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

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.

💡 Research Summary

The paper tackles a class of problems where learning (i.e., reducing uncertainty about a system) and optimization (i.e., achieving a performance goal) must be performed simultaneously. Traditional Bayesian Optimization (BO) focuses on goal‑directed sampling to locate the maximum of an unknown objective, while Bayesian Experimental Design (BED) concentrates on information‑directed sampling to maximize expected information gain about parameters of interest. Because BO and BED treat their objectives as separate, they provide limited guidance for hybrid scenarios such as constrained system identification, targeted active search, or composite optimization with hidden user preferences.

The authors propose “pragmatic curiosity,” a unified learning‑optimization paradigm derived from active inference (AIF). In AIF, actions are selected by minimizing Expected Free Energy (EFE), a single scalar that decomposes into two terms:

1. Epistemic value – the expected reduction in uncertainty about latent variables (mutual information between parameters s and future observations (x, y)). This term coincides with the classic expected information gain used in BED.
2. Pragmatic value – the expected surprisal under a prior preference distribution p(y). Low surprisal corresponds to outcomes that align with user‑specified goals, mirroring the utility function in BO.

Mathematically, with a surrogate posterior q(s|Dₜ) = p(s|Dₜ), the EFE can be written as
G(x) = – E₍q(y|x)₎