Learning a Formula of Interpretability to Learn Interpretable Formulas

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.

Machine learning for automatic construction of pseudo-realistic pediatric abdominal phantoms

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.

On Explaining Machine Learning Models by Evolving Crucial and Compact Features

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 four ML algorithms, including support vector machines and random forest. Our results on 20 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.

A Model-based Genetic Programming Approach for Symbolic Regression of Small Expressions

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 randomly, 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.