Symbolic Regression is a powerful data-driven technique that searches for mathematical expressions that explain the relationship between input variables and a target of interest. Due to its efficiency and flexibility, Genetic Programming can be seen as the standard search technique for Symbolic Regression. However, the conventional Genetic Programming algorithm requires storing all data in a central location, which is not always feasible due to growing concerns about data privacy and security. While privacy-preserving research has advanced recently and might offer a solution to this problem, their application to Symbolic Regression remains largely unexplored. Furthermore, the existing work only focuses on the horizontally partitioned setting, whereas the vertically partitioned setting, another popular scenario, has yet to be investigated. Herein, we propose an approach that employs a privacy-preserving technique called Secure Multiparty Computation to enable parties to jointly build Symbolic Regression models in the vertical scenario without revealing private data. Preliminary experimental results indicate that our proposed method delivers comparable performance to the centralized solution while safeguarding data privacy.

The scores of distance-based outlier detection methods are difficult to interpret, making it challenging to determine a cut-off threshold between normal and outlier data points without additional context. We describe a generic transformation of distance-based outlier scores into interpretable, probabilistic estimates. The transformation is ranking-stable and increases the contrast between normal and outlier data points. Determining distance relationships between data points is necessary to identify the nearest-neighbor relationships in the data, yet, most of the computed distances are typically discarded. We show that the distances to other data points can be used to model distance probability distributions and, subsequently, use the distributions to turn distance-based outlier scores into outlier probabilities. Our experiments show that the probabilistic transformation does not impact detection performance over numerous tabular and image benchmark datasets but results in interpretable outlier scores with increased contrast between normal and outlier samples. Our work generalizes to a wide range of distance-based outlier detection methods, and because existing distance computations are used, it adds no significant computational overhead.

Vectorial Genetic Programming (Vec-GP) extends GP by allowing vectors as input features along regular, scalar features, using them by applying arithmetic operations component-wise or aggregating vectors into scalars by some aggregation function. Vec-GP also allows aggregating vectors only over a limited segment of the vector instead of the whole vector, which offers great potential but also introduces new parameters that GP has to optimize. This paper formalizes an optimization problem to analyze different strategies for optimizing a window for aggregation functions. Different strategies are presented, included random and guided sampling, where the latter leverages information from an approximated gradient. Those strategies can be applied as a simple optimization algorithm, which itself ca be applied inside a specialized mutation operator within GP. The presented results indicate, that the different random sampling strategies do not impact the overall algorithm performance significantly, and that the guided strategies suffer from becoming stuck in local optima. However, results also indicate, that there is still potential in discovering more efficient algorithms that could outperform the presented strategies.

Particle-based modeling of materials at atomic scale plays an important role in the development of new materials and understanding of their properties. The accuracy of particle simulations is determined by interatomic potentials, which allow to calculate the potential energy of an atomic system as a function of atomic coordinates and potentially other properties. First-principles-based ab initio potentials can reach arbitrary levels of accuracy, however their aplicability is limited by their high computational cost. Machine learning (ML) has recently emerged as an effective way to offset the high computational costs of ab initio atomic potentials by replacing expensive models with highly efficient surrogates trained on electronic structure data. Among a plethora of current methods, symbolic regression (SR) is gaining traction as a powerful "white-box" approach for discovering functional forms of interatomic potentials. This contribution discusses the role of symbolic regression in Materials Science (MS) and offers a comprehensive overview of current methodological challenges and state-of-the-art results. A genetic programming-based approach for modeling atomic potentials from raw data (consisting of snapshots of atomic positions and associated potential energy) is presented and empirically validated on ab initio electronic structure data.

This work provides the exact expression of the probability distribution of the hypervolume improvement (HVI) for bi-objective generalization of Bayesian optimization. Here, instead of a single-objective improvement, we consider the improvement of the hypervolume indicator concerning the current best approximation of the Pareto front. Gaussian process regression models are trained independently on both objective functions, resulting in a bi-variate separated Gaussian distribution serving as a predictive model for the vector-valued objective function. Some commonly HVI-based acquisition functions (probability of improvement and upper confidence bound) are also leveraged with the help of the exact distribution of HVI. In addition, we show the superior numerical accuracy and efficiency of the exact distribution compared to the commonly used approximation by Monte-Carlo sampling. Finally, we benchmark distribution-leveraged acquisition functions on the widely applied ZDT problem set, demonstrating a significant advantage of using the exact distribution of HVI in multi-objective Bayesian optimization.

Real-world project scheduling often requires flexibility in terms of the selection and the exact length of alternative production activities. Moreover, the simultaneous scheduling of multiple lots is mandatory in many production planning applications. To meet these requirements, a new flexible resource-constrained multi-project scheduling problem is introduced where both decisions (activity selection flexibility and time flexibility) are integrated. Besides the minimization of makespan, two alternative objectives inspired by a steel industry application case are presented: maximization of balanced length of selected activities (time balance) and maximization of balanced resource utilization (resource balance). New mixed integer and constraint programming (CP) models are proposed for the developed integrated flexible project scheduling problem. The real-world applicability of the suggested CP models is shown by solving large steel industry instances with the CP Optimizer of IBM ILOG CPLEX. Furthermore, benchmark instances on flexible resource-constrained project scheduling problems (RCPSP) are solved to optimality.