Active Learning and Bayesian Optimization: a Unified Perspective to Learn with a Goal

TL;DR

This paper unifies Bayesian optimization (BO) and pool-based active learning (AL) under a goal-driven adaptive-sampling framework, formalizing the analogy between Bayesian infill criteria and AL learning criteria. It introduces a three-way classification of adaptive sampling (adaptive probing, adaptive modeling, adaptive learning) and demonstrates how BO and multifidelity BO (MFBO) can be viewed as goal-driven learners that iteratively refine a surrogate and select informative queries via acquisition functions. By mapping EI, PI, ES/MES to informativity and representativeness criteria, and extending these to multi-oracle and multifidelity settings (MFPI, MFEI, MFMES, etc.), the authors provide a cohesive perspective on how to accelerate learning and optimization. Extensive experiments on benchmark problems—including Forrester, Rosenbrock, Rastrigin, ALOS, and a spring-mass system with noise—show that balanced exploration-exploitation and the use of multiple fidelities yield substantial speedups under realistic budgets, with guidance on selecting learning criteria for different problem regimes. The work advances practical guidance for applying goal-driven surrogates in complex, resource-constrained engineering and scientific optimization tasks, while clarifying the conceptual overlap and distinctions between BO and AL.

Abstract

Science and Engineering applications are typically associated with expensive optimization problems to identify optimal design solutions and states of the system of interest. Bayesian optimization and active learning compute surrogate models through efficient adaptive sampling schemes to assist and accelerate this search task toward a given optimization goal. Both those methodologies are driven by specific infill/learning criteria which quantify the utility with respect to the set goal of evaluating the objective function for unknown combinations of optimization variables. While the two fields have seen an exponential growth in popularity in the past decades, their dualism and synergy have received relatively little attention to date. This paper discusses and formalizes the synergy between Bayesian optimization and active learning as symbiotic adaptive sampling methodologies driven by common principles. In particular, we demonstrate this unified perspective through the formalization of the analogy between the Bayesian infill criteria and active learning criteria as driving principles of both the goal-driven procedures. To support our original perspective, we propose a general classification of adaptive sampling techniques to highlight similarities and differences between the vast families of adaptive sampling, active learning, and Bayesian optimization. Accordingly, the synergy is demonstrated mapping the Bayesian infill criteria with the active learning criteria, and is formalized for searches informed by both a single information source and multiple levels of fidelity. In addition, we provide guidelines to apply those learning criteria investigating the performance of different Bayesian schemes for a variety of benchmark problems to highlight benefits and limitations over mathematical properties that characterize real-world applications.

Figures (14)

• Figure 1: Citations of Bayesian Optimization (BO), Active Learning (AL), Adaptive Sampling (AS) and the three terms combined (BO+AL+AS).
• Figure 2: Where adaptive sampling and active learning meet: this work focuses on the synergies between Bayesian optimization and active learning as goal-driven learning procedures driven by common learning principles.
• Figure 3: Learning criteria: watering optimization problem.
• Figure 4: Mapping methodologies to learning criteria.
• Figure 5: Mapping of the learning criteria in active learning and infill criteria in Bayesian optimization.