arXiv subjects

Wolfgang Banzhaf

Publications and source records attributed to Wolfgang Banzhaf.

At least 19 recordsLinked to original sources

arXiv2026

Adaptive Protection for Evolutionary Feature Construction in Symbolic Regression with Application to Credit Classification

Evolutionary feature construction has shown strong promise in symbolic regression by automatically discovering informative transformations of input features that enhance a simple base learner. However, existing approaches often lack explicit mechanisms to preserve important constructed features discovered during evolution, and valuable genetic material can be lost when genetic operators disrupt effective features. This paper introduces an adaptive protection mechanism that leverages feature importance metrics to selectively preserve constructed features during evolution. The mechanism provides stronger protection for more important constructed features while still allowing less important features to be modified and to incorporate useful building blocks from more important features. We evaluate the approach using multiple feature importance calculation methods and demonstrate its robustness across different base learners. Experimental results on 98 regression benchmark datasets show that the proposed mechanism consistently improves solution quality over baseline approaches, and experiments on two credit classification datasets demonstrate that the method also extends effectively to improve search effectiveness beyond symbolic regression.

cs.LG↗

arXiv2026

Benchmarking Zero-Shot LLM-Generated Parent Selection in Genetic Programming for Symbolic Regression

Parent selection significantly affects exploration, exploitation, and complexity control in genetic programming (GP) for symbolic regression. It is unclear whether large language models (LLMs) can synthesize effective operators in a zero-shot setting without iterative meta-evolution. Here, zero-shot means that the model receives only the task description, with no reference operators or iterative feedback. In this work, we benchmark zero-shot synthesis of parent-selection operators across eight LLMs within a standard GP framework for symbolic regression. Each model receives the same natural-language prompt to generate a parent-selection operator, which is then evaluated in a standard GP framework with only the parent-selection operator replaced, while all other components and the evolutionary-search budget are held constant. For each LLM, ten independent zero-shot operators are evaluated on twelve OpenML regression benchmarks and compared against automatic lexicase and tournament selection baselines. Claude Sonnet~4.6 and Gemini~3.1 Pro stand out for consistently strong performance on both training and held-out test $R^2$. The strongest operator in our benchmark---a Kimi~K2.5 zero-shot synthesis---surpasses the automatic lexicase and tournament baselines in search effectiveness. These results suggest that zero-shot LLM synthesis is a viable approach to generating competitive GP selection operators. Analysis shows that many generated operators use semantics to guide selection, suggesting that LLMs can produce non-trivial search heuristics from the task description alone. We also examine the relationship between public LLM leaderboard rankings and GP performance. Widely used benchmarks, such as Humanity's Last Exam and SWE-bench Verified, strongly correlate with training $R^2$, while their relationship to held-out test $R^2$ is weaker and less clear.

cs.NE↗

arXiv2026

Open Questions about Time and Self-reference in Living Systems

Living systems exhibit a range of fundamental characteristics: they are active, self-referential, self-modifying systems. This paper explores how these characteristics create challenges for conventional scientific approaches and why they require new theoretical and formal frameworks. We introduce a distinction between 'natural time', the continuing present of physical processes, and 'representational time', with its framework of past, present and future that emerges with life itself. Representational time enables memory, learning and prediction, functions of living systems essential for their survival. Through examples from evolution, embryogenesis and metamorphosis we show how living systems navigate the apparent contradictions arising from self-reference as natural time unwinds self-referential loops into developmental spirals. Conventional mathematical and computational formalisms struggle to model self-referential and self-modifying systems without running into paradox. We identify promising new directions for modelling self-referential systems, including domain theory, co-algebra, genetic programming, and self-modifying algorithms. There are broad implications for biology, cognitive science and social sciences, because self-reference and self-modification are not problems to be avoided but core features of living systems that must be modelled to understand life's open-ended creativity.

cs.ET↗

arXiv2026

The Effects of Population Size on the Performance of BEAGLE GPU-Based Genetic Programming Runs

The Beagle framework, through GPU-based Genetic Programming, enables population dynamics previously unattainable (within practical time frames) by CPU-constrained Genetic Programming systems. This work explores how GPU-enabled population sizes impact the success of training for symbolic regression problems. Specifically, when using constant population sizes, we see benefits of using very narrow and deep searches (as narrow as 1000 individuals) for some problems, while other problems benefit from very broad and shallow searches (as broad as 10 million individuals). We also explore stepped population sizes that start with large populations and drop to small populations to balance the breadth and depth of search.

cs.NE↗

arXiv2026

ECLIPSE: An Evolutionary Computation Library for Instrumentation Prototyping in Scientific Engineering

Designing scientific instrumentation often requires exploring large, highly constrained design spaces using computationally expensive physics simulations. These simulators pose substantial challenges for integrating evolutionary computation (EC) into scientific design workflows. EC typically requires numerous design evaluations, making the integration of slow, low-throughput simulators challenging, as they are optimized for accuracy and ease of use rather than throughput. We present ECLIPSE, an evolutionary computation framework built to interface directly with complex, domain-specific simulation tools while supporting flexible geometric and parametric representations of scientific hardware. ECLIPSE provides a modular architecture consisting of (1) Individuals, which encode hardware designs using domain-aware, physically constrained representations; (2) Evaluators, which prepare simulation inputs, invoke external simulators, and translate the simulator's outputs into fitness measures; and (3) Evolvers, which implement EC algorithms suitable for this domain. We evolve solutions for two novel space-science applications: 3D antennas optimized for directional sensitivity and spacecraft geometries optimized for drag reduction. Notably, we identify antennas with directional sensitivity roughly comparable to the expected sensitivity of two-antenna interferometric arrays, representing potential cost-savings. ECLIPSE enables interdisciplinary teams of physicists, engineers, and EC researchers to collaboratively explore designs for scientific hardware while leveraging existing domain-specific simulation software.

cs.NE↗

arXiv2026

LLM-Meta-SR: In-Context Learning for Evolving Selection Operators in Symbolic Regression

Large language models (LLMs) have revolutionized algorithm development, yet their application in symbolic regression, where algorithms automatically discover symbolic expressions from data, remains limited. In this paper, we propose a meta-learning framework that enables LLMs to automatically design selection operators for evolutionary symbolic regression algorithms. We first identify two key limitations in existing LLM-based algorithm evolution techniques: lack of semantic guidance and code bloat. The absence of semantic awareness can lead to ineffective exchange of useful code components, while bloat results in unnecessarily complex components; both can hinder evolutionary learning progress or reduce the interpretability of the designed algorithm. To address these issues, we enhance the LLM-based evolution framework for meta-symbolic regression with two key innovations: a complementary, semantics-aware selection operator and bloat control. Additionally, we embed domain knowledge into the prompt, enabling the LLM to generate more effective and contextually relevant selection operators. Our experimental results on symbolic regression benchmarks show that LLMs can devise selection operators that outperform nine expert-designed baselines, achieving state-of-the-art performance. Moreover, the evolved operator can further improve a state-of-the-art symbolic regression algorithm, achieving the best performance among 28 symbolic regression and other machine learning algorithms across 116 regression datasets. This demonstrates that LLMs can exceed expert-level algorithm design for symbolic regression.

cs.NE↗

arXiv2026

EvoIQA - Explaining Image Distortions with Evolved White-Box Logic

Traditional Image Quality Assessment (IQA) metrics typically fall into one of two extremes: rigid, hand-crafted mathematical models or "black-box" deep learning architectures that completely lack interpretability. To bridge this gap, we propose EvoIQA, a fully explainable symbolic regression framework based on Genetic Programming that Evolves explicit, human-readable mathematical formulas for image quality assessment (IQA). Utilizing a rich terminal set from the VSI, VIF, FSIM, and HaarPSI metrics, our framework inherently maps structural, chromatic, and information-theoretic degradations into observable mathematical equations. Our results demonstrate that the evolved GP models consistently achieve strong alignment between the predictions and human visual preferences. Furthermore, they not only outperform traditional hand-crafted metrics but also achieve performance parity with complex, state-of-the-art deep learning models like DB-CNN, proving that we no longer have to sacrifice interpretability for state-of-the-art performance.

cs.CV↗

arXiv2026

GPU-Accelerated Genetic Programming for Symbolic Regression with Beagle Framework

Beagle is a new software framework that enables execution of Genetic Programming tasks on the GPU. Currently available for symbolic regression, it processes individuals of the population and fitness cases for training in a way that maximizes throughput on extant GPU platforms. In this contribution, we report on the benchmarking of Beagle on the Feynman Symbolic Regression dataset and compare its performance with a fast CPU system called StackGP and the widely available PySR system under the same wall clock budget. We also report on the use of two different fitness functions, one a point-to-point error function, the other a correlation fitness function. The results demonstrate that the Beagle's GPU-aided Symbolic Regression significantly outperforms leading CPU-based frameworks.

cs.NE↗

arXiv2026

Enhancing Generalization in Evolutionary Feature Construction for Symbolic Regression through Vicinal Jensen Gap Minimization

Genetic programming-based feature construction has achieved significant success in recent years as an automated machine learning technique to enhance learning performance. However, overfitting remains a challenge that limits its broader applicability. To improve generalization, we prove that vicinal risk, estimated through noise perturbation or mixup-based data augmentation, is bounded by the sum of empirical risk and a regularization term-either finite difference or the vicinal Jensen gap. Leveraging this decomposition, we propose an evolutionary feature construction framework that jointly optimizes empirical risk and the vicinal Jensen gap to control overfitting. Since datasets may vary in noise levels, we develop a noise estimation strategy to dynamically adjust regularization strength. Furthermore, to mitigate manifold intrusion-where data augmentation may generate unrealistic samples that fall outside the data manifold-we propose a manifold intrusion detection mechanism. Experimental results on 58 datasets demonstrate the effectiveness of Jensen gap minimization compared to other complexity measures. Comparisons with 15 machine learning algorithms further indicate that genetic programming with the proposed overfitting control strategy achieves superior performance.

cs.LG↗

arXiv2025

Node Preservation and its Effect on Crossover in Cartesian Genetic Programming

While crossover is a critical and often indispensable component in other forms of Genetic Programming, such as Linear- and Tree-based, it has consistently been claimed that it deteriorates search performance in CGP. As a result, a mutation-alone $(1+λ)$ evolutionary strategy has become the canonical approach for CGP. Although several operators have been developed that demonstrate an increased performance over the canonical method, a general solution to the problem is still lacking. In this paper, we compare basic crossover methods, namely one-point and uniform, to variants in which nodes are ``preserved,'' including the subgraph crossover developed by Roman Kalkreuth, the difference being that when ``node preservation'' is active, crossover is not allowed to break apart instructions. We also compare a node mutation operator to the traditional point mutation; the former simply replaces an entire node with a new one. We find that node preservation in both mutation and crossover improves search using symbolic regression benchmark problems, moving the field towards a general solution to CGP crossover.

cs.NE↗

arXiv2025

Genetic Programming for the Nuclear Many-Body Problem: a Guide

Genetic Programming is an evolutionary algorithm that generates computer programs, or mathematical expressions, to solve complex problems. In this Guide, we demonstrate how to use Genetic Programming to develop surrogate models to mitigate the computational costs of modeling atomic nuclei with ever increasing complexity. The computational burden escalates when uncertainty quantification is pursued, or when observables must be globally computed for thousands of nuclei. By studying three models in which the mean field depends on the total particle density self-consistently, we show that by constructing reduced order models supported by Genetic Programming one can speed up many-body computations by several orders of magnitude with a negligible loss in accuracy

nucl-th↗

arXiv2025

Bridging Fitness With Search Spaces By Fitness Supremums: A Theoretical Study on LGP

Genetic programming has undergone rapid development in recent years. However, theoretical studies of genetic programming are far behind. One of the major obstacles to theoretical studies is the challenge of developing a model to describe the relationship between fitness values and program genotypes. In this paper, we take linear genetic programming (LGP) as an example to study the fitness-to-genotype relationship. We find that the fitness expectation increases with fitness supremum over instruction editing distance, considering 1) the fitness supremum linearly increases with the instruction editing distance in LGP, 2) the fitness infimum is fixed, and 3) the fitness probabilities over different instruction editing distances are similar. We then extend these findings to explain the bloat effect and the minimum hitting time of LGP based on instruction editing distance. The bloat effect happens because it is more likely to produce better offspring by adding instructions than by removing them, given an instruction editing distance from the optimal program. The analysis of the minimum hitting time suggests that for a basic LGP genetic operator (i.e., freemut), maintaining a necessarily small program size and mutating multiple instructions each time can improve LGP performance. The reported empirical results verify our hypothesis.

cs.NE↗

arXiv2024

Multi-Representation Genetic Programming: A Case Study on Tree-based and Linear Representations

Existing genetic programming (GP) methods are typically designed based on a certain representation, such as tree-based or linear representations. These representations show various pros and cons in different domains. However, due to the complicated relationships among representation and fitness landscapes of GP, it is hard to intuitively determine which GP representation is the most suitable for solving a certain problem. Evolving programs (or models) with multiple representations simultaneously can alternatively search on different fitness landscapes since representations are highly related to the search space that essentially defines the fitness landscape. Fully using the latent synergies among different GP individual representations might be helpful for GP to search for better solutions. However, existing GP literature rarely investigates the simultaneous effective use of evolving multiple representations. To fill this gap, this paper proposes a multi-representation GP algorithm based on tree-based and linear representations, which are two commonly used GP representations. In addition, we develop a new cross-representation crossover operator to harness the interplay between tree-based and linear representations. Empirical results show that navigating the learned knowledge between basic tree-based and linear representations successfully improves the effectiveness of GP with solely tree-based or linear representation in solving symbolic regression and dynamic job shop scheduling problems.

cs.NE↗

arXiv2024

Sharpness-Aware Minimization in Genetic Programming

Sharpness-Aware Minimization (SAM) was recently introduced as a regularization procedure for training deep neural networks. It simultaneously minimizes the fitness (or loss) function and the so-called fitness sharpness. The latter serves as a measure of the nonlinear behavior of a solution and does so by finding solutions that lie in neighborhoods having uniformly similar loss values across all fitness cases. In this contribution, we adapt SAM for tree Genetic Programming (TGP) by exploring the semantic neighborhoods of solutions using two simple approaches. By capitalizing upon perturbing input and output of program trees, sharpness can be estimated and used as a second optimization criterion during the evolution. To better understand the impact of this variant of SAM on TGP, we collect numerous indicators of the evolutionary process, including generalization ability, complexity, diversity, and a recently proposed genotype-phenotype mapping to study the amount of redundancy in trees. The experimental results demonstrate that using any of the two proposed SAM adaptations in TGP allows (i) a significant reduction of tree sizes in the population and (ii) a decrease in redundancy of the trees. When assessed on real-world benchmarks, the generalization ability of the elite solutions does not deteriorate.

cs.NE↗

arXiv2024

Sharpness-Aware Minimization for Evolutionary Feature Construction in Regression

In recent years, genetic programming (GP)-based evolutionary feature construction has achieved significant success. However, a primary challenge with evolutionary feature construction is its tendency to overfit the training data, resulting in poor generalization on unseen data. In this research, we draw inspiration from PAC-Bayesian theory and propose using sharpness-aware minimization in function space to discover symbolic features that exhibit robust performance within a smooth loss landscape in the semantic space. By optimizing sharpness in conjunction with cross-validation loss, as well as designing a sharpness reduction layer, the proposed method effectively mitigates the overfitting problem of GP, especially when dealing with a limited number of instances or in the presence of label noise. Experimental results on 58 real-world regression datasets show that our approach outperforms standard GP as well as six state-of-the-art complexity measurement methods for GP in controlling overfitting. Furthermore, the ensemble version of GP with sharpness-aware minimization demonstrates superior performance compared to nine fine-tuned machine learning and symbolic regression algorithms, including XGBoost and LightGBM.

cs.LG↗

arXiv2024

On The Nature Of The Phenotype In Tree Genetic Programming

In this contribution, we discuss the basic concepts of genotypes and phenotypes in tree-based GP (TGP), and then analyze their behavior using five benchmark datasets. We show that TGP exhibits the same behavior that we can observe in other GP representations: At the genotypic level trees show frequently unchecked growth with seemingly ineffective code, but on the phenotypic level, much smaller trees can be observed. To generate phenotypes, we provide a unique technique for removing semantically ineffective code from GP trees. The approach extracts considerably simpler phenotypes while not being limited to local operations in the genotype. We generalize this transformation based on a problem-independent parameter that enables a further simplification of the exact phenotype by coarse-graining to produce approximate phenotypes. The concept of these phenotypes (exact and approximate) allows us to clarify what evolved solutions truly predict, making GP models considered at the phenotypic level much better interpretable.

cs.NE↗

arXiv2023

Discovering Adaptable Symbolic Algorithms from Scratch

Autonomous robots deployed in the real world will need control policies that rapidly adapt to environmental changes. To this end, we propose AutoRobotics-Zero (ARZ), a method based on AutoML-Zero that discovers zero-shot adaptable policies from scratch. In contrast to neural network adaptation policies, where only model parameters are optimized, ARZ can build control algorithms with the full expressive power of a linear register machine. We evolve modular policies that tune their model parameters and alter their inference algorithm on-the-fly to adapt to sudden environmental changes. We demonstrate our method on a realistic simulated quadruped robot, for which we evolve safe control policies that avoid falling when individual limbs suddenly break. This is a challenging task in which two popular neural network baselines fail. Finally, we conduct a detailed analysis of our method on a novel and challenging non-stationary control task dubbed Cataclysmic Cartpole. Results confirm our findings that ARZ is significantly more robust to sudden environmental changes and can build simple, interpretable control policies.

cs.RO↗

arXiv2023

Active Learning in Genetic Programming: Guiding Efficient Data Collection for Symbolic Regression

This paper examines various methods of computing uncertainty and diversity for active learning in genetic programming. We found that the model population in genetic programming can be exploited to select informative training data points by using a model ensemble combined with an uncertainty metric. We explored several uncertainty metrics and found that differential entropy performed the best. We also compared two data diversity metrics and found that correlation as a diversity metric performs better than minimum Euclidean distance, although there are some drawbacks that prevent correlation from being used on all problems. Finally, we combined uncertainty and diversity using a Pareto optimization approach to allow both to be considered in a balanced way to guide the selection of informative and unique data points for training.

cs.NE↗