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

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.

Hyperparameter selection remains a key challenge in Bayesian optimization (BO) and Bayesian active learning (AL), as model misspecification can lead to suboptimal performance, while more accurate fully Bayesian treatments typically rely on computationally expensive MCMC sampling. This paper proposes a unified framework, KENDO (Kernel ENsemble Disagreement-aware Operator), that integrates Ensemble Gaussian Processes (EGP) with disagreement-aware acquisition strategies. The central idea is to replace hyperparameter sampling with a kernel ensemble and adaptive Bayesian weighting, combined with disagreement-aware acquisition strategies. Within this unified framework, we instantiate KENDO-BO for BO and KENDO-AL for Bayesian AL, demonstrating that both arise from a common self-correcting mechanism with task-specific acquisition objectives. We further extend the approach to multi-objective optimization via random scalarization that preserves the single-optimizer conditioning structure. Thorough numerical tests on synthetic and real-world benchmarks across single-objective optimization, multi-objective optimization, and active learning demonstrate that (i) KENDO-BO achieves competitive or superior optimization performance compared to state-of-the-art methods while reducing computational overhead by up to $5\times$ and (ii) KENDO-AL achieves superior predictive calibration over MCMC-based active learning baselines with up to $27\times$ speedup.

Efficient Gaussian process (GP) inference is critical for sequential decision-making tasks such as active learning, online prediction, and Bayesian optimization. Iterative approaches of approximating the GP posterior using solvers like conjugate gradients, stochastic gradient descent, or alternating projections avoid cubic costs, but often require many iterations to converge, limiting their efficacy when the posterior is updated frequently with new data. To address this, we introduce three warm-start strategies that exploit solutions of smaller linear systems to substantially speed-up convergence when updating the posterior with new data. Our methods are supported by theoretical analysis showing reduced initialization error in reproducing kernel Hilbert space (RKHS) distance, and by empirical results on regression benchmarks and Bayesian optimization tasks. Across solvers, warm-starting achieves speed-ups of up to 19x when solving to tolerance, and produces more accurate posterior estimates under fixed compute budgets, directly improving optimization performance. These results establish warm-starting as a simple, effective, and broadly applicable tool for scaling Gaussian processes in sequential settings.