Pragmatic Curiosity: A Hybrid Learning-Optimization Paradigm via Active Inference

TL;DR

This work introduces pragmatic curiosity, an Active Inference–based hybrid learning-optimization framework that unifies goal-directed optimization and information gathering for expensive black-box evaluations. By minimizing the expected free energy, the method couples pragmatic preferences with epistemic information gain, recovering BO- and BED-like behavior as limiting cases. The authors demonstrate strong empirical gains across constrained system identification, targeted active search, and composite optimization with unknown preferences, including settings with evolving or implicit goals. The approach yields improved estimation accuracy, broader region coverage, and higher-quality final solutions within fixed budgets, with broad implications for environmental monitoring, autonomous systems, and energy-resource design.

Abstract

Many engineering and scientific workflows depend on expensive black-box evaluations, requiring decision-making that simultaneously improves performance and reduces uncertainty. Bayesian optimization (BO) and Bayesian experimental design (BED) offer powerful yet largely separate treatments of goal-seeking and information-seeking, providing limited guidance for hybrid settings where learning and optimization are intrinsically coupled. We propose "pragmatic curiosity," a hybrid learning-optimization paradigm derived from active inference, in which actions are selected by minimizing the expected free energy--a single objective that couples pragmatic utility with epistemic information gain. We demonstrate the practical effectiveness and flexibility of pragmatic curiosity on various real-world hybrid tasks, including constrained system identification, targeted active search, and composite optimization with unknown preferences. Across these benchmarks, pragmatic curiosity consistently outperforms strong BO-type and BED-type baselines, delivering higher estimation accuracy, better critical-region coverage, and improved final solution quality.

Figures (5)

• Figure 1: Performance evaluation for constrained system identification on environmental monitoring in 2d plume fields. Error bars represent $\pm 1$ std over 5 seeds.
• Figure 2: Performance evaluation for targeted active search on failure discovery in autonomous driving scenarios. Error bars represent $\pm 1$ std over 4 seeds.
• Figure 3: Performance evaluation for composite BO with unknown preferences. Error bars represent $\pm 1$ std over 20 seeds for vehicle safety, penicillin, and 5 for energy resource allocation.
• Figure 4: Examples of missed object detection by YOLO due to two reasons considered in perception failure case study in Section \ref{['appendix: carla']}. Fig (left to right): example of Failure-1 (distance), Failure-2 (poor light) and Failure-1 and Failure-2 both in one scene (distance and poor light), respectively. Bounding boxes for detected objects (misdetections) shown in yellow (red) with detection confidence numbers. Each scene has two cars and a pedestrian.
• Figure 5: Extended baseline comparison with BOPE for energy resource allocation. Error bars represent $\pm 1$ standard deviation over 5 seeds.

Theorems & Definitions (9)

• Theorem 3.1
• proof
• Lemma 3.2
• proof
• proof
• Lemma 3.1
• proof
• Definition 3.2
• proof