Correcting Boundary Bias and Observation Independence in Bayesian Experimental Design
Authors: Sanna Jarl, Jens Sjölund, Jonathan J. S. Scragg, Maria Bånkestad
Abstract
In many experimental settings, active learning can improve sample efficiency by sequentially selecting where to measure, which is particularly valuable when experiments are expensive. Gaussian processes with variance-based acquisition criteria are widely used for this purpose, but have two limitations. First, they are observation-independent: their posterior variance depends only on where samples are acquired, not on what is measured, impairing their sensitivity to the structure of the acquired data. Second, they inflate the variance near boundaries, leading to excessive sampling at the edges of the space compared to the interior. These limitations undermine the gains in sampling efficiency expected from sequential acquisition. We address both limitations. We derive a reconstruction-driven design density and use the posterior mean to build a training-free warp that places more measurements where the target function varies rapidly. A geometric equalizer separately corrects boundary bias. Across sixteen synthetic and two real-data benchmarks, the geometric equalizer consistently improves function reconstruction by correcting boundary bias, while the reconstruction warp provides further gains by concentrating measurements where the posterior mean varies rapidly.
Gaussian processes with stationary kernels on bounded domains exhibit inflated posterior variance near the boundary. Despite being a long-recognized artifact in geostatistics and a source of over-exploration in Bayesian optimization, the causes and effects of boundary-induced acquisition bias are underexplored. We trace the root cause to a simple geometric mechanism: the truncation of the kernel correlation neighborhood at the domain boundary creates an observation-independent distortion that worsens with dimensionality. We show how this distortion manifests across three acquisition classes: variance maximization concentrates selections at the corners, whereas negative integrated posterior variance and expected predictive information gain move selections inward to axis-aligned interior shells. These patterns arise without reference to any objective function, meaning that acquisition behavior can be dominated by kernel geometry rather than the desired task-specific uncertainty. To quantify this, we introduce a function-free selection-profile diagnostic for arbitrary acquisitions, kernels, and bounded-domain geometries.
The effectiveness of active learning hinges on the choice of the acquisition criterion by which a learning algorithm selects potentially informative data points whose label is subsequently queried. This paper proposes a novel gradient-based acquisition criterion, derived from a generalization bound introduced by Luo et al. (2022). This criterion can be applied in lieu of uncertainty measures in uncertainty sampling, or incorporated into diversity-based methods that consider the spread of sampled points in addition to the uncertainty of their labels. We provide a theoretical justification of the proposed acquisition criterion, and demonstrate its effectiveness in an empirical evaluation.
We consider the active learning problem where the goal is to learn an unknown function with low prediction error under an unknown Boltzmann distribution induced by the function itself. This self-induced weighting arises naturally in problems such as potential energy surface (PES) modeling in computational chemistry, yet poses unique challenges as the target distribution is unknown and its partition function is intractable. We propose \texttt{AB-SID-iVAR}, a Gaussian Process-based acquisition function that approximates the intractable Bayesian target distribution in closed form while avoiding partition function estimation, and is applicable to both discrete and continuous input domains. We also analyze a Thompson sampling alternative (\texttt{TS-SID-iVAR}) as a higher variance Monte Carlo variant. Despite the unknown target, under mild conditions, we establish that the terminal prediction error vanishes with high probability, and provide a tighter average-case guarantee. We demonstrate consistent improvements over existing approaches in this setting on synthetic benchmarks and real-world PES modeling and drug discovery tasks.