Active Learning for Gaussian Process Regression Under Self-Induced Boltzmann Weights

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.

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.

Gaussian Processes (GPs) are widely used as surrogates for black-box functions in sequential decision-making problems such as Bayesian optimization (BO), level set estimation (LSE), and Bayesian active learning (BAL). GP performance critically depends on kernels, and standard kernels can lead to suboptimal decisions under misspecification. To address this, we introduce HACK GPs (Hedge Adaptive Cumulative Kernels), a method that views kernel selection as an online learning with expert advice problem. HACK treats each candidate kernel as a GP "expert" and updates a distribution over experts online using AdaHedge, based on a loss received as a proxy for their ability to fit the function and align with the task objective. We provide two variants of HACK: (i) Mixture of Gaussians (MoG) and (ii) categorical sampling. We establish general guarantees showing that, under a loss-gap condition, the weight concentrates on the best kernel and the resulting acquisition function is close to that of the best expert. Empirically, we observe robust performance across BO, LSE, and BAL compared to standard kernels such as Squared Exponential and Matern-5/2, as well as simple ensemble baselines.