of the article. Namely, Anton Bouter, who is affiliated with the Centrum Wiskunde & Informatica as described above in this erratum. In table 1, a space is missing after the character “<” in the line `Cervix -> Sigmoid' in column `Sparing criteria'. In table 3, the `%' signs in first column (DVI) should also be in subscript, similar to how it is written in the first column of Table 4. The supplementary material should state that needle contribution is limited to up to 40% of the total given dwell time (in applicator and needles). Lastly, the article should be seen as a research article and the authors would like to be cited as “L.R.M. Dickhoff and R.J. Scholman et al”.

The publisher would like to apologise for any inconvenience caused.

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.

Real-world problems are often dependent on multiple data modalities, making multimodal fusion essential for leveraging diverse information sources. In high-stakes domains, such as in healthcare, understanding how each modality contributes to the prediction is critical to ensure trustworthy and interpretable AI models. We present MultiFIX, an interpretability-driven multimodal data fusion pipeline that explicitly engineers distinct features from different modalities and combines them to make the final prediction. Initially, only deep learning components are used to train a model from data. The black-box (deep learning) components are subsequently either explained using post-hoc methods such as Grad-CAM for images or fully replaced by interpretable blocks, namely symbolic expressions for tabular data, resulting in an explainable model. We study the use of MultiFIX using several training strategies for feature extraction and predictive modeling. Besides highlighting strengths and weaknesses of MultiFIX, experiments on a variety of synthetic datasets with varying degrees of interaction between modalities demonstrate that MultiFIX can generate multimodal models that can be used to accurately explain both the extracted features and their integration without compromising predictive performance.

Sharing synthetic medical images is a promising alternative to sharing real images that can improve patient privacy and data security. To get good results, existing methods for medical image synthesis must be manually adjusted when they are applied to unseen data. To remove this manual burden, we introduce a Hyperparameter-Free distributed learning method for automatic medical image Synthesis, Sharing, and Segmentation called HyFree-S3. For three diverse segmentation settings (pelvic MRIs, lung X-rays, polyp photos), the use of HyFree-S3 results in improved performance over training only with site-specific data (in the majority of cases). The hyperparameter-free nature of the method should make data synthesis and sharing easier, potentially leading to an increase in the quantity of available data and consequently the quality of the models trained that may ultimately be applied in the clinic.

When it comes to solving optimization problems with evolutionary algorithms (EAs) in a reliable and scalable manner, detecting and exploiting linkage information, that is, dependencies between variables, can be key. In this paper, we present the latest version of, and propose substantial enhancements to, the gene-pool optimal mixing evolutionary algorithm (GOMEA): an EA explicitly designed to estimate and exploit linkage information. We begin by performing a large-scale search over several GOMEA design choices to understand what matters most and obtain a generally best-performing version of the algorithm. Next, we introduce a novel version of GOMEA, called CGOMEA, where linkage-based variation is further improved by filtering solution mating based on conditional dependencies. We compare our latest version of GOMEA, the newly introduced CGOMEA, and another contending linkage-aware EA, DSMGA-II, in an extensive experimental evaluation, involving a benchmark set of nine black-box problems that can be solved efficiently only if their inherent dependency structure is unveiled and exploited. Finally, in an attempt to make EAs more usable and resilient to parameter choices, we investigate the performance of different automatic population management schemes for GOMEA and CGOMEA, de facto making the EAs parameterless. Our results show that GOMEA and CGOMEA significantly outperform the original GOMEA and DSMGA-II on most problems, setting a new state of the art for the field.

Machine learning models can be made more performant and their predictions more consistent by creating an ensemble. Each neural network in an ensemble commonly performs its own feature extraction. These features are often highly similar, leading to potentially many redundant calculations. Unifying these calculations (i.e., reusing some of them) would be desirable to reduce computational cost. However, splicing two trained networks is non-trivial because architectures and feature representations typically differ, leading to a performance breakdown. To overcome this issue, we propose to employ stitching, which introduces new layers at crossover points. Essentially, a new network consisting of the two basis networks is constructed. In this network, new links between the two basis networks are created through the introduction and training of stitches. New networks can then be created by choosing which stitching layers to (not) use, thereby selecting a subnetwork. Akin to a supernetwork, assessing the performance of a selected subnetwork is efficient, as only their evaluation on data is required. We experimentally show that our proposed approach enables finding networks that represent novel trade-offs between performance and computational cost compared to classical ensembles, with some new networks even dominating the original networks.

The transformation model is an essential component of any deformable image registration approach. It provides a representation of physical deformations between images, thereby defining the range and realism of registrations that can be found. Two types of transformation models have emerged as popular choices: B-spline models and mesh models. Although both models have been investigated in detail, a direct comparison has not yet been made, since the models are optimized using very different optimization methods in practice. B-spline models are predominantly optimized using gradient-descent methods, while mesh models are typically optimized using finite-element method solvers or evolutionary algorithms. Multi-objective optimization methods, which aim to find a diverse set of high-quality trade-off registrations, are increasingly acknowledged to be important in deformable image registration. Since these methods search for a diverse set of registrations, they can provide a more complete picture of the capabilities of different transformation models, making them suitable for a comparison of models. In this work, we conduct the first direct comparison between B-spline and mesh transformation models, by optimizing both models with the same state-of-the-art multi-objective optimization method, the Multi-Objective Real-Valued Gene-pool Optimal Mixing Evolutionary Algorithm (MO-RV-GOMEA). The combination with B-spline transformation models, moreover, is novel. We experimentally compare both models on two different registration problems that are both based on pelvic CT scans of cervical cancer patients, featuring large deformations. Our results, on three cervical cancer patients, indicate that the choice of transformation model can have a profound impact on the diversity and quality of achieved registration outcomes.

PURPOSE: Without a clear definition of an optimal treatment plan, no optimization model can be perfect. Therefore, instead of automatically finding a single “optimal” plan, finding multiple, yet different near-optimal plans, can be an insightful approach to support radiation oncologists in finding the plan they are looking for. METHODS AND MATERIALS: BRIGHT is a flexible AI-based optimization method for brachytherapy treatment planning that has already been shown capable of finding high-quality plans that trade-off target volume coverage and healthy tissue sparing. We leverage the flexibility of BRIGHT to find plans with similar dose-volume criteria, yet different dose distributions. We further describe extensions that facilitate fast plan adaptation should planning aims need to be adjusted, and straightforwardly allow incorporating hospital-specific aims besides standard protocols. RESULTS: Results are obtained for prostate (n = 12) and cervix brachytherapy (n = 36). We demonstrate the possible differences in dose distribution for optimized plans with equal dose-volume criteria. We furthermore demonstrate that adding hospital-specific aims enables adhering to hospital-specific practice while still being able to automatically create cervix plans that more often satisfy the EMBRACE-II protocol than clinical practice. Finally, we illustrate the feasibility of fast plan adaptation. CONCLUSIONS: Methods such as BRIGHT enable new ways to construct high-quality treatment plans for brachytherapy while offering new insights by making explicit the options one has. In particular, it becomes possible to present to radiation oncologists a manageable set of alternative plans that, from an optimization perspective are equally good, yet differ in terms of coverage-sparing trade-offs and shape of the dose distribution.

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.

Purpose: Deformable image registration (DIR) can benefit from additional guidance using corresponding landmarks in the images. However, the benefits thereof are largely understudied, especially due to the lack of automatic landmark detection methods for three-dimensional (3D) medical images. Approach: We present a deep convolutional neural network (DCNN), called DCNN-Match, that learns to predict landmark correspondences in 3D images in a self-supervised manner. We trained DCNN-Match on pairs of computed tomography (CT) scans containing simulated deformations. We explored five variants of DCNN-Match that use different loss functions and assessed their effect on the spatial density of predicted landmarks and the associated matching errors. We also tested DCNN-Match variants in combination with the open-source registration software Elastix to assess the impact of predicted landmarks in providing additional guidance to DIR. Results: We tested our approach on lower abdominal CT scans from cervical cancer patients: 121 pairs containing simulated deformations and 11 pairs demonstrating clinical deformations. The results showed significant improvement in DIR performance when landmark correspondences predicted by DCNN-Match were used in the case of simulated (p = 0e0) as well as clinical deformations (p = 0.030). We also observed that the spatial density of the automatic landmarks with respect to the underlying deformation affect the extent of improvement in DIR. Finally, DCNN-Match was found to generalize to magnetic resonance imaging scans without requiring retraining, indicating easy applicability to other datasets. Conclusions: DCNN-match learns to predict landmark correspondences in 3D medical images in a self-supervised manner, which can improve DIR performance.

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.

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).

Real-world problems are often multi-objective, with decision-makers unable to specify a priori which trade-off between the conflicting objectives is preferable. Intuitively, building machine learning solutions in such cases would entail providing multiple predictions that span and uniformly cover the Pareto front of all optimal trade-off solutions. We propose a novel approach for multi-objective training of neural networks to approximate the Pareto front during inference. In our approach, we train the neural networks multi-objectively using a dynamic loss function, wherein each network’s losses (corresponding to multiple objectives) are weighted by their hypervolume maximizing gradients. Experiments on different multi-objective problems show that our approach returns well-spread outputs across different trade-offs on the approximated Pareto front without requiring the trade-off vectors to be specified a priori. Further, results of comparisons with the state-of-the-art approaches highlight the added value of our proposed approach, especially in cases where the Pareto front is asymmetric.

Neural Architecture Search (NAS) has recently become a topic of great interest. However, there is a potentially impactful issue within NAS that remains largely unrecognized: noise. Due to stochastic factors in neural network initialization, training, and the chosen train/validation dataset split, the performance evaluation of a neural network architecture, which is often based on a single learning run, is also stochastic. This may have a particularly large impact if a dataset is small. We therefore propose to reduce this noise by evaluating architectures based on average performance over multiple network training runs using different random seeds and cross-validation. We perform experiments for a combinatorial optimization formulation of NAS in which we vary noise reduction levels. We use the same computational budget for each noise level in terms of network training runs, i.e., we allow less architecture evaluations when averaging over more training runs. Multiple search algorithms are considered, including evolutionary algorithms which generally perform well for NAS. We use two publicly available datasets from the medical image segmentation domain where datasets are often limited and variability among samples is often high. Our results show that reducing noise in architecture evaluations enables finding better architectures by all considered search algorithms.

Genetic Programming (GP) can make an important contribution to explainable artificial intelligence because it can create symbolic expressions as machine learning models. Nevertheless, to be explainable, the expressions must not become too large. This may, however, limit their potential to be accurate. The re-use of subexpressions has the unique potential to mitigate this issue. The Genetic Programming Gene-pool Optimal Mixing Evolutionary Algorithm (GP-GOMEA) is a recent model-based GP approach that has been found particularly capable of evolving small expressions. However, its tree representation offers no explicit mechanisms to re-use subexpressions. By contrast, the graph representation in Cartesian GP (CGP) is natively capable of re-use. For this reason, we introduce CGP-GOMEA, a variant of GP-GOMEA that uses graphs instead of trees. We experimentally compare various configurations of CGP-GOMEA with GP-GOMEA and find that CGP-GOMEA performs on par with GP-GOMEA on three common datasets. Moreover, CGP-GOMEA is found to produce models that re-use subexpressions more often than GP-GOMEA uses duplicate subexpressions. This indicates that CGP-GOMEA has unique added potential, allowing to find even smaller expressions than GP-GOMEA with similar accuracy.

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.

Gene-pool Optimal Mixing Evolutionary Algorithms (GOMEAs) have been shown to achieve state-of-the-art results on various types of optimization problems with various types of problem variables. Recently, a GOMEA for permutation spaces was introduced by leveraging the random keys encoding, obtaining promising first results on permutation flow shop instances. A key cited strength of GOMEAs is linkage learning, i.e., the ability to determine and leverage, during optimization, key dependencies between problem variables. However, the added value of linkage learning was not tested in depth for permutation GOMEA. Here, we introduce a new version of permutation GOMEA, called qGOMEA, that works directly in permutation space, removing the redundancy of using random keys. We additionally consider various linkage information sources, including random noise, in both GOMEA variants, and compare performance with various classic genetic algorithms on a wider range of problems than considered before. We find that, although the benefits of linkage learning are clearly visible for various artificial benchmark problems, this is far less the case for various real-world inspired problems. Finally, we find that qGOMEA performs best, and is more applicable to a wider range of permutation problems.