Loss-Driven Bayesian Active Learning

Zhuoyue Huang; Freddie Bickford Smith; Tom Rainforth

cs.LGApr 13, 2026

Loss-Driven Bayesian Active Learning

Authors: Zhuoyue Huang, Freddie Bickford Smith, Tom Rainforth

Organizations: University of Oxford

Abstract

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	$\mathcal{Z}$	$T(z)$	$\phi(z)$	$D_{\phi}(T(z),T(b))$
Squared error	Berger (1985)	$\mathbb{R}$	$z$	$z^{2}$	$(z-b)^{2}$
Box-Cox squared error	Box and Cox (1964)	$\mathbb{R}^{+}$	$(z^{\lambda}-1)/\lambda,\lambda\neq 0$	$z^{2}$	$(z^{\lambda}-b^{\lambda})^{2}/\lambda^{2}$
Linex	Zellner (1986)	$\mathbb{R}^{+}$	$\exp(-\alpha z),\alpha>0$	$-\log z$	$\exp(\alpha(b-z))-\alpha(b-z)-1$

Table 1: Standard predictive losses can be expressed as a Bregman divergence in a transformed space,

D_{\phi}(T(z),T(b))

, where

z\in\mathcal{Z}

is a world state,

b\in\mathcal{Z}

is an in-space action,

\phi

is a convex function and

T:\mathcal{Z}\to\mathrm{dom}(\phi)

is measurable.

Figure 1: Expected variance reduction ( EVR ) vs. a variant with weighting

w(z)=\exp(z)

(

\text{EVR}_{w}

). Row 1: prediction with data acquired by EVR . Row 2: values of EVR and

\text{EVR}_{w}

given the Row 1 training set (the maximiser is labelled next). Row 3: prediction with data acquired by

\text{EVR}_{w}

. Row 4: values of EVR and

\text{EVR}_{w}

given the Row 3 training set.

Figure 2: Top to bottom: performance (mean

\pm

SEM) on Slump , Yacht and Estate with

w(z)=\exp(-z)

Method	$\mathrm{SEL}\downarrow$		$\mathrm{SEL}_{w}\downarrow$
Random	0.9238 $\pm$	0.0940	343.4 $\pm$	109.5
EVR	0.3449 $\pm$	0.0025	107.1 $\pm$	1.971
$\text{EVR}_{w}$	1.5872 $\pm$	0.1158	72.06 $\pm$	1.508

Table 2: Test losses in 1d regression. Here

\mathrm{SEL}

denotes squared-error loss and

\mathrm{SEL}_{w}

denotes a weighted variant. For both test losses, the corresponding acquisition objective (

\mathrm{EVR}

for the former;

\mathrm{EVR}_{w}

for the latter) achieves the lowest loss. We report mean

\pm

SEM over 25 runs. Bold indicates the best mean for each loss.

Figure 3: Evolution of performance (mean

\pm

SEM) under downstream losses (SEL and Linex) on Estate .

Figure 4: Top to bottom: mean

\pm

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]

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Project	Citation	License	URL
NumPy	Harris et al. (2020)	BSD (3-clause)	numpy.org
SciPy	Virtanen et al. (2020)	BSD (3-clause)	scipy.org
Scikit-learn	Pedregosa et al. (2011)	BSD (3-clause)	scikit-learn.org
Matplotlib	Hunter (2007)	PSF-based	matplotlib.org
pandas	McKinney (2010)	BSD (3-clause)	pandas.pydata.org
tqdm	da Costa-Luis (2019)	MIT	github.com/tqdm/tqdm

Appendix

Table 3: Third‑party software used in this work.

Dataset	Citation	URL
Vehicle	Mowforth and Shepherd (1993)	archive.ics.uci.edu/dataset/149/statlog+vehicle+silhouettes
Landsat	Srinivasan (1993)	archive.ics.uci.edu/dataset/146/statlog+landsat+satellite
Vowel	Deterding et al. (1988)	archive.ics.uci.edu/dataset/152/connectionist+bench+vowel+recognition+deterding+data
Slump	Yeh (2007)	archive.ics.uci.edu/dataset/182/concrete+slump+test
Yacht	Gerritsma et al. (1981)	archive.ics.uci.edu/dataset/243/yacht+hydrodynamics
Estate	Yeh (2018)	archive.ics.uci.edu/dataset/477/real+estate+valuation+data+set

Appendix

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)

(

\text{EVR}_{w}

). Row 1: prediction with data acquired by EVR . Row 2: values of EVR and

\text{EVR}_{w}

given the Row 1 training set (the maximiser is labelled next). Row 3: prediction with data acquired by

\text{EVR}_{w}

. Row 4: values of EVR and

\text{EVR}_{w}

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]

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Department of Computer Science, Georgia Institute of Technology, Atlanta, GA 30332

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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-

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