Seven Cost Functions Tested Across 20 Domains

Inductive logic programming learns logical hypotheses from data, but choosing an objective remains a challenge. In a recent issue of the Journal of Artificial Intelligence Research (JAIR), Céline Hocquette and Andrew Cropper extended a constraint‑based ILP system to evaluate seven standard cost functions on over 20 diverse domains and 1,000 learning tasks, including game playing, program synthesis, and image reasoning. Their work was published on 2026/07/20 in JAIR ¹.

The paper describes how each ILP system minimizes a chosen cost function—such as minimizing training error, textual complexity, or hypothesis description length—to produce optimal hypotheses for the training data. The authors implemented these objectives within their framework and empirically compared the resulting generalization errors across all tasks. They found that no single cost function dominated; however, minimising training error or description length yielded the best overall performance, while simply reducing hypothesis size did not always improve generalization.

Expert context highlights that this study builds on earlier work by Kodituwaka et al., who first championed description length minimization in 2005. By evaluating multiple cost terms across many domains, Hocquette and Cropper provide much needed guidance for practitioners often uncertain which loss term to adopt. Yet they caution that results may not translate directly into other machine learning paradigms such as deep neural networks, where inductive biases differ substantially. Future research should explore how hypothesis complexity interacts with generalization performance within each domain studied, potentially extending the current methodology to broader AI fields.

Methodologically, the authors extended an existing constraint‑based ILP engine to support seven distinct objective functions without altering the underlying search algorithm. Each task involved a set of background facts, positive examples, and negative examples; the system then generated optimal hypotheses under each cost metric. The evaluation measured test accuracy on unseen data, revealing consistent patterns: training‑error minimization produced robust models, whereas description‑length minimization also achieved high scores but sometimes at the expense of slightly larger rule sets. In contrast, minimizing hypothesis size alone frequently led to over-simplified rules that failed to capture necessary structure.

The implications are practical. For developers building ILP systems—whether for knowledge base construction or automated theorem proving—the findings suggest prioritizing either empirical error reduction or concise descriptions rather than merely shrinking rule counts. Moreover, the study underscores the importance of representative training data.

Overall, Hocquette and Cropper’s comparative analysis offers a clear roadmap for selecting cost functions in inductive logic programming, backed by extensive experimentation across diverse application areas. Their results will help shape future ILP tool design and inform researchers about the trade‑offs inherent in different loss formulations. This work marks a significant step toward more reliable and interpretable AI reasoning systems.