k-Volume: a Unified Active Learning Strategy

Eliot Tron; Alain Celisse; Christophe Denis; Alice Le Brigant

cs.LGNov 6, 2026

k-Volume: a Unified Active Learning Strategy

Authors: Eliot Tron, Alain Celisse, Christophe Denis, Alice Le Brigant

Organizations: SAMM, Université Paris 1 · IRIT, INP Toulouse

Abstract

In the classification context where revealing the label of a sample is costly, we propose a new Active Learning (AL) strategy named k-Volume (kVol). It randomly selects a sample to be labeled according to the volume of the smallest ball containing its k nearest neighbors. kVol is highly versatile: it can be applied either in the input space or in the output space, and it enjoys the desirable properties of the three main families of AL strategies: density, diversity, and (when performed in the output space) uncertainty. From a theoretical perspective, we establish that kVol is asymptotically equivalent to uniform sampling over the support of the (input or output) data distribution. Based on experiments on various synthetic and real-life tabular datasets and images, we observe that kVol is surprisingly stable: whatever the classifier or the experimental setup, it still lies among the best AL strategies unlike other competitors that fail dramatically. In particular, kVol outperforms state-of-the-art competitors such as CoreSet, TypiClust or Uncertainty on several real-life datasets, at a low computational cost.

Figures & tables

Figure 1: Illustration of

\kvol_{k}

with

k=4

Scenario	Balanced	Dimension
No. 1	Yes	2
No. 2	No	2
No. 3	Yes	20
No. 4	No	20

Table 1: Parameters for the numerical experiments on synthetic data.

Panel of Figure 2 — Figure 2: Visualization of the synthetic training set in the 2D scenarios.

Panel of Figure 3 — Figure 3: Accuracy comparison during active training. Initial batch size $b_{0}=50$ , step batch size $b=10$ and budget $B=500$ . The dimension of the data space is 2.

Dataset Id	Name	Number of classes	Number of points	Input dimension
46	splice	3	3190	60
182	satimage	6	6430	36
846	elevators	2	16599	18

Table 2: Characteristics of the tabular datasets used in Section 4.2 .

Panel of Figure 4 — Figure 4: Accuracy comparison during active training. Initial batch size $b_{0}=10$ , step batch size $b=10$ and budget $B=500$ . The dataset id is 182.

Name	Number of classes	Number of points	Image dimension	Embedding dimension	Reference
MNIST	10	60000	784	2048	[ 24 ]
CIFAR10	10	50000	3072	2048	[ 23 ]
CIFAR100	100	50000	3072	2048	[ 23 ]

Table 3: Characteristics of the datasets of images used in Section 4.3 .

Panel of Figure 5 — Figure 5: Accuracy comparison during active training. Initial batch size $b_{0}=10$ , step batch size $b=50$ and budget $B=760$ .

Figure 6: CIFAR100. Accuracy comparison during active training. Initial batch size

b_{0}=500

, step batch size

b=500

and budget

B=5000

Appendix figures & tables10 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 7: Difficult case for the uncertainty family. The red line represents the decision boundary of the trained classifier

N

, the colors represent the probabilities of each class, and the shape of the points represent the true labels.

Panel of Figure 8 — Figure 8: Accuracy comparison during active training. Initial batch size $b_{0}=50$ , step batch size $b=10$ and budget $B=500$ . The dimension of the data space is 20.

Panel of Figure 9 — Figure 9: Accuracy comparison during active training. Initial batch size $b_{0}=10$ , step batch size $b=10$ and budget $B=500$ .

Panel of Figure 10 — Figure 10: Accuracy comparison during active training. Initial batch size $b_{0}=100$ , step batch size $b=10$ and budget $B=500$ .

Panel of Figure 11 — Figure 11: Accuracy comparison during active training. Initial batch size $b_{0}=10$ , step batch size $b=10$ and budget $B=500$ . The dataset id is 182.

Panel of Figure 12 — Figure 12: Accuracy comparison during active training. Initial batch size $b_{0}=10$ , step batch size $b=10$ and budget $B=500$ . The dataset id is 46.

Panel of Figure 13 — Figure 13: Accuracy comparison during active training. Initial batch size $b_{0}=10$ , step batch size $b=10$ and budget $B=500$ . The dataset id is 846.

Figure 14: Accuracy comparison during active training. Initial batch size

b_{0}=500

, step batch size

b=500

and budget

B=10000

Figure 15: Accuracy comparison during active training for different values of

k

. Initial batch size

b_{0}=10

, step batch size

b=50

and budget

B=760

. The dataset is CIFAR10.

	Uniform	Entropy	CoreSet	TypiClust	kVolOut ( $k=4$ )	kVolIn ( $k=4$ )
Figure 6	1.70e+03s	1.71e+03s	3.06e+03s	1.07e+04s	1.73e+03s	1.90e+03s
Figure 14	6.54e+03s	6.54e+03s	-	-	6.58e+03s	6.87e+03s
	kVolOut1	kVolOut10	kVolOut100	kVolOut500
Figure 15	1.14e+02s	1.37e+02s	2.01e+02s	3.43e+02s

Appendix

Table 4: Average computing time per seed (over 10 repetitions) for each strategy (in seconds).

Active Learning

Active learning selects the examples, labels, features, or experiments most useful for improving a model under limited budgets.

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University of California, Berkeley

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Open paper AskPDF arXiv

With rapid advancement over the last few years, many different methods are now widely used for classification. However, training these models requires substantial labeled data. Active Learning is a potential solution to this problem. Pool-based active learning minimizes costs by querying only the most informative samples from an unlabeled dataset. Diversity-based approaches, on the other hand, attempt to select a representative subset of the data. There are many different objectives for determining the selection process, including exact K-center, exact K-median, and Greedy K-center. In this paper, we will focus on evaluating the performance of Greedy K-center across a variety of metric spaces: the raw feature space, a Linear Discriminant Analysis (LDA) space, and a model-derived probability space (with and without entropy-based weighting). Using Random Forest classifiers as a baseline evaluator, our empirical results on synthetic and real-world datasets demonstrate that mapping unlabeled instances into a predictive probability space and weighting the result by entropy often dominates the other options for active learning selection with Greedy K-center.

Abstract

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