Marco Virgolin

Pierre-Alexandre Kamienny, Guillaume Lample, Sylvain Lamprier et al. · meta-ai

Symbolic regression (SR) is the problem of learning a symbolic expression from numerical data. Recently, deep neural models trained on procedurally-generated synthetic datasets showed competitive performance compared to more classical Genetic Programming (GP) algorithms. Unlike their GP counterparts, these neural approaches are trained to generate expressions from datasets given as context. This allows them to produce accurate expressions in a single forward pass at test time. However, they usually do not benefit from search abilities, which result in low performance compared to GP on out-of-distribution datasets. In this paper, we propose a novel method which provides the best of both worlds, based on a Monte-Carlo Tree Search procedure using a context-aware neural mutation model, which is initially pre-trained to learn promising mutations, and further refined from successful experiences in an online fashion. The approach demonstrates state-of-the-art performance on the well-known \texttt{SRBench} benchmark.

Marco Virgolin, Solon P. Pissis

Symbolic regression (SR) is the task of learning a model of data in the form of a mathematical expression. By their nature, SR models have the potential to be accurate and human-interpretable at the same time. Unfortunately, finding such models, i.e., performing SR, appears to be a computationally intensive task. Historically, SR has been tackled with heuristics such as greedy or genetic algorithms and, while some works have hinted at the possible hardness of SR, no proof has yet been given that SR is, in fact, NP-hard. This begs the question: Is there an exact polynomial-time algorithm to compute SR models? We provide evidence suggesting that the answer is probably negative by showing that SR is NP-hard.

Thomas Uriot, Marco Virgolin, Tanja Alderliesten et al.

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.

Davide Tugnoli, Andrea De Lorenzo, Marco Virgolin et al.

Synthetic tabular data generation addresses data scarcity and privacy constraints in a variety of domains. Tabular Prior-Data Fitted Network (TabPFN), a recent foundation model for tabular data, has been shown capable of generating high-quality synthetic tabular data. However, TabPFN is autoregressive: features are generated sequentially by conditioning on the previous ones, depending on the order in which they appear in the input data. We demonstrate that when the feature order conflicts with causal structure, the model produces spurious correlations that impair its ability to generate synthetic data and preserve causal effects. We address this limitation by integrating causal structure into TabPFN's generation process through two complementary approaches: Directed Acyclic Graph (DAG)-aware conditioning, which samples each variable given its causal parents, and a Completed Partially Directed Acyclic Graph (CPDAG)-based strategy for scenarios with partial causal knowledge. We evaluate these approaches on controlled benchmarks and six CSuite datasets, assessing structural fidelity, distributional alignment, privacy preservation, and Average Treatment Effect (ATE) preservation. Across most settings, DAG-aware conditioning improves the quality and stability of synthetic data relative to vanilla TabPFN. The CPDAG-based strategy shows moderate improvements, with effectiveness depending on the number of oriented edges. These results indicate that injecting causal structure into autoregressive generation enhances the reliability of synthetic tabular data.

Luigi Rovito, Marco Virgolin

Survival Regression (SuR) is a key technique for modeling time to event in important applications such as clinical trials and semiconductor manufacturing. Currently, SuR algorithms belong to one of three classes: non-linear black-box -- allowing adaptability to many datasets but offering limited interpretability (e.g., tree ensembles); linear glass-box -- being easier to interpret but limited to modeling only linear interactions (e.g., Cox proportional hazards); and non-linear glass-box -- allowing adaptability and interpretability, but empirically found to have several limitations (e.g., explainable boosting machines, survival trees). In this work, we investigate whether Symbolic Regression (SR), i.e., the automated search of mathematical expressions from data, can lead to non-linear glass-box survival models that are interpretable and accurate. We propose an evolutionary, multi-objective, and multi-expression implementation of SR adapted to SuR. Our empirical results on five real-world datasets show that SR consistently outperforms traditional glass-box methods for SuR in terms of accuracy per number of dimensions in the model, while exhibiting comparable accuracy with black-box methods. Furthermore, we offer qualitative examples to assess the interpretability potential of SR models for SuR. Code at: https://github.com/lurovi/SurvivalMultiTree-pyNSGP.

Marco Virgolin, Saverio Fracaros

Counterfactual explanations (CEs) are a powerful means for understanding how decisions made by algorithms can be changed. Researchers have proposed a number of desiderata that CEs should meet to be practically useful, such as requiring minimal effort to enact, or complying with causal models. We consider a further aspect to improve the usability of CEs: robustness to adverse perturbations, which may naturally happen due to unfortunate circumstances. Since CEs typically prescribe a sparse form of intervention (i.e., only a subset of the features should be changed), we study the effect of addressing robustness separately for the features that are recommended to be changed and those that are not. Our definitions are workable in that they can be incorporated as penalty terms in the loss functions that are used for discovering CEs. To experiment with robustness, we create and release code where five data sets (commonly used in the field of fair and explainable machine learning) have been enriched with feature-specific annotations that can be used to sample meaningful perturbations. Our experiments show that CEs are often not robust and, if adverse perturbations take place (even if not worst-case), the intervention they prescribe may require a much larger cost than anticipated, or even become impossible. However, accounting for robustness in the search process, which can be done rather easily, allows discovering robust CEs systematically. Robust CEs make additional intervention to contrast perturbations much less costly than non-robust CEs. We also find that robustness is easier to achieve for the features to change, posing an important point of consideration for the choice of what counterfactual explanation is best for the user. Our code is available at: https://github.com/marcovirgolin/robust-counterfactuals.

William La Cava, Patryk Orzechowski, Bogdan Burlacu et al.

Many promising approaches to symbolic regression have been presented in recent years, yet progress in the field continues to suffer from a lack of uniform, robust, and transparent benchmarking standards. In this paper, we address this shortcoming by introducing an open-source, reproducible benchmarking platform for symbolic regression. We assess 14 symbolic regression methods and 7 machine learning methods on a set of 252 diverse regression problems. Our assessment includes both real-world datasets with no known model form as well as ground-truth benchmark problems, including physics equations and systems of ordinary differential equations. For the real-world datasets, we benchmark the ability of each method to learn models with low error and low complexity relative to state-of-the-art machine learning methods. For the synthetic problems, we assess each method's ability to find exact solutions in the presence of varying levels of noise. Under these controlled experiments, we conclude that the best performing methods for real-world regression combine genetic algorithms with parameter estimation and/or semantic search drivers. When tasked with recovering exact equations in the presence of noise, we find that deep learning and genetic algorithm-based approaches perform similarly. We provide a detailed guide to reproducing this experiment and contributing new methods, and encourage other researchers to collaborate with us on a common and living symbolic regression benchmark.

Marco Virgolin

Learning ensembles by bagging can substantially improve the generalization performance of low-bias, high-variance estimators, including those evolved by Genetic Programming (GP). To be efficient, modern GP algorithms for evolving (bagging) ensembles typically rely on several (often inter-connected) mechanisms and respective hyper-parameters, ultimately compromising ease of use. In this paper, we provide experimental evidence that such complexity might not be warranted. We show that minor changes to fitness evaluation and selection are sufficient to make a simple and otherwise-traditional GP algorithm evolve ensembles efficiently. The key to our proposal is to exploit the way bagging works to compute, for each individual in the population, multiple fitness values (instead of one) at a cost that is only marginally higher than the one of a normal fitness evaluation. Experimental comparisons on classification and regression tasks taken and reproduced from prior studies show that our algorithm fares very well against state-of-the-art ensemble and non-ensemble GP algorithms. We further provide insights into the proposed approach by (i) scaling the ensemble size, (ii) ablating the changes to selection, (iii) observing the evolvability induced by traditional subtree variation. Code: https://github.com/marcovirgolin/2SEGP.

Dazhuang Liu, Marco Virgolin, Tanja Alderliesten et al.

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.

Marco Virgolin, Andrea De Lorenzo, Tanja Alderliesten et al.

Emotion recognition in children can help the early identification of, and intervention on, psychological complications that arise in stressful situations such as cancer treatment. Though deep learning models are increasingly being adopted, data scarcity is often an issue in pediatric medicine, including for facial emotion recognition in children. In this paper, we study the application of data augmentation-based contrastive learning to overcome data scarcity in facial emotion recognition for children. We explore the idea of ignoring generational gaps, by adding abundantly available adult data to pediatric data, to learn better representations. We investigate different ways by which adult facial expression images can be used alongside those of children. In particular, we propose to explicitly incorporate within each mini-batch adult images as augmentations for children's. Out of $84$ combinations of learning approaches and training set sizes, we find that supervised contrastive learning with the proposed training scheme performs best, reaching a test accuracy that typically surpasses the one of the second-best approach by 2% to 3%. Our results indicate that adult data can be considered to be a meaningful augmentation of pediatric data for the recognition of emotional facial expression in children, and open up the possibility for other applications of contrastive learning to improve pediatric care by complementing data of children with that of adults.

Mattias Wahde, Marco Virgolin

In this chapter, we provide a review of conversational agents (CAs), discussing chatbots, intended for casual conversation with a user, as well as task-oriented agents that generally engage in discussions intended to reach one or several specific goals, often (but not always) within a specific domain. We also consider the concept of embodied conversational agents, briefly reviewing aspects such as character animation and speech processing. The many different approaches for representing dialogue in CAs are discussed in some detail, along with methods for evaluating such agents, emphasizing the important topics of accountability and interpretability. A brief historical overview is given, followed by an extensive overview of various applications, especially in the fields of health and education. We end the chapter by discussing benefits and potential risks regarding the societal impact of current and future CA technology.

Arkadiy Dushatskiy, Marco Virgolin, Anton Bouter et al.

When it comes to solving optimization problems with evolutionary algorithms (EAs) in a reliable and scalable manner, detecting and exploiting linkage information, i.e., dependencies between variables, can be key. In this article, we present the latest version of, and propose substantial enhancements to, the Gene-pool Optimal Mixing Evoutionary 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 9 black-box problems that can only be solved efficiently 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.

Mattias Wahde, Marco Virgolin

In this position paper, we present five key principles, namely interpretability, inherent capability to explain, independent data, interactive learning, and inquisitiveness, for the development of conversational AI that, unlike the currently popular black box approaches, is transparent and accountable. At present, there is a growing concern with the use of black box statistical language models: While displaying impressive average performance, such systems are also prone to occasional spectacular failures, for which there is no clear remedy. In an effort to initiate a discussion on possible alternatives, we outline and exemplify how our five principles enable the development of conversational AI systems that are transparent and thus safer for use. We also present some of the challenges inherent in the implementation of those principles.

Marco Virgolin, Andrea De Lorenzo, Francesca Randone et al.

High-stakes applications require AI-generated models to be interpretable. Current algorithms for the synthesis of potentially interpretable models rely on objectives or regularization terms that represent interpretability only coarsely (e.g., model size) and are not designed for a specific user. Yet, interpretability is intrinsically subjective. In this paper, we propose an approach for the synthesis of models that are tailored to the user by enabling the user to steer the model synthesis process according to her or his preferences. We use a bi-objective evolutionary algorithm to synthesize models with trade-offs between accuracy and a user-specific notion of interpretability. The latter is estimated by a neural network that is trained concurrently to the evolution using the feedback of the user, which is collected using uncertainty-based active learning. To maximize usability, the user is only asked to tell, given two models at the time, which one is less complex. With experiments on two real-world datasets involving 61 participants, we find that our approach is capable of learning estimations of interpretability that can be very different for different users. Moreover, the users tend to prefer models found using the proposed approach over models found using non-personalized interpretability indices.

Marco Virgolin, Andrea De Lorenzo, Eric Medvet et al.

Many risk-sensitive applications require Machine Learning (ML) models to be interpretable. Attempts to obtain interpretable models typically rely on tuning, by trial-and-error, hyper-parameters of model complexity that are only loosely related to interpretability. We show that it is instead possible to take a meta-learning approach: an ML model of non-trivial Proxies of Human Interpretability (PHIs) can be learned from human feedback, then this model can be incorporated within an ML training process to directly optimize for interpretability. We show this for evolutionary symbolic regression. We first design and distribute a survey finalized at finding a link between features of mathematical formulas and two established PHIs, simulatability and decomposability. Next, we use the resulting dataset to learn an ML model of interpretability. Lastly, we query this model to estimate the interpretability of evolving solutions within bi-objective genetic programming. We perform experiments on five synthetic and eight real-world symbolic regression problems, comparing to the traditional use of solution size minimization. The results show that the use of our model leads to formulas that are, for a same level of accuracy-interpretability trade-off, either significantly more or equally accurate. Moreover, the formulas are also arguably more interpretable. Given the very positive results, we believe that our approach represents an important stepping stone for the design of next-generation interpretable (evolutionary) ML algorithms.

Marco Virgolin, Ziyuan Wang, Tanja Alderliesten et al.

Machine Learning (ML) is proving extremely beneficial in many healthcare applications. In pediatric oncology, retrospective studies that investigate the relationship between treatment and late adverse effects still rely on simple heuristics. To assess the effects of radiation therapy, treatment plans are typically simulated on phantoms, i.e., virtual surrogates of patient anatomy. Currently, phantoms are built according to reasonable, yet simple, human-designed criteria. This often results in a lack of individualization. We present a novel approach that combines imaging and ML to build individualized phantoms automatically. Given the features of a patient treated historically (only 2D radiographs available), and a database of 3D Computed Tomography (CT) imaging with organ segmentations and relative patient features, our approach uses ML to predict how to assemble a patient-specific phantom automatically. Experiments on 60 abdominal CTs of pediatric patients show that our approach constructs significantly more representative phantoms than using current phantom building criteria, in terms of location and shape of the abdomen and of two considered organs, the liver and the spleen. Among several ML algorithms considered, the Gene-pool Optimal Mixing Evolutionary Algorithm for Genetic Programming (GP-GOMEA) is found to deliver the best performing models, which are, moreover, transparent and interpretable mathematical expressions.

Marco Virgolin, Tanja Alderliesten, Peter A. N. Bosman

Feature construction can substantially improve the accuracy of Machine Learning (ML) algorithms. Genetic Programming (GP) has been proven to be effective at this task by evolving non-linear combinations of input features. GP additionally has the potential to improve ML explainability since explicit expressions are evolved. Yet, in most GP works the complexity of evolved features is not explicitly bound or minimized though this is arguably key for explainability. In this article, we assess to what extent GP still performs favorably at feature construction when constructing features that are (1) Of small-enough number, to enable visualization of the behavior of the ML model; (2) Of small-enough size, to enable interpretability of the features themselves; (3) Of sufficient informative power, to retain or even improve the performance of the ML algorithm. We consider a simple feature construction scheme using three different GP algorithms, as well as random search, to evolve features for five ML algorithms, including support vector machines and random forest. Our results on 21 datasets pertaining to classification and regression problems show that constructing only two compact features can be sufficient to rival the use of the entire original feature set. We further find that a modern GP algorithm, GP-GOMEA, performs best overall. These results, combined with examples that we provide of readable constructed features and of 2D visualizations of ML behavior, lead us to positively conclude that GP-based feature construction still works well when explicitly searching for compact features, making it extremely helpful to explain ML models.

Marco Virgolin, Tanja Alderliesten, Cees Witteveen et al.

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, i.e., 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.