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 glassbox 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 glassbox 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.

Symbolic regression searches for analytic expressions that accurately describe studied phenomena. The main promise of this approach is that it may return an interpretable model that can be insightful to users, while maintaining high accuracy. The current standard for benchmarking these algorithms is SRBench, which evaluates methods on hundreds of datasets that are a mix of real-world and simulated processes spanning multiple domains. At present, the ability of SRBench to evaluate interpretability is limited to measuring the size of expressions on real-world data, and the exactness of model forms on synthetic data. In practice, model size is only one of many factors used by subject experts to determine how interpretable a model truly is. Furthermore, SRBench does not characterize algorithm performance on specific, challenging sub-tasks of regression such as feature selection and evasion of local minima. In this work, we propose and evaluate an approach to benchmarking SR algorithms that addresses these limitations of SRBench by 1) incorporating expert evaluations of interpretability on a domain-specific task, and 2) evaluating algorithms over distinct properties of data science tasks. We evaluate 12 modern symbolic regression algorithms on these benchmarks and present an in-depth analysis of the results, discuss current challenges of symbolic regression algorithms and highlight possible improvements for the benchmark itself.

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.

Interpretability is a critical aspect to ensure a fair and responsible use of machine learning (ML) in high-stakes applications. Genetic programming (GP) has been used to obtain interpretable ML models because it operates at the level of functional building blocks: if these building blocks are interpretable, there is a chance that their composition (i.e., the entire ML model) is also interpretable. However, the degree to which a model is interpretable depends on the observer. Motivated by this, we study a recently-introduced human-in-the-loop system that allows the user to steer GP’s generation process to their preferences, which shall be online-learned by an artificial neural network (ANN). We focus on the generation of ML models as analytical functions (i.e., symbolic regression) as this is a key problem in interpretable ML, and propose a two-fold contribution. First, we devise more general representations for the ML models for the ANN to learn upon, to enable the application of the system to a wider range of problems. Second, we delve into a deeper analysis of the system’s components. To this end, we propose an incremental experimental evaluation, aimed at (1) studying the effectiveness by which an ANN can capture the perceived interpretability for simulated users, (2) investigating how the GP’s outcome is affected across different simulated user feedback profiles, and (3) determining whether humans participants would prefer models that were generated with or without their involvement. Our results pose clarity on pros and cons of using a human-in-the-loop approach to discover interpretable ML models with GP.

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.

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) ones. 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 SRBench benchmark.

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. In this paper, we consider the interplay between the desiderata of robustness (i.e., that enacting CEs remains feasible and cost-effective even if adverse events take place) and sparsity (i.e., that CEs require only a subset of the features to be changed). In particular, we study the effect of addressing robustness separately for the features that are recommended to be changed and those that are not. We provide definitions of robustness for sparse CEs that are workable in that they can be incorporated as penalty terms in the loss functions that are used for discovering CEs. To carry out our experiments, 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.

We present a detailed implementation of five core principles for transparent and acccountable conversational AI, namely interpretability, inherent capability to explain, independent data, interactive learning, and inquisitiveness. This implementation is a dialogue manager called DAISY that serves as the core part of a conversational agent. We show how DAISY-based agents are trained with human-machine interaction, a process that also involves suggestions for generalization from the agent itself. Moreover, these agents are capable to provide a concise and clear explanation of the actions required to reach a conclusion. Deep neural networks (DNNs) are currently the de facto standard in conversational AI. We therefore formulate a comparison between DAISY-based agents and two methods that use DNNs, on two popular data sets involving multi-domain task-oriented dialogue. Specifically, we provide quantitative results related to entity retrieval and qualitative results in terms of the type of errors that may occur. The results show that DAISY-based agents achieve superior precision at the price of lower recall, an outcome that might be preferable in task-oriented settings. Ultimately, and especially in view of their high degree of interpretability, DAISY-based agents are a fundamentally different alternative to the currently popular DNN-based methods.

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

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.

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.

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.

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 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 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.

There is an increasing interest in the development of techniques for automatic relation extraction from unstructured text. The biomedical domain, in particular, is a sector that may greatly benefit from those techniques due to the huge and ever increasing amount of scientific publications describing observed phenomena of potential clinical interest. In this paper, we consider the problem of automatically identifying sentences that contain interactions between genes and proteins, based solely on a dictionary of genes and proteins and a small set of sample sentences in natural language. We propose an evolutionary technique for learning a classifier that is capable of detecting the desired sentences within scientific publications with high accuracy. The key feature of our proposal, that is internally based on Genetic Programming, is the construction of a model of the relevant syntax patterns in terms of standard part-of-speech annotations. The model consists of a set of regular expressions that are learned automatically despite the large alphabet size involved. We assess our approach on two realistic datasets and obtain 74% accuracy, a value sufficiently high to be of practical interest and that is in line with significant baseline methods.