Observation-dependent Bayesian active learning via input-warped Gaussian processes

TL;DR

The paper identifies a fundamental limitation in standard Gaussian-process-based active learning: the posterior variance, and hence variance-based acquisitions, do not condition on observed function values under fixed hyperparameters. It introduces input-warped Gaussian processes, where a learned monotone warp $T_\phi$ reshapes the input geometry for acquisition while keeping the predictive model unchanged, enabling observation-dependent exploration; the warp is parameterized by conditional rational quadratic splines and trained either via a self-supervised objective or marginal likelihood. The authors demonstrate, across synthetic non-stationary benchmarks and a real-world solar-cell imaging task, that a self-supervised warp yields systematic gains in sample efficiency and robustness to non-stationarity, outperforming unwarped GPs and likelihood-trained warps. This geometry-centric approach shows that improved active learning can be achieved without more expressive surrogates, simply by learning how to measure uncertainty on a data-driven input manifold with a decoupled acquisition design.

Abstract

Bayesian active learning relies on the precise quantification of predictive uncertainty to explore unknown function landscapes. While Gaussian process surrogates are the standard for such tasks, an underappreciated fact is that their posterior variance depends on the observed outputs only through the hyperparameters, rendering exploration largely insensitive to the actual measurements. We propose to inject observation-dependent feedback by warping the input space with a learned, monotone reparameterization. This mechanism allows the design policy to expand or compress regions of the input space in response to observed variability, thereby shaping the behavior of variance-based acquisition functions. We demonstrate that while such warps can be trained via marginal likelihood, a novel self-supervised objective yields substantially better performance. Our approach improves sample efficiency across a range of active learning benchmarks, particularly in regimes where non-stationarity challenges traditional methods.

Figures (6)

• Figure 1: An example demonstrating the necessity of warping the input space. On the left, we see the scenario of an unwarped GP; on the right, a warped GP with the locations of two selected new points. We see that complex regions are stretched out, while flatter regions are compressed, facilitating the selection of points in complex regions.
• Figure 2: Predictive distribution and EIG for a warped and unwarped GP. The middle figure shows the warped $x^w$, using rational quadratic splines, relative to the unwarped $x$.
• Figure 3: Average relative decrease in each performance metric on the synthetic benchmark functions Gramacy-Lee08 ($D=2$), Peaks ($D=2$), Box ($D=2,3$) and Hartmann ($D=4$). In all tasks our proposed C-RQS warp achieves superior performance, with four out five cases trained in a self-supervised (SS) fashion.
• Figure 4: Illustrative example of geometry-aware Gaussian process modeling on synthetic two-dimensional benchmark functions. Results are shown for the Glree08 function (top row) and the Box-2D function (bottom row). (a) Ground-truth function evaluated on a dense grid, together with the current set of training points. (b) Predictive variance of a standard Gaussian process in the original input space, reflecting uncertainty induced primarily by sparse sampling. (c) Predictive variance of the input-warped Gaussian process evaluated in the original input space (pullback), illustrating how the learned geometry redistributes uncertainty toward regions of localized variation. (d) The same predictive variance visualized in the warped input space, together with the deformed input mesh and warped training points. (e) Deformation of the two-dimensional input domain induced by the learned warp, demonstrating how resolution is adaptively concentrated in regions where the target function exhibits rapid local changes.
• Figure 5: Average relative decrease in each performance metric on two photoluminescence datasets (IA & IB shown to the left). Our proposed method C-RQS trained via self-supervised learning achieves superior performance across all tasks.