Learning from Label Proportions with Dual-proportion Constraints

Tianhao Ma; Ximing Li; Changchun Li; Renchu Guan

cs.LGMar 22, 2026

Learning from Label Proportions with Dual-proportion Constraints

Authors: Tianhao Ma, Ximing Li, Changchun Li, Renchu Guan

Organizations: The University of Tokyo, Japan · College of Computer Science and Technology, Jilin University, China · Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education, Jilin University, China · RIKEN Center for Advanced Intelligence Project, Japan

Abstract

Learning from Label Proportions (LLP) is a weakly supervised problem in which the training data comprise bags, that is, groups of instances, each annotated only with bag-level class label proportions, and the objective is to learn a classifier that predicts instance-level labels. This setting is widely applicable when privacy constraints limit access to instance-level annotations or when fine-grained labeling is costly or impractical. In this work, we introduce a method that leverages Dual proportion Constraints (LLP-DC) during training, enforcing them at both the bag and instance levels. Specifically, the bag-level training aligns the mean prediction with the given proportion, and the instance-level training aligns hard pseudo-labels that satisfy the proportion constraint, where a minimum-cost maximum-flow algorithm is used to generate hard pseudo-labels. Extensive experimental results across various benchmark datasets empirically validate that LLP-DC consistently improves over previous LLP methods across datasets and bag sizes. The code is publicly available at https://github.com/TianhaoMa5/CV PR2026_Findings_LLP_DC.

Figures & tables

Figure 1: Training an instance-level classifier from a dataset with only bag-level class proportions.

Figure 2: A toy example of minimum-cost maximum-flow problem with 3 instances and 2 labels. Given a labeled bag $\{\mathbf{x}_{i1},\mathbf{x}_{i2},\mathbf{x}_{i3},\bm{\alpha}_{i}\}$ and the current predicted outputs of instances $\{\mathbf{p}_{i1},\mathbf{p}_{i2},\mathbf{p}_{i3}\}$ , we can form a corresponding directed multistage graph, and apply any off-the-shelf tool to solve for the optimal flow $\mathbf{g}^{*}$ . It corresponds to the optimal candidate label assignment $\mathbf{\hat{y}}^{*}$ with the highest probability, indicated by the bold amounts for the bold edges between instance and label nodes. Best viewed in color.

Figure 3: Overview of L lp-dc : the illustrated pipeline depicts the training process for a single bag. Given weakly augmented instances $\big(\mathbf{x}_{1}^{\mathrm{w}},\dots,\mathbf{x}_{m}^{\mathrm{w}}\big)$ and strongly augmented instances $\big(\mathbf{x}_{1}^{\mathrm{s}},\dots,\mathbf{x}_{m}^{\mathrm{s}}\big)$ , the model $g(\cdot)$ predicts on the weak views $g(\mathbf{x}_{1}^{\mathrm{w}}),\dots,g(\mathbf{x}_{m}^{\mathrm{w}})$ ; their average is aligned with the label proportion $\bm{\alpha}$ to compute the bag-level loss $\mathcal{L}_{\mathrm{bag}}$ . Using these weak predictions and $\bm{\alpha}$ , we build a graph and solve a minimum-cost maximum-flow problem to obtain pseudo-labels $\mathbf{y}^{*}$ that satisfy the proportion constraints. Finally, we compute the instance-level loss $\mathcal{L}_{\mathrm{ins}}$ on the strong views $g(\mathbf{x}_{1}^{\mathrm{s}}),\dots,g(\mathbf{x}_{m}^{\mathrm{s}})$ against $\mathbf{y}^{*}$ .

Dataset	Model	Bag Size				Fully Supervised
Dataset	Model	16	32	64	128	Fully Supervised
CIFAR-10	DLLP	91.59 ± 1.52	88.61 ± 0.90	79.76 ± 1.45	64.95 ± 0.01	96.05 ± 0.33
	LLP-VAT	91.80 ± 0.08	89.11 ± 0.22	78.75 ± 0.46	63.89 ± 0.19
	ROT	94.86 ± 0.68	94.34 ± 0.65	93.97 ± 0.96	92.23 ± 0.81
	SoftMatch	95.24 ± 0.12	95.25 ± 0.14	94.23 ± 0.18	93.87 ± 0.22
	FLMm*	92.34	92.00	91.74	91.54

Table 1: Classification accuracy (mean ± std) on CIFAR-10, CIFAR-100, SVHN, Fashion-MNIST and MiniImageNet for different bag sizes. The highest accuracy is highlighted in bold. The symbol “*” indicates results for FLMm are reproduced directly from its original publication [ 55 ] , while the dash symbol “-” signifies the absence of reported results for SVHN, Fashion-MNIST, and MiniImageNet in FLMm’s experiments.

Figure 4: Pseudo-label accuracy (left) and pseudo-label ratio (right) during training on CIFAR-10 (left two panels) and CIFAR-100 (right two panels) under different bag sizes (16, 32, 64, 128).

Figure 5: Sensitivity analysis of the instance-level loss weight

\lambda

\mathcal{L}_{\mathrm{ins}}

and the threshold

\tau

parameter on CIFAR-10 and CIFAR-100. For

\lambda\in\{2,1,0.75,0.5,0.25,0\}

, performance remains stable over a moderate range, indicating robustness to the choice of

\lambda

. Similarly, for

\tau

\{0.95,0.8,0.6,0.3,0\}

, performance is consistent across a broad range, demonstrating robustness to threshold selection. Setting

\lambda=0

or the

\tau

to 1 reduces the method to DLLP.

	CIFAR100				MiniImageNet
Method	16	32	64	128	16	32	64	128
DLLP	$18.45$	$18.35$	$18.23$	$17.95$	$21.74$	$21.66$	$21.01$	$20.22$
LLP-VAT	$62.95$	$61.58$	$60.44$	$60.86$	$70.04$	$69.02$	$67.52$	$67.46$
ROT (iter = $3$ )	$40.63$	$38.50$	$36.97$	$36.65$	$47.23$	$43.91$	$43.09$	$42.39$
ROT (iter = $75$ )	$60.03$	$48.50$	$40.08$	$37.26$	$66.07$	$53.28$	$46.07$	$43.21$
SoftMatch	$39.45$	$38.70$	$36.45$	$36.40$	$46.03$	$42.95$	$42.23$	$41.73$

Table 2: Execution time results on CIFAR100 and MiniImageNet (bag size

\in\{16,32,64,128\}

). Units are seconds per epoch. Reported numbers are the mean over 10 epochs under the same settings on a single NVIDIA A100 GPU. For ROT, iter denotes the number of Sinkhorn iterations.

Weakly Supervised Learning

Weakly supervised learning trains models from incomplete, noisy, coarse, or indirect labels instead of dense ground truth.

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Rethinking Learning from Label Proportions via Moment Matching

Tianhao Ma, Wei Wang, Yivan Zhang, Dong-Dong Wu+3

The University of Tokyo · RIKEN AIP

Weakly Supervised Learning

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Learning from label proportions (LLP) is a weakly supervised setting where training data are grouped into bags with only bag-level label proportions observed, and the goal is to learn a classifier that predicts labels for individual instances. Recently, a large body of methods has emerged; however, most of them rely on the assumption that instance-label pairs are sampled i.i.d. and randomly assembled into bags, an assumption often violated in real-world scenarios. Our experimental results indicate that these methods perform suboptimally under non-random settings. In this work, we adopt a more realistic assumption: bags are sampled i.i.d., and instance labels are conditionally independent given the instances. Under this setting, we show that the label counts follow a Poisson multinomial distribution. Motivated by this observation, we propose LLP via moment matching (LLP-MM), a simple yet effective approach which leverages multi-order factorial moments as training objectives, encouraging the classifier predictions to match these moments and thereby more fully exploit the underlying statistical structure of the data. Extensive experiments on benchmark datasets under various bag construction strategies demonstrate the effectiveness of our approach while maintaining high computational efficiency. Our code is available at https://github.com/TianhaoMa5/LLP-MM.

Abstract

Weakly Supervised Learning

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