The central goal of active learning is to gather data that maximises downstream predictive performance, but popular approaches have limited flexibility in customising this data acquisition to different downstream problems and losses. We propose a rigorous loss-driven approach to Bayesian active learning that allows data acquisition to directly target the loss associated with a given decision problem. In particular, we show how any loss can be used to derive a unique objective for optimal data acquisition. Critically, we then show that any loss taking the form of a weighted Bregman divergence permits analytic computation of a central component of its corresponding objective, making the approach applicable in practice. In regression and classification experiments with a range of different losses, we find our approach reduces test losses relative to existing techniques.
Figures & tables
Name
Example
Z
T(z)
ϕ(z)
Dϕ(T(z),T(b))
Squared error
Berger (1985)
R
z
z2
(z−b)2
Box-Cox squared error
Box and Cox (1964)
R+
(zλ−1)/λ,λ=0
z2
(zλ−bλ)2/λ2
Linex
Zellner (1986)
R+
exp(−αz),α>0
−logz
exp(α(b−z))−α(b−z)−1
Table 1: Standard predictive losses can be expressed as a Bregman divergence in a transformed space, Dϕ(T(z),T(b)) , where z∈Z is a world state, b∈Z is an in-space action, ϕ is a convex function and T:Z→dom(ϕ) is measurable.
Figure 1: Expected variance reduction ( EVR ) vs. a variant with weighting w(z)=exp(z) ( EVRw ). Row 1: prediction with data acquired by EVR . Row 2: values of EVR and EVRw given the Row 1 training set (the maximiser is labelled next). Row 3: prediction with data acquired by EVRw . Row 4: values of EVR and EVRw given the Row 3 training set.
Figure 2: Top to bottom: performance (mean ± SEM) on Slump , Yacht and Estate with w(z)=exp(−z) .
Method
SEL↓
SELw↓
Random
0.9238 ±
0.0940
343.4 ±
109.5
EVR
0.3449 ±
0.0025
107.1 ±
1.971
EVRw
1.5872 ±
0.1158
72.06 ±
1.508
Table 2: Test losses in 1d regression. Here SEL denotes squared-error loss and SELw denotes a weighted variant. For both test losses, the corresponding acquisition objective ( EVR for the former; EVRw for the latter) achieves the lowest loss. We report mean ± SEM over 25 runs. Bold indicates the best mean for each loss.
Figure 3: Evolution of performance (mean ± SEM) under downstream losses (SEL and Linex) on Estate .
Figure 4: Top to bottom: mean ± SEM on Vehicle ( w=[50,1,1,50] ), Landsat ( w=[1,1,1,1,50,50] ) and Vowel ( w=[1,1,1,1,1,1,50,50,1,1,1] ).
Table 4: UCI datasets used in this work. All are available under Creative Commons Attribution 4.0 (CC BY 4.0).
Figure 5: Expected variance reduction ( EVR ) vs. a variant with weighting w(z)=exp(−z) ( EVRw ). Row 1: prediction with data acquired by EVR . Row 2: values of EVR and EVRw given the Row 1 training set (the maximiser is labelled next). Row 3: prediction with data acquired by EVRw . Row 4: values of EVR and EVRw given the Row 3 training set.
Figure 6: From top to bottom: evolution of acquired-class proportions on Vehicle ( w=[50,1,1,50] ), Landsat ( w=[1,1,1,1,50,50] ) and Vowel ( w=[1,1,1,1,1,1,50,50,1,1,1] ).
Over the past couple of decades, many active learning acquisition functions have been proposed, leaving practitioners with an unclear choice of which to use. Bayesian active learning offers principled objectives with explainable intuition, including Expected Error Reduction (EER), Expected Predictive Information Gain (EPIG), and Bayesian Active Learning by Disagreements (BALD). A key challenge of such methods is the difficulty of extending to the batch setting, leading to either computational challenges (BatchBALD) or dramatic performance drops (top-B selection). In this work, we extend active learning's Bayesian Decision Theory framework to derive Partial Batch Label Sampling (ParBaLS) for the EPIG algorithm. We show experimentally that ParBaLS EPIG gives superior performance for a variety of datasets with different budgets using Bayesian logistic regression on embeddings from large pre-trained models. Our code is available at https://github.com/ADDAPT-ML/ParBaLS.
Kangping Hu, Stephen Mussmann
Department of Computer Science, Georgia Institute of Technology, Atlanta, GA 30332
Active data acquisition is central to many learning and optimization tasks in deep neural networks, yet remains challenging because most approaches rely on predictive uncertainty estimates that are difficult to obtain reliably. To this end, we propose Goal-Oriented Influence- Maximizing Data Acquisition (GOIMDA), an active acquisition algorithm that avoids explicit posterior inference while remaining uncertainty-aware through inverse curvature. GOIMDA selects inputs by maximizing their expected influence on a user-specified goal functional, such as test loss, predictive entropy, or the value of an optimizer-recommended design. Leveraging first-order influence functions, we derive a tractable acquisition rule that combines the goal gradient, training-loss curvature, and candidate sensitivity to model parameters. We show theoretically that, for generalized linear models, GOIMDA approximates predictive-entropy minimization up to a correction term accounting for goal alignment and prediction bias, thereby, yielding uncertainty-aware behavior without maintaining a Bayesian posterior. Empirically, across learning tasks (including image and text classification) and optimization tasks (including noisy global optimization benchmarks and neural-network hyperparameter tuning), GOIMDA consistently reaches target performance with substantially fewer labeled samples or function evaluations than uncertainty-based active learning and Gaussian-process Bayesian optimization baselines.
Weichi Yao, Bianca Dumitrascu, Bryan R. Goldsmith +1
Active learning studies the fundamental question: what data should we choose to observe? The greedy algorithm in optimal experiment design is a common heuristic and also equivalent to myopic Bayesian active learning for linear regression, the common framework where long-term planning is replaced with the one-step optimal choice. In this work, we prove a first-of-its-kind approximation ratio for the greedy algorithm's risk that is tight up to an absolute constant. The approximation ratio is linear in the maximum initial leverage score (MILS), a newly identified quantity fundamental to the greedy algorithm's performance. Finally, we illustrate the results with simple numerical simulations.
Stephen Mussmann
School of Computer Science Georgia Institute of Technology