Robust Label Proportions Learning
Authors: Jueyu Chen
Wantao Wen
Yeqiang Wang
Erliang Lin
Yemin Wang
Yuheng Jia
Venue: NeurIPS 2025 poster ()
Keywords: Label Proportions Learning
Optimal Transport
Primary area: general_machine_learning

Abstract: Learning from Label Proportions (LLP) is a weakly-supervised paradigm that uses bag-level label proportions to train instance-level classifiers, offering a practical alternative to costly instance-level annotation. However, the weak supervision makes effective training challenging, and existing methods often rely on pseudo-labeling, which introduces noise. To address this, we propose RLPL, a two-stage framework. In the first stage, we use unsupervised contrastive learning to pretrain the encoder and train an auxiliary classifier with bag-level supervision. In the second stage, we introduce an LLP-OTD mechanism to refine pseudo labels and split them into high- and low-confidence sets. These sets are then used in LLPMix to train the final classifier. Extensive experiments and ablation studies on multiple benchmarks demonstrate that RLPL achieves comparable state-of-the-art performance and effectively mitigates pseudo-label noise.

Robust Label Proportions Learning
Jueyu Chen', Wantao Wen'; Yeqiang Wang"; Erliang Lin'; Yemin Wang', Yuheng Jia"
School of Artificial Intelligence, Southeast University, Nanjing, China
Wortthwest A&F University, Xianyang, China
School of Informatics, Xiamen University, Xiamen, China "School Computer Science and Engineering, Southeast University, Nanjing, China (jueyuc, 213211883 21 213223797) @seu. adu. en, wangyeminéstu, xmu, edu. en, yhjia@seu. edu.
com,
Abstract
Learning from Label Proportions (LLP) a weakly-superrised paradigm that uses bag-level label proportions to train instance-level classifiers, offering a practical alternative to costly instance-level annotation. However, the weak supervision makes effective training challenging, and existing methods often rely on pscudo- labeling, which introduces noise. To address this, we propose RLPL, two- stage framework. In the first stage, we use unsupervised contrastive learning to pretrain the encoder and train an auxiliary classifier with bag-level supervision. the second stage, we introduce an LLP-OTD mechanism to refine pseudo- labels and split them into high- and low-confidence sets. These sets are then used LLPMix to train the final classifier. Extensive experiments and ablation studies on multiple benchmarks demonstrate that RLPL achieves comparable state-of-the-art performance and effectively mitigates pscudo-label noise.
1
Introduction
Learning from Label Proportions (LLP), a significant weakly-supervised learning paradigm I 36
15
addresses scenarios where individual instance labels are costly or inaccessible due to privacy concerns 10 Instead, LL.P leverages more readily available label proportions within bags of instances. This practical advantage has led to LLP's application in diverse fields such as medical analysis [T] e-commerce 233 political science (27] and remote sensing ( The core task in LLP is to train an instance-level classifier using only these bag-level label proportions, a distinct challenge compared to traditional instance-level supervision.
13
The weak supervision from bag-level proportions poses significant challenges for learning accurate instance-level classifiers. As observed by Yu et al. (34), insufficient class separation in bag proportions can severely degrade performance. Many recent LLP methods therefore turn to pscudo-labeling: Ma al. [20] re-weight high-contidence labels for an auxiliary instance loss; PLOT [19] alternates between begelevel and pseudo-label training; Liu et al. [18] employ self-ensembling. However, noisy pseudo-labels remain major bottleneck.
We propose Robust Label Proportions Learning (RLPL), a two-stage framework. In Stage 1, we pre-train encoder via contrastive learning and train classifier head using only bag proportions. In Stage 2, we refine the resulting pseudo-labels with LLP-OTD (L.. ...e-eenllzed Optimal Transport- based Label Dividing), splitting data into high-confidence labeled set and an unlabeled set. Finally, LLPMiX, inspired by MixMatch, integrates LLP constraints into a semi semi-supervised pipeline to train the main classifier. Experiments standard benchmarks show that RLPL possesses comparable performance to prior methods, and ablations confirm the effectiveness of LLP-OTD in filtering noise.
The primary contributions of this paper are:
'These authors contributed equally.
1 CCoresponding author.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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We propose RLPL, novel two-stage LLP framework that effectively leverages both bag-level proportions and instance-level pseudo-labels, demonstrating robust performance, particularly in challenging large-bag scenarios.
We introduce LLP-OTD, LLP-constrained optimal transport-based mechanism to refine noisy pseudo-labels by extracting high-confidence labels. Experiments confirm its superior performance pseudo-label refinement.
We develop LLPMix, training mechanism that utilizes the LLP-OTD refined dataset and label proportion information within a semi-supervised framework, achieving excellent experimental results.
Comprehensive experiments validate RLPL's comparable of-te-of-the-ae performance against current leading LLP models, supported by thorough ablation studies. Meanwhile, our model exhibits bag-inseniitive robustmess.
Related Work
2.1 Learning from Label Proportions(LLP)
Learning from Label Proportions (LLP) is weakly supervised paradigm in which only aggregated label proportions for predefined bags are available, and instance-level labels are inaccessible [23 34
12 16 22].
Early approaches matched predicted bag proportions ground truth (34, but this coarse
supervision admits multiple instance-label configurations, degrading classification performance. To recover finer supervision, recent methods generate pseudo-labels- for example, L2P-AHIL [20] re-weights high-confidence labels via entropy measures, and LLPFC [36 treats LLP as a label-noise problem.
Our RLPL framework advances LLP by introducing LLP-OTD, an Optimal Transport-baut based de- noising mechanism with a post-optimization consistency heuristic to filter reliable pseudo-labels, and LLPMix, which treats low-confidence labels as unlabeled and enforces an explicit LLP consistency loss within MitMatth-ttyle semi-supervised pipeline. Together, these components yield superior instance-level classifiers from bag-level proportions.
2.2 Optimal Transport in LLP
Traditional LLP methods that minimize KI. divergence between predicted and bag-level proportions often yield high-entropy, flat distributions lacking discriminative power [19]. To enforce more structured alignment, Optimal Transport (OT) has been introduced into LLP. Liu et al.
[17] proposed OT-LLP, using entropically regularized Sinkhorn solver match proportions exactly, thereby boosting accuracy. La et al. [24] combined OT with prototypical contrastive learning to align embeddings with class prototypes. Liu et al. [19] also proposed PLOT (Pseudo-Label Optimization via OT), which alternates based label assignment with model updates to suppress noise.
While these OT-based methods improve proportion matching, they often struggle with pseudo-label noise and reliable confidence assessment. Our LLP-OTD mechanism offers distinct advantages by iteratively correcting pseudo-labels via an OT process incorporating an L.IP-specific penalty in its cost function, directly enforeing bag constraints during refinement. Crucially, L.PP-TT employs novel "post-optimization consistency" metric robustly distinguish high-confidence pseudo-labels from unreliable ones, surpassing common heuristic criteria. This focus on advanced pseudo-label correction and confidence assessment allows LLP-OTD significantly enhance the quality instance-leve supervision for more accurate LLP classifiers.
2.3
Semi-Supervised Learning
Semi-Supe Semi-SSperviied Learning (SSL) aims to leverage both labeled and unlabeled data during training
37
29 4.
Common SSL approaches include consistency regularization,
and
entropy minimization, often combined with strong data augmentations prominent SSL method that unifies data augmentation, label guessing with sharpening, and MixUp
[32] MixMatch [2] is a
35] to effectively utilize all available data. Given that the dataset refined by our L.IP-OTD is partially
labeled, resembling an SSL setting, we adapt SSL principles for LLP. However, standard SSL methods

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Classifier head
Encoder
High CONE Peeudo Labela
LLPMix
LLP-OTD
Auxiliary Classifier
Proportions
Noisy Pseudo Labels
LLP-Proportion Penalty
LLP LLP-Conisstency
GGt
dog
bird
First Stage Training
Second Stage Training
Robust Label Proportions Learning
Figure Overview Methodology
like MixMatch do not inherently account for bag-level label proportion constraints crucial LLP. Therefore, LLPMix mechanism enhances MixMatch-like framework by integrating explicit LLP consistency loss. This ensures that the model's 's predictions adhere to the known bag label proportions while benefiting from the robust semi- semi-supervised training strategy, effectively bridging SSL with the specific demands of LLP.
Preliminary: Problem Setting
In the Learning from Label Proportions (LLP) setting, seek to train instance-level classifier without access to individual labels. Instead, the training data = {x; h C is partitioned into
M disjoint bags Bj. each containing ny instances. For each bag B3 (Tgl Fz,nz) we observe only the class proportions
Pj Pp........ P.,k]
Py. Σ
1 =c),
where y/ (cl...., ] and Ee
1. Each instance x; has an unknown true label yı y and feature embedding (x;,) E IRdd produced by encoder f.
Pj.A Pj.k
Our goal is to learn a classifier h Rd + A that, given f(;;) and the bag-level proportions {py}. recovers accurate estimates of the hidden labels. In other words, h f should predict y; for each J;; by leveraging only the aggregated supervision provided by the P.j-
4 Method
this section, we detail our proposed framework. Beginning with overview of the whole methodology and formal definition of problems, we give a detailed illustration of each component in our method.
4.1 Overview
To solve the LLP problem, we propose our RLPL framework as illustrated in Fig. Our framework possesses two training stages. For the first stage, we utilize unsupervised contrastive representation learning strategy train encoder and leverage bag label proportions guide classifier head training. After first-stage training, we obtain the initial naive classifier pseudo- labells for next-stage training. Since the initial classifier is trained under bag-leved supervi- sion, the aseudo-labels generated by this classifier imply the label proportions knowledge. These meaningful labels are sent to the second stage. However, these labels are usually noisy since initial classifier cannot give full-correct pseudo-labeling. To cope with these noises and exploit the information brought by pseudo-labels, we propose our L.LP-OTD mechanism distinguish high- confident pscudo-labels out of the pscudo-label set and discard low-confident label set. This robust
auxiliary classifier to generate

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distinguishing mechanism provides a strong support for the following training. After LLP-OTD divides the pseudo-label dataset into high-confident pseudo-label set and unlabeled set, we establish LLPMix to train on the LLP-OTTD-divin
dataset. Given the similarity between LLP-OTD-diviee dataset and semi-supervised dataset, we organically integrate I.I.P constraints and MixMatch 2 learning into LLPMix, which not only make full use of bag-level information but also leverage instance-level high-confident labels and samples themselves, providing a instance-bag dual-level supervision for main classifier training.
4.2 First Stage: Bag-level Supervised Naive Classifier
The primary goal of the first stage is to obtain initial, albeit potentially naive, instance- classifier by leveraging the available weak supervision signal, i.c., the bag-level label proportions. This stage consists of two steps: representation learning and initial classifier training.
Firstly, to effectively capture the underlying structure and semantics of the instance data, we employ an unsupervised contrastive representation learning strategy, such as SimCLR 77. to pre-train the feature encoder f X + Rd. This step learns instance representations f(x;) for all xy E without using any label or proportion information, providing foundation for subsequent tasks.
Secondly, we train an initial classifier head, denoted as himit R AK where A is the K- dimensional probability simplex. This classifier takes the learned feature representation f(x,) as input and outputs a probability distribution over the K classes, q. (f (=:)), where qi I x and a represents the predicted probability that instance x, belongs class ck.
The training of himi is guided solely by the bag-level label proportions {P, Specifically, for each bag Bj. we can estimate the predicted label proportions P. [. p by averaging the predicted instance probabilities within the bag:
PJ.k
Σ qi,k
TiJ Z.EB;
The classifier head mit is then trained by minimizing the discrepancy between the predicted bag proportions Py and the true bag proportions Pj across all bags. A common choice for the loss function is the Kullback-Leibler (KL) divergence:
M
Latagel
Σ M
DKs(p,
J=1
Minimizing this loss encourages the classifier's average predictions within each bag to align with the known ground-truth -tuut proportions.
After training, this initial classifier Ainit can be used to generate initial pseudo-labels it for each instance vi, typically by selecting the class with the highest predicted probability:
D: arg max G..k
eyEy
This initial stage provides pseudo-labels (ú. I that incorporate LLP prior knowledge by respecting bag proportions. However, due to the inherently ambiguous nature of bag-level supervision, the classifier Ami is trained with weak signals, potentially leading to unreliable predictions and noisy pseudo-labels. Consequently, these initial pseudo-labels require further refinement and robust handling in a subsequent stage to effectively train an accurate instance-level classifier, motivating the introduction of our second stage (detailed in Section 4.3)
4.3 Second stage: Instance-Bag Dual-level Guided Robust Main Classifier
The initial classifier uses only bag-level supervision, but as shown by Yu et al. 34 even perfect alignment with bag proportions does not ensure correct instance-level predictions. Consequently, its pseudo-labels tthelh though informed by LLP priors priors-remain noisy. To address this, we propose LLP-Optimal Transport Denoising (LLP-OTD) which separates initial lseudo-labels into high- confidence and low-confidence subsets, retaining only the former for training. This yields refined, semi-labeled dataset of reliable pseudo-labels. Building on this, we introduce I.L Match- inspired[2] superviee framework that enforces begel consistency while leveraging the clean pseudo-labels to train the final instance-level classifier.

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4.3.1 LLP-OTD: LLP-guided Optimal Transport Dividing
The LLP-OTD mechanism aims to refine the initial pseudo-labels (i by employing an iterative optimal transport process. This process is guided by both the instance feature geometry, derived from the main classifier's encoder (x,,n and the known bag-level label proportions Pj. LLP- OTD consists of two main steps: iterative pseudo-label refinement via OT, and confident sample partitioning.
Iterative Pseudo-Label Refinement This refinement performed in two OT passes. Let Smain (x:) be the feature representation of instance z; from the current main encoder. The initial pseudo-labels YJ are obtained from Eq.
The first OT pass commences with the calculation of initial class barycenters. For each class ca E V, its barycenter 2/ H is computed as the mean of features fmiin(;;) (z,) for instances 2; initially assigned the pseudo-label ÿ = Cg:
(0) Eisi.ee Smain (x4)
(4)
Kil0: = =c
Should any class lack assigned instances, its barycenter can be initialized using random values or alternative heuristics. Subsequently, an L.P--aaare cost matrix ((0) is constructed. The cost C of associating instance 2; (from bag By(6) with true proportions Py()) with class cą (represented by
(0) is defined as: Mk
C = 11 (2:)
U4 1 + AOTD - P5(3),k)
(0)
(5)
Here, Pj().* denotes the true proportion of class Cj in bag By(g- and AOTD 2 0 is a hyperparameter that balances the Euclidean feature distance against the LLP proportion penalty. This penalty term serves to discourage the assignment of an instance to a class that is known to be rare or absent within its originating bag. Embedding the LLP prior directly into the cost matrix Ck,i in this manner fundamentally reshapes the matching landscape, enforcing bag-level consistency at the individual instance-prototype level rather than only at the final aggregated proportion level. With the cost matrix established, an entropy-rege mngropyrregularized optimal transport problem is solved to find an optimal transport plan T E IR- x
(1) arg min (el((a.)
ΣΣ
Te. c (T(T)
(6)
where H(T) = E.. Tlog 1) is the entropy regularization, Y 0 its strength, and Ma, b) represents the set valid transport plans satisfying marginal constraints a E g and b E - (typically uniform vectors, c.g., $1 1x and b 1NN). The entropy-regularized OT problem in biis strictly convex optimization over a compact convex set, which guarantees a unique optimal solution T that can be efficiently found using the Sinkhorn Kinkhorn-Knopp algorithm. A formal proof the solution's existence, uniqueness, and algorithmic convergence, along with an analysis of the L.P-Propootion Penalty's role, is provided in Appendix. The pseudo-label for each instance v, then updated to o by selecting the class Cg that receives the maximum "mass" from T()):
0 arg max (1))
on eyEy k.
The second OT pass aims further refine these pseudo-labels. It begins by recalculating class barycenters, u using the updated pseudo-labels f1(1) and the same instance features fmain (z;):
(1) Σne =ca fmain( (2xe)
(8)
(il
ca =cx
new cost matrix (()) then constructed using these refined barycenters 4 following the same formulation as Eq. 5
C,K [l.fmain (24) -14 11113 AOTD P)())k)
(9)
5

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Another entropy-reeullarized OT problem (analogous to Eq. 6 is solved with C to yield new transport plan T The final OT-refined pseudo-labels, D1 are then determined from T(?);
D
arg max (2(2) CLEY
(10)
This two-pass iterative process allows the barycenter representations and pseudo-label assignments to mutually refine each other, guided by both feature similarity and LLP constraints, thereby enhancing the quality of the pseudo-labels.
Confident Sample Partitioning After obtaining the final OT-refined pseudo-labels 7 we partition the dataset D into high-confidence labeled set Dr and a low-confidence unlabeled set Dy. The partitioning is based on the agreement between the initial pseudo-labels D: (from Eq. 3 and the OT-refined pseudo-labels o.
Ffgh-CConfidene Labeled Set D,: Instances where the initial pseudo-label and the OT-refined pseudo-label agree are considered high-confidence. Their labels taken j.
Dz = { [ E D. = is}
(11)
low-Confidence Unlabeled Set Du: Instances where the pseudo-labels disagree are considered low-confidence. These instances treated unlabeled in the subsequent L.PMixx training stage.
Du = { E D, U. + ū.}
(12)
This partitioning strategy aims to select more reliable pseudo-labela for supervised training while leveraging the remaining instances in an unsupervised or semi suni--upereieed manner, thus mitigating the impact of noise from the initial pseudo-laheling. The sets Dr and Du are then used the L.PPMxx framework.
4.3.2 LLPMix: Semi-Supervised Learning with LLP Consistency
Building on the high high-confidence labeled set Dr and the unlabeled set Dy produced by LLP-OTD, LLPMix integrates standard semi-supervisee learning with an explicit bag-level consistency con- straint. First, for each example in Drr, we generate several weak augmentations, collect the model's predictions, and sharpen their average to obtain soft pseudo-labels. Next, we mix labeled and unlabeled sxamples-includiig
both their inputs and labels-using the MixUp approach, thereby creating a unified training batch that blends reliable OT-refined labels with guessed labels. Finally, optimize combined objective comprising supervised classification loss on the mixed labeled data, an unsupervised consistency loss on the mixed unlabeled data, and a KL divergence-baser term that ensures the model's aggregated predictions over each original bag adhere the known bag proportions. This streamlined LLPMix procedure effectively harnesses both high-confidence pseudo-labele and legelevel supervision to drive robust instance-level learning under the LLP setting.
The core of LLPMix lies in loss funetion, which combines standard supervised cross-entropy loss Ls for labeled data (from Diix originating from Dr.), an unsupervised consistency loss Lu for unlabeled data (from D Dmnx originating from Dt;), and our novel LLP consistency term L...P.Cons. Cons:
CLLPMIX = Ls + wy Lu + WLLL
(13)
where wo, WLLP are weighting coefficients.
The crucial LLP consistency term, LLLP.COMS: ensures that the model's predictions adhere to the original bag-level proportions. This term calculated *before* the MixUp operation. Specifically, let Bori be the set of original instances (from D,, UDr that form the basis of the current mini-batch. For each original bag By represented in Buigg, we calculate the predicted proportion phatch by averaging the predictions Immin [main( (Aug(r;))) for all instances æ; Borig that originated from Bj. For a; E Dr. Aug(z:) is x; itself; for x; E Dr, Aug(x:) is one of its augmented versions used for label guessing:
batch Pjk
Σ
Amain (fmai (Aug(x:)))
(14)
Borig 1x, E B₁H Mxi

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The LLP consistency loss is then the KL divergence between these batch-wise predicted proportions and the true bag proportions Pj-
LLLP-COnn
Σ
DKL hatch)
(15)
[Brags|
B, BraB
where Bags is the set of unique original bags represented in Brig. This term guides the main classifier to produce instance-level predictions that, when aggregated at the bag-level from pre-Mix samples, align with the known ground-truth proportions. This injects the LLP prior directly into the semi- supervised learning phase, complementing the instance-level signals.
4.4 Overall Training Algorithm
The proposed RLPL framework integrates two-stage training to leverage instance-bag dual-level information to form robust label proportions learning model. The algorithm summarized in the appendix and outlines the complete procedure.
Experiment
5.1 Experimental Setup
Dataset We utilized four standard benchmark datasets commonly employed in Learning from
Label Proportions (LLP) research. These datasets are CIFAR-IO, CIFAR-I00 [4, SVHN (2I] and Mini- Mini-ImageNet [30]. Both the CIFAR-10 and CIFAR-100 100 datasets 14 contain 50,000 training images and 10,000 test images. Each image 32 32 color natural scene, categorized into 10 and 100 classes, respectively. The SVHN dataset consists of 32 x 32 RGB images of digits, with 73,257 images for training and 26,032 for testing: additional training samples were not used in our experiments. Imagee, subset of the ImageNet dataset, includes 100 classes, each with 80 images for training and 20 for testing, all resized to 64 64 pixels.
Baseline We compare RLPL against seven representative LLP approaches. LLPFC formulates learning from label proportions by minimizing the KL divergence between predicted and true bag proportions within a fuzzy-clustering framework [23). DLLP employs an end-to-end convolutional network that integrates labeled samples and bag-level proportions via a reshaped cross-entropy loss (25] LLP-VAT augments virtual adversarial training with consistency regularization to enforce smoothness in instance predictions under local perturbations (28). OT-LLP leverages entropically regularized optimal transport to impose exact proportion constraints on the classifier [17] SoftMatch overcomes the quantity-quality trade-off by weighting pseudo-labels using truncated Gaussian function combined with uniform alignment [5] FLMm derives mneanooperatorbbasee sufficient statistic for proper scoring losses, enabling learning from bag proportions without instance labels (33) Finally, L'P-AHIL introduces dual antropy-based weights to form auxiliary high-confidence instance- level losses, jointly optimized with bag-level supervision 200
5.2 Implementation Details
Bag Partition For each dataset, bags of a specified size M were formed by randomly selecting M samples from the training set, ensuring that samples distinct bags do not overlap. The class proportion information within each bag guided the training process, without the use of true instance labels. Following established practices [20], we selected M from the set 16, 32, 64, 128, 256. Since each dataset contains a balanced number of samples per class, this bag generation method yields relatively balanced class proportions.
Results and Analysis Table I presents classification accuracies of RLPL and satte-of-the-ann baselines on CIFAR-10. CIFAR- 100, SVHN, and MinilmageNett under five different bag sizes (based on (20). Across all datasets, RLPL demonstrates competitive or superior performance compared to prior methods. On MinilmageNer, RLPL achieves the best average accuracy (54.52%) and maintains a low coefficient of variation (CV=0.171), outperforming methods such as DLLP, LLP-VAT, and IHP--AHI,, which exhibit significant drops as bag size increases. For CIFAR-10, 10, RIPI. achieves an average accuracy of 93.7 71% ,, slightly behind L L'PILHIL (94.21%), while maintaining very low
7

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Table 1: Performance comparison of various methods on MinilmageNett CIFAR-I0, CIFAR-100, and SVHN datasets for different bag sizes. For each dataset, results across bag sizes [16, 32, 64, 128. 256] are reported, along with their average performance (f) and coefficient of variation (CV, J). The best performing method each column is highlighted in bold, and the second best is underlined. Baseline methods, excluding RLPL(Ours), are reproduced experimental results from Ma et al. [20 The dash "-" signifies missing or inapplicable results.
Bag Size
Dataset
Model
Average (1) Coeff. of Var. (4)
16
32
64
128
256
LLPFC
DLLP
64.53
55.37
27.57
9.06
3.40
31.99
0.762
LLP-VAT
64.17
54.36
30.96
9.69
4.90
32.82
0.717
ROT
67.02
27.49
6.01
3.50
1.75
21.15
1.170
MinilmageNet
SoftMatch
2.02
1.86
1.95
1.72
1.87
1.88
0.053
FLMm
L'P-AHIL
70.26 59.81
37.51
16.91
7.46
38.39
0.627
RLPL (Ours)
62.63
61.11
58.42
53.46
36.98
54.52
0.171
LLPFC
84.10 71.70 52.71
20.78
18.79
49.62
0,531
DLLP
91.59 88.61 79.76 64.95
44.87
73,96
0,233
LLP-VAT
91.80 89.11 78.75 63.89
46.93
74.10
0,226
ROT
94.86 94.34 93.97 92.23
63.10
87.70
0.141
CIFAR-10
SoftMatch
95.24 95.25 94.23 93.87
48.20
85.36
0.218
FLMm
92.34 92.00 91.74 91.54
91.29
91.78
0.004
L'P-AHIL
94.96 95.00 94.58 93.64
92.88
94.21
0.009
RLPL (Ours)
92.54 94.02 93.50
94.53
93.95
93.71
0.007
LLPFC
DLLP
71.28 69.92
53.58
25.86
8.82
45.89
0.539
LLP-VAT
73.85 71.62
65.31
37.36
2.79
50.19
0,539
ROT
72.74 69.31
17.48
11.02
2.86
34.68
0.867
CIFAR-100
SoftMatch
80.14 2.40
2.04
2.12
1.98
17.74
1.759
FLMm
66.16 65.59
64.07
61.25
57.10
62.83
0.053
L'P-AHIL
78.65
77.30
76.52
72.21
23.56
65.65
0.322
RLPL (Ours)
68.96
68.88
68.39
66.73
65.41
67.67
0.021
LLPFC
93.04 23.26 21.28 20.54
19.58
35.54
0,810
DLLP
96.90 96.93 96.64 95.51
94.34
96.06
0.010
LLP-VAT
96.88 96.68 96.38 95.29
92.18
95.48
0,018
ROT
95.54
94.78 96.75 26.00
12.15
65.04
0.581
SVHN
SoftMatch
22.39 19.68 19.60 19.64
19.57
20.18
0.055
FLMm
L'P-AHIL
97.91 97.88
97.74
97.67
96.98
97.64
0.003
RLPL (Ours)
94.64
94.83 95.02 94.92 95.18
94.92
0.002
CV of 0.007, indicating strong robustness. On the more challenging CIFAR-100 dataset, RLPL outperforms all baselines with the highest average accuracy (67.67%) and lowest variability (CV=0.021), showcasing its resistance to label dilution. Similarly, on SVHN, RIPI. achieves stable and high performance (Avg=94.92%. CV=0.002 0020,, comparable to L'P-AHIL and significantly better than conventional baselines such as LLPFC and ROT. Overall, RLPL consistently ranks among the top performers across datasets and bag sizes, demonstrating that it is less sensitive to the bag size setting. The consistently low coefficients of variation further verify RLPL's robustness, highlighting its capacity to maintain stable and reliable performance across varying weak supervision levels.
We also observe from Table T that RLPL's performance advantage is particularly pronounced in large-bag scenarios (e.g., bag sizes and 256). hypothesize this stems from the degree of label ambiguity. With smaller bags, the label proportions provide relatively strong and unambiguous supervisory signal, allowing simpler methods to perform reasonably well. Conversely, bag size increases, the label ambiguity escalates significantly; single proportion vector correspond to

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Table 2: Ablation Study
Method
RLPL
w/o LLP- Proporrroo Penalty
w/o LLP-OTD
w/o LLPMix
Accuracy (%)
93.95
93.36
84.98
92.33
Table 3: Performance comparison the UCI Adult tabular dataset.
Method
RLPL
ROT
L2P-AHIL
Accuracy (%)
77.57
72.82
75.99
Table 4: Performance (%) on CIFAR-10 10 (bag size 256) with noisy label proportions.
Noise Type
Gaussian (Mod.)
Gaussian (Heavy)
Uniform (Mod.)
Uniform (Heavy)
Accuracy (%)
94.05
93.84
93.95
93.94
Table 5: Performance (%) long-tailed CIFAR-10 10 (bag size 256) with varying imbalance ratios.
Imbalance Ratio (IR)
5
10
15
50
100
Accuracy (%)
93.55
91.00
89.72
81.61
70.28
vast number of potential instance-level label configurations. In these high-ambiguity settings, the robustness of our LLP-OTD refinement process becomes critical. Its ability to effectively denoise pseudo-labels from a highly ambiguous signal allows RI RI.PI. to excel and significantly outperform baselines, whereas other methods may struggle with the diluted supervision.
All experiments were conducted using a single NVIDIA RTX 4090 GPU with 24GB memory. For each setting, we repeated the experiment five times with different random seeds and report the mean and standard deviation of results. More detailed hyperparameter configurations are provided Appendix.
5.3 Ablation Study
As illustrated in Table 2 we conducted series of ablation studies to validate the effectiveness of key components within our proposed RLPL. model on CIFAR-10 Dataset setting bag size as 256. Removing the entire LLP-guided Optimal Transport Denoising (LLP-OTD) module (RLPL w/o LLP-OTD) results in the most significant performance drop, with accuracy decreasing from 93.95% to 84.98%. This underscores the critical role of the LLP-OTD module refining pseudo-labels and substantially boosting model performance. When the LLP proportion penalty term is excluded from the OT cost function within the LLP-OTD module (RLPI. w/o LLP I.L--Propottion Penalty), the accuracy falls to 93.36%, demonstrating the importance of integrating true bag-level proportion information to guide the pseudo-label correction process effectively. Furthermore, omitting the subsequent LLPMix (MixMatch-basse' semi-supervised Icarning) stage (RLPL w/o LLPMix) leads 92.33%, indicating that the semi-supervised learning component successfully leverages the data refined by LLP-OTD (both the reliable labeled set and the distinguished unlabeled set) further enhance the model's generalization capabilities. Collectively, these results clearly demonstrate that the LLP proportion penalty, the core LLP-OTD refinement module, and the LLPMix strategy all contribute positively to the final performance of RLPL, with the LLP-OTD module exhibiting a particularly pronounced impact.
accuracy of
5.4 Robustness and Generalization Analysis
We conducted further experiments to evaluate RLPL's generalization beyond vision tasks and its robustness under challenging data conditions, including noisy proportions and class imbalance.
Generalization Tabular Data assess the applicability of RIPLL beyond image modalities, we performed experiments on the widely-used UCI Adult tabular dataset. As shown in Table 3 RLPL

---

achieves an accuracy of 77.57%, outperforming strong LLP baselines. This result demonstrates that our framework effective in the tabular data domain and generalizes well to non-vision data.
Robustness Noisy Proportions We evaluated RLPL's resilience to imperfect supervision by injecting noise into the bag proportions CIFAR-10 (bag size 256). We applied two types of noise: Gaussian (p' clip(p + + (0, ))) and Uniform (p' = cp + (rr r))) at moderate and heavy levels. Table 4 shows that RLPL's performance remains remarkably stable, with only minimal accuracy drop even under heavy noise (e.g., 93.84% with heavy Gaussian noise). This highlights the robustness of the LLP-OTD mechanism in handling ambiguous and noisy supervisory signals.
Robustness Class Imbalance test performance in non-uniform data distributions, we con- structed long-tailed version of CIFAR-10, parameterized by the imbalance ratio (IR)-ther ratio of sample sizes between the most and least frequent classes. As detailed in Table while accuracy naturally degrades the imbalance becomes more extreme, RLPL maintains strong performance, achieving 70.28% even at a severe IR of 100. This demonstrates its robustness handling highly imbalanced class distributions.
6
Conclusion
This paper addresses the issue of noisy pscudo-labels in Learning from Label Proportions (LLP) by proposing the Robust Label Proportions Learning (RLPL) framework. This two-stage framework first pretrains encoder via contrastive learning and trains initial classifier using bag proportion information. It then introduces the core LLP-OTD (LLP-pent) -ene-lenl Optimal Transport- Dividing) mechanism to refine pseudo-labels, dividing data into a high-confidence labeled and an unlabeled set. Finally, the LLPMix strategy, inspired by MixMatch, integrates the refined pseudo- labels and bag proportion constraints within semi-supervised pipeline train the main classifier. Extensive experiments on standard LLP benchmark datasets demonstrate that RLPL's performance comparable current Mate-of-theear methods, exhibiting stronger robustness, particularly in challenging large-bag scenarios. Ablation studies also validate the effectiveness of each component, especially LLP-OTD in filtering noise. Future research directions include extending RLPL to more complex data modalities, exploring adaptive mechanisms for LLP-OTD, and theoretically investigating its noise- filtering capabilities. We will discuss the limitations of our current work in the appendix.
Label
Acknowledgments
This work was supported by the National Natural Science Foundation of China under Grants U24A20322 and 62576094. This work is also supported by the Big Data Computing Center of Southeast University.
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A Experiment
A.1 Detailed Experiment Results
This section provides a more comprehensive presentation of our experimental findings. In comparison to the performance summary presented in the main paper, Table this appendix additionally incorporates the standard deviation (std) for each reported metric. The inclusion of these standard deviations is intended enhance the statistical rigor the stability of the RLPL model's performance across various datasets and bag size configurations. The average performance (Avg) and coefficient of variation (CV) values reported herein, along with the newly included standard deviations, are derived from five independent experimental runs conducted. This repetition of experiments serves to validate the consistency and reliability of our model's 's reported performance.
results and more thoroughly demonstrate
Table 6: Performance comparison of various methods on MinilmageNet, CIFAR-10. 10, CIFAR-I00, and SVHN datasets for different bag sizes (mean + std). For each dataset, results across bag sizes [16, 32, 64, 128, 256] are reported, along with their average performance (1) and coefficient of variation (CV, D). The best performing method in each column is highlighted in bold, and the second best is underlined. The dash '-' signifies missing or inapplicable results.
Bag Size
Dataset
Model
Average (1) Coeff. of Var. (1)
16
32
64
128
256
LLPFC
DLLP
64.5320.1
55.37
27.57
9.06gau
3.40
31,99
0.762
LLP-VAT
64.17=0.4
54.36
30,96-024
9.69
4,90g0m
32.82
0.717
MinilmageNet
ROT
67.02-2014
27.49
6.01
3.50ga
1.75
21.15
1.170
SoftMutch
2.02.am
1.86
1.95
1.72ga30
1.87
1.88
0,053
FLMm
I.PI-HII.
70.26-0.6
59.81
37,51
16.91
7.46
38.39
0.627
RLPL(OUrS)
62.63
61.11
58.42
53,46.0.26
36.98
54.52
0.171
LLPFC
84.10_o.9
71.70ga78
52.71
20.78
18. 18.794021
49.62
0.531
DLLP
91.5920.52
88
61 4.00
79.76_1.45
64.95.0.01
44.87
73.96
0.233
LLP-VAT
91,80
89.11
78.75=046
63.89
46.93
74.10
0.226
ROT
94.86_0.4
94.34
93.97
92.23
63.101084
87.70
0.141
CIFAR-10
SoftMatch
95.24_..12 95.25
94.23
93.87,0.22
48
48.30
85.36
0.218
FLMm
92.34
92.00
91.74
91.54
91.29
91.78
0.004
L'P-AHIL
94.96
95.00
94.58
93.64
92.88
94.21
0.009
RLPL(OUrS)
_0.54_000
94.02
93,50_oas
94.5310.12
93.95
93.71
0.007
LLPFC
DLLP
71.28
69.92
69.
53.58_1.00
25.8612.15
8.82
45.89
0.539
ILP-VAT
73.85_012
71.62
65.31
37.361043
2.79
50.19
0.539
CIFAR-100
ROT
72.74
69.3 4022
17.48
11.02.20.9
2.86
34.68
0.867
SoftMatch
80.14
2.40
2.04
2.12:am
1.98
17.74
1.759
FLMM
66.16
65.59
64.07
61.25
$7.10
62.83
0.053
L'P-AHIL
78.65
77.30
76.52
72.21
10.37
23.56
65.65
0.322
RLPL(OUrs)
68.96_0.09
68. s009
68.
68.39-0.18
66.73+a.11
65.41 20.15
67.67
0.021
LLPFC
93.04_021
23.261n65
21.28
20.5410.37
19.58
35.54
0.810
DLLP
96.90-0.0
96.93
96.64_a.0
95.51
94.34
96.06
0.010
LLP-VAT
96.88 96.68
96.38
95.2910.17
92.18
95.48
0.018
SVHN
ROT
95.54
94.78
96.751o11
26.00
12.15
65.04
0.581
SoftMatch
22.39
19.68
19.60
19.64.n.14 19
20.18
0.055
FLMm
L'P-AHIL
97.91=aaz
97.88
97.74=a06
97.67-aa.7
96.98
97.64
0.003
RIPr((Our))
94.64_0.8
94.83
95.022200s
94.9210.17
95.18
94.92
0.002
A.2 Hyperparameter Configurations
To ensure the reproducibility of our experimental results, we detail the key hyperparameter configura- tions for our Robust Label Proportions Learning (RLPL) framework below.
First Stage Training (Auxiliary Classifier) In the first stage, for training the encoder via unsuper- vised contrastive representation learning, we employ SimCLR strategy. The encoder backbone ResNet-18. The projection head in SimCLR consists of single linear layer. For SimCLR training,
13

---

we utilize the Adam optimizer, setting the learning rate to 1 x 10 and the weight decay to 1 x 10 Subsequently, the auxiliary classifier head, also a single MLP layer, trained using the Adam optimizer with learning rate of x 10 and a weight decay of 1 x 10-". This stage utilizes only bag-level supervision from the bag dataset, minimizing the Kullback-Leiille (KI.) divergence between the true bag proportions Pj and the predicted bag proportions P.j.
Second Stage Training (Main Classifier) During the second stage for training the main classifier, the L. P.P-Prroortion Penalty coefficient, denoted as λOTD. in the LLP-OTD module's ' cost function set to 0.1. This coefficient balances the Euclidean feature distance against the LLP proportion penalty. For the LLPMix training, the weight for the Consisssst loss, denoted as WLLP in the combined objective function LLLP is set to 1 10-4, The main classifier is trained using the SGD optimizer. The initial learning rate is set to 0.02, with a momentum of 0.9 and a weight decay of 5 10
Backbone Architectures for Main Classifier To facilitate fair comparison with baseline methods, we adopt specific backbone architectures for the main classifier across different datasets. For the CIFAR-10 and SVHN datasets, both our model and the baselines utilize WRN-28-2 as the backbone. For the CIFAR-100 dataset, WRN-28-8 is employed. On the Mini-ImageNet dataset, ResNet-18 the backbone.
Hyperparameter Sensitivity Analysis. To validate the robustness choices, we performed a sensitivity analysis for the two key coefficients: the LLP-Proportion Penalty ÀOTD (Eq. 5) and the JLP-Consistency loss weight WLLP (Eq. 13). We evaluated various combinations on the CIFAR-10 dataset (bag size 256), recording the accuracy at 100 epochs. The results, presented in Table 7 demonstrate that RLPL achieves high and stable accuracy (ranging from 90.82% to 91.44%) across a wide range of values for both parameters, This indicates that our framework is not overly sensitive their specific settings, confirming robustness.
model hyperparameter
Table 7: Hyperparameter sensitivity analysis on CIFAR-I0 10 (bag size 256). report accuracy (%) at 100 epochs for different combinations of λOTD and WLLP.
WLLP
0.05
0.1
0.2
0.5
1.0
AOTD
0.05
91.06%
90.83%
91.44%
90.92%
90.82%
0.1
90.94%
90.95%
91.33%
91.17%
91.32%
0.2
90.97%
91.28%
90.92%
90.94%
91.20%
0.5
91.15%
90.98%
91.06%
91.12%
91.02%
A.3 Additional Baseline Comparisons
To further contextualize the performance of RLPL, conducted additional experiments CIFAR- 10 comparing our method against other notable weakly-supervised frameworks, including Count Loss (6) and GLWS As shown in Table RLPI. consistently outperforms these baselines across various bag sizes, particularly demonstrating significant advantage as the bag size increases.
Table 8: Performance comparison (%)
against Count Loss and GLWS on CIFAR-10 10.
Bag Size
Count Loss
GLWS
RLPL (Ours)
16
87.5%
85.46%
92.54%
32
83.61%
81.1 1%
94.02 %
64
68.35%
64.64%
93.50%
A.4 Computational Cost and Scalability Analysis
We provide detailed analysis of the computational requirements of our framework.
14

---

Theoretical Complexity. The computational cost of our LLP-OTD module is dominated by two steps: 1) Constructing the x N cost matrix, which takes O(KNd) time, where K is classes, N is instances, and d is the feature dimension; 2) Solving the entropyy-reguarrized OT problem using the Sinkhorn- SatioorrnKKoppppppp Knopp algorithm for I. iterations, which takes (TLKN) time. The total complexity is therefore O K (d + L)). As this complexity is linear with respect to the number of instances N, our method is highly scalable and efficient for large-scale LLP problems, distinguishing it from classical OT solvers that can have ( complexity.
Impact of Sinkhorn Iterations. The complexity depends on the number of Sinkhorn iterations, L. We evaluated its impact on accuracy on CIFAR-10 (hag size=256) epoch 50. As shown in Table 9 performance saturates quickly; a small number of iterations (e.g., I, = 5) is sufficient to achieve strong results. This confirms that can be treated as a small constant, keeping the practical cost low.
Table 9: Accuracy (%) vs. Number of Sinkhorn Iterations (L) on CIFAR-10 (bag size=256).
Iterations (L)
2
3
5
10
20
50
Accuracy (%)
89.24
89.48
89.96
90.00
89.61
89.85
89.68
Empirical Training Time. We profiled the single-epoch training time of RLPL against baselines on CIFAR-10 10 (bag size=256) on an NVIDIA RTX 4090. As shown in Table 10. RLPL's runtime is comparable to other methods like LLP-VAT, and the performance gains justify the modest increase over methods like L2P-AHIL. We also analyzed the prohibitive cost of replacing our Estagel KL divergence with a Count Loss. which involves a dynamic programming step with O(n)) complexity per bag. As shown in Table [T] the training time for Count Loss scales quadratically with bag size, becoming intractable, whereas our KL divergence loss remains highly efficient.
Table 10: Single-epoch training time (seconds) on CIFAR-10 (bag size=256).
Model
L2P-AHIL
ROT
LLP-VAT
RLPL (Ours)
Time (s)
10.61
10.89
16.09
16.11
Table 11: Training time (s/epoch) of Stage I loss: KL Divergence vs. Count Loss.
Bag Size
16
32
64
128
Count Loss (s/epoch)
2245
5012
9215
19964
KL Divergence (s/epoch)
17
9
6
6
A.5 Adherence to Bag Proportions
To quantitatively verify that our final classifier respects the original bag-level constraints, we measured the Mean Absolute Error (MAE) between the classifier's aggregated instance-level predictions and the ground-truth bag proportions. A lower MAE indicates better adherence. We compared RLPI. against the strong 1.2P-AHIL baseline on CIFAR-10 across all bag sizes. As shown Table RI.PL consistently achieves lower MAE, demonstrating superior adherence to the bag-level supervision. This effect is particularly notable in more challenging small-bag scenarios (e.g., 30.6% lower MAE than L2P-AHIL at bag size 16), confirming that the LLLLP-COS term in our LLPMix stage (Eq. 15) effectively preserves the proportion constraints throughout the semi-supervised training phase.
B Theoretical Guarantees for LLP-OTD
In the main paper, we stated that the entropy-regularizer optimal transport (OT) problem at the core of LLP-OTD is well-posed and efficiently solvable. Here, we provide the formal theoretical guarantees, addressing the existence and uniqueness of the solution, the convergence of the algorithm, and the role of our LLP- AL--Propotionn Penalty.
15

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Table 12: Mean Absolute Error (MAE) (lower is better) of predicted bag proportions on CIFAR-10.
Method
Bag Size 16
Bag Size 32
Bag Size 64
Bag Size 128
Bag Size 256
L2P-AHIL
0.015280 0.010600
0.013498 0.009505
0.010868 0.008061
0.008544
0.006758
RLPL
0.006725
0.005273
B.1 Existence and Uniqueness of the Optimal Solution
We first establish that the core optimization problem in our LLP-OTD mechanism has a unique global minimizer. The problem is defined
min (au(a.))
(C, -TH(T),
(16)
where (C, = k., Tk. is the transport cost, H (T) = - Te. :
Tk, -1. is the entropy term, 7 > 0 is the regularization strength, and (a, b) is the transport polytope defined by marginal constraints (as defined in the main text).
Proof: Let F(T) = (C, T -T) We prove the strict convexity of the objective function.
1. The term (C, T is linear in T, and hence convex.
2.
For the negative entropy term (T = Σn,, T Tk, - 1), its Hessian matrix is diagonal with entries ((-H) L.
ØT? Tv.s Since Tk,s > O within the domain, the Hessian is positive definite, implying - (T) is strictly convex.
3. Therefore, F(T) = (C,T) + 7( (-H(TT) (with 7 > 0) the sum convex function and strictly convex function, and is thus convex.
Furthermore, the feasible set (e, b) ( TTERKNN E. - a Σx Tx.; - bi) is defined by 20
linear equalities and non-negativity constraints, making it convex polytope. Since all constraints are closed (equalities or non-strict inequalities) and the marginals sum to (i.e., 0 5 Tk.i S 1), ((a,)) b) is a non-empty, compact, and convex set.
By standard results in convex optimization, a convex function F(T) optimized over non-empty, compact, and convex set (a, b) has unique global minimizer T*.
B.2 Convergence of the Sinkhorn-Knopi Algorithm
Next, we demonstrate that the Sinkhorn- Knopp algorithm, used to solve the OT problem, converges to the unique optimal solution T" The optimal transport plan T is known to have a specific structure:
Ck.
Th uk KE Uby where KR - exp
(17)
The Sinkhorn-Knoph algorithm is an iterative procedure to find the scaling vectors u R and N that satisfy the marginal constraints.
The convergence can be shown by interpreting the algorithm as an alternating projection procedure using the Kullback-Leibhe (KL) divergence (a specific type of Bregman divergence). Let the of matrices satisfying the row and column constraints be:
Proof:
C, T
R
K 20
Σ. TR. ax} (Row constraints)
C2 = T KXN R 20
Σ Tk, = by} (Column constraints)
Each iteration of the Sinkhorn SinkhornKnoop algorithm (which corresponds to alternating updates of u and v) can be viewed as performing the following alternating KL projections:
(1+ arg min KL(T))
T(").
(18)
TeC
(1+1) arg min KL ( #)), TeC,
(19)
16

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Since C1 and C2 are convex sets, and their intersection C1 nC2 (a, b) is non-empty (containing the unique optimal solution ), the theory of alternating Bregman projections guarantees that this iterative process converges 88] Therefore, the sequence of iterates T produced by the Sinkhorn- Knopp algorithm is guaranteed to converge to the unique optimal solution T
B.3
Role of the LLP-Proportion Penalty
Finally, we provide the intuition for why our innovative LLP-Proportion Penalty helps align the pseudo-label distribution with the true LLP prior. Our effective cost function (Eq. in the main paper) is:
C = (x. - msll2 1 + λOTD P)()).)
(20)
The key component is the penalty term λOTD(1 P.))).)) To understand its effect, analyze the part of the objective function corresponding to this penalty, which we denote LLLP-Poualty (T):
LLLP-Penally Cermall((1 (T)
The (orD(1 P)()).)
(21)
Grouping by bags By. this is equivalent to:
K
LLLP-Penalty (T) ÀOTD
Σ B
Te -Ppka)
(22)
Minimizing this term is equivalent to maximizing the alignment between the transport plan's implied label proportions and the true bag proportions. Let Pr B,) Σie, Th,s the total mass assigned to class from bag Bj by the transport plan T. (Note: this not a normalized distribution yet, but proportional to it). Minimizing LLLP-Penatty (T) is equivalent to:
M
min
Σ Tkis (1 p2An) min Σ iE
Σ
(Pr(k) B₁) Pr Bj) P.AA)
(23)
M
K
min
Pr(k B2) Σ
Pr(k B2) Ps.k .s.k
(24)
#=1
Since E Pr
Ek= This = Eien, by (where b, the marginal for instance
i, typically 1/N), the first term Ek= Pr(k Bj) is constant with respect to the distribution of assignments within the bag. Therefore, minimizing the penalty term becomes equivalent
K ΣV
max T
Pr(k Bj) PYA & max [(Pr(-1B).P( B,))
(25)
==
where Pr( Bj) represents the vector of implied proportions from bag B3 derived from T. Maximizing this dot product encourages the transport plan's 's implied distribution Pr to align with the true prior which is analogous to minimizing statistical divergences like the KL divergence.
Therefore, by introducing the LLP- Proportion Penalty into the cost matrix, directly enforee that the optimal transport plan the unique solution to the optimization prohlem-favon assignments that are consistent with the known bag-level supervision.
C Algorithm
Our overall algorithm is summarized in Algorithm [ where the notations and definitions used are consistent with those introduced in the main paper.
17

---

Algorithm RLPL: Robust Label Proportions Learning
1: Input: Dataset = {(2,, Bx()))= Bag proportions {p,1y Number of classes K.
2: Output: Trained main classifier maaio (fain (main
3: procedure STAGE 1: INITIAL NAIVE CLASSIFIER
4: Pre-train feature encoder f + Rd using SimCLR on D.
5: Initialize classifier head Amm IRd A
6: while not converged do
7:
For each instance 2i, compute predicted probability distribution q: hinin (x())). For each bag Bj. compute predicted proportions Pÿ.b- = m, Ea, Uik-
8:
J
Update hinit by minimizing Litagel 10: end while
9:
Σラ Dxr(P,)le.)
Generate initial pseudo-labels ( using ús arg max o eyyl end procedure
12:
13:
procedure STAGE 2: MAIN CLASSIFIER TRAINING
14:
Initialize main encoder maann (e.g., with weights from /) and main classifier head main- LLP-OTD: Pseudo-Label Denoising and Partitioning
15:
16:
for each training epoch = 1..... Emain do For each instance 24, extract features fmain (2:4).
17:
Compute initial class barycenters 14 (0) 21:0, S
18:
Construct cost matrix ((O) kgj fm (2:) - 11 (0) 113 + AOTO - P)(t),))
19:
20:
Solve OT problem T(1) = arg minreu((, Σn, T - (T) to get T()+
21:
Update pscudo-labels to 9; arg max Ey k,
Σ 1,0 Smm((ze)
Compute refined class barycenters 14
1610 1610)
Construct cost matrix c(!) k.j = [l - H 112 + AOTD - P)()),)
(1)
23:
24:
Solve OT using (()) get T22), then final OT-refined pseudo-labels of
(2(2)x arg m k.
25:
Partition data into Dr, ((r,, )p) E D, Dt (.) and Du (0i/2 E D, 0tTh +
26:
LLPMix: LLP-Consistent Semi-Supervised Learning for each mini-batch B from Dr, Du do For Tu E DDr, generate soft pseudo-label Tu. by averaging sharpened predictions from multiple weak augmentations of zu.
28:
29:
Apply MixUp to inputs and labels (both 6."T for Dz and guessed Tu. for Du) to form
Dmnx-
30:
Compute supervised classification loss Ls mixed labeled samples.
Compute unsupervised consistency loss Lv on mixed unlabeled samples.
32:
Compute predicted bag proportions for the mini-bater (pre-MixUp) as P
shatch
[ar E Bung B,Jl Ex,EBuynt EE (f (Aug(::)))s.
33:
Compute LLP consistency loss LIIP BBand EBJEBOyt (RI(Pilii)
34:
Compute total loss LLIPMIX = Ls + ww Lu + WLILLLLCCOnS Update fmain and Amain by minimizing LILPMIX.
35:
36:
end for
37:
end for
38:
end procedure
39: return /amain ( fmain (-))
18

---

D Limitations and Future Work
While RLPL demonstrates strong performance, its multi-stage framework is inherently more struc- tured than single-stage end-to-end methods. Although our computational analysis (Section (A.4) confirms that the runtime comparable baselines and scalable, future work could explore simpli- fying this refinement pipeline.
Furthermore, although RLPL is competitive in terms of average performance and robustness, other highly specialized methods might exhibit slight advantage on specific datasets or bag sizes. Future research could also focus extending the framework other data modalities, such text, adapting it for the multi-label LLP setting.
19

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Reviews:
[{"review_id": "x7fgY8Ga2B", "reviewer": "Reviewer_oaZQ", "summary": "The authors introduced a novel LLP approach to handle multi-label LLP data. Their approach consists of two stages: 1) contrastive loss is used to train an encoder based on only feature information and then fit a second model based on the contrastive embedding where bag level information is used. This second model can be viewed as a PropotionMatching approach with KL loss. 2) The model from the first stage is then fine-tuned. First this model is used to assign so-called pseudo labels to each instance. And then this pseudo-labeling is updated in an expectation maximization (EM) framework where the centers are updated by using an Optimal Transport (OT) loss instead of L2  like it is done in the original EM.  If the labels are mismatching between two such an update then the labeling is considered not to be confident. \n\n\nThe proposed approach is tested on four image dataset and its superiority is shown with respect to the state of the art. Especially, there is an experiment which shows the impact of an OT update.\n\n\n\n\nEven if this paper does not show any theoretical result, and the approach is quite complex, I am landed on the positive side with this paper. The approach is very natural, like when I was reading it, I was thinking that this approach is what I personally want to try and see how it performs on my LLP problems. The contrastive loss mode with finetuning is a nice idea in my opinion. The experiments however are quite limited, even if the impact of OT loss is nicely demonstrated.", "questions": "I have one major concern, beside the experiments. The approach is multi-label, however the pseudo-labeling is multi-class. This feels a kind of discrepancy between the model and setup. Do the author have some comment on the extension to make this approach truly multi-label?", "limitations": "I think I addressed this in the question.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strength: many different new notions are nicely mixed in an LLP approach. \nWeakness: experiments are limited to image processing data.", "was_revised": false, "final_justification": "", "rebuttal": "# On Theoretical Justification \n\nIn the response, we provide three **theoretical guarantees** for the proposed method. We will add them to the final version of our paper. First, we prove the existence and uniqueness of the **optimal solution** for our entropy-regularized optimal transport (OT) problem. Second, we demonstrate that the Sinkhorn-Knopp algorithm, which we employ, is **guaranteed to converge to this unique solution**. Finally, we explain how our innovative LLP-Proportion Penalty, incorporated into the cost function, **guides the pseudo-label distribution to align with the true LLP prior**. \n\n## **1. Existence and Uniqueness of the Optimal Solution**\n\nWe first establish that the core optimization problem in our LLP-OTD mechanism has a unique global minimizer. The problem is defined as:\n$$\n\\min_{T \\in U(a, b)} \\langle C, T \\rangle - \\gamma H(T),\n$$\nwhere $\\langle C, T \\rangle = \\sum_{k,i} C_{k,i} T_{k,i}$ is the transport cost, $H(T) = -\\sum_{k,i} T_{k,i} (\\log T_{k,i} - 1)$ is the entropy term, $\\gamma > 0$ is the regularization strength, and $U(a, b)$ is the transport polytope defined by marginal constraints.\n\n**Proof:** \n\nLet $F(T) = \\langle C, T \\rangle - \\gamma H(T)$. Then we will prove the strict convexity of the objective function.  For the term $\\langle C, T \\rangle$, it is linear in $T$, hence convex. For the entropy term $-H(T) = \\sum_{k,i} T_{k,i} (\\log T_{k,i} - 1)$, we can compute its Hessian matrix:\n$$\n\\frac{\\partial^2 (-H)}{\\partial T_{k,i}^2} = \\frac{1}{T_{k,i}} > 0 \\quad \\text{since } T_{k,i} > 0,\n$$\nimplying strict convexity of $-H(T)$. Therefore, $F(T) = \\langle C, T \\rangle - \\gamma H(T)$ is the sum of a convex and strictly convex function, and thus strictly convex. Meanwhile, for $U(a,b)= \\left\\{ T \\in \\mathbb{R}_{\\geq 0}^{K \\times N} \\, \\middle| \\, \\sum_i T_{k,i} = a_k, \\sum_k T_{k,i} = b_i \\right\\}$, we will prove it to be convex, closed and bounded. As $U(a,b)$ is defined by linear equalities and nonnegativity constraints, it is a convex polytope. Because all constraints are equation or non-strict inequality, $U(a,b)$ is closed. Since $\\sum a_k = \\sum b_i = 1$, we have $0 \\le T_{k,i} \\le 1$, and $U(a,b)$ is a compact convex set. Finally, by standard results in convex optimization, a strictly convex function over a non-empty compact convex set has a **unique global minimizer** $T^*$.\n\n## **2. Convergence of the Sinkhorn-Knopp Algorithm**\n\nNext, we prove that the Sinkhorn-Knopp algorithm, used to solve the OT problem, converges to the unique optimal solution $T^*$. The optimal transport plan $T^*$ is known to have a specific structure:\n$$\nT^*_{k,i} = u_k \\cdot K_{k,i} \\cdot v_i, \\quad \\text{where } K_{k,i} = \\exp\\left(-\\frac{C_{k,i}}{\\gamma}\\right)\n$$\nThe Sinkhorn-Knopp algorithm is an iterative procedure to find the scaling vectors $u \\in \\mathbb{R}^K$ and $v \\in \\mathbb{R}^N$ that satisfy the marginal constraints.\n\n**Proof:**\n\nThe convergence can be shown by interpreting the algorithm as an alternating projection procedure using the Kullback-Leibler (KL) divergence. Let the sets of matrices satisfying the row and column constraints be:\n\n- $\\mathcal{C}_1 = \\{ T \\in \\mathbb{R}_{\\geq 0}^{K \\times N} \\mid \\sum_i T_{k,i} = a_k \\}$ (Row constraints)\n- $\\mathcal{C}_2 = \\{ T \\in \\mathbb{R}_{\\geq 0}^{K \\times N} \\mid \\sum_k T_{k,i} = b_i \\}$ (Column constraints)\nEach iteration of the Sinkhorn-Knopp algorithm can be viewed as performing the following alternating KL projections:\n$$\n\\begin{aligned}\nT^{(l+\\frac{1}{2})} &= \\arg\\min_{T \\in \\mathcal{C}_1} \\mathrm{KL}(T \\parallel T^{(l)}), \\\\\nT^{(l+1)} &= \\arg\\min_{T \\in \\mathcal{C}_2} \\mathrm{KL}(T \\parallel T^{(l+\\frac{1}{2})}),\n\\end{aligned}\n$$\nwhere the KL divergence is a specific type of Bregman divergence. Since $\\mathcal{C}_1$ and $\\mathcal{C}_2$ are convex sets, the theory of alternating Bregman projections guarantees that this iterative process converges [1]. The intersection $\\mathcal{C}_1 \\cap \\mathcal{C}_2 = U(a,b)$ is non-empty and contains the unique optimal solution $T^*$. Therefore, the sequence of iterates $T^{(l)}$ produced by the Sinkhorn-Knopp algorithm is guaranteed to **converge to the unique optimal solution $T^*$**.\n\n## **3. Role of the LLP-Proportion Penalty**\n\nFinally, we provide the intuition for why our innovative LLP-Proportion Penalty helps align the pseudo-label distribution with the true LLP prior. Our effective cost function is:\n$$\nC_{\\text{eff}}(c_k, z_i) = d(c_k, z_i) + \\lambda_{\\text{LLP}}(1 - p(k \\mid B_j(i)))\n$$\nThe key component is the penalty term. To understand its effect, we analyze the part of the objective function corresponding to this penalty, which we denote $\\mathcal{L}_{\\text{LLP}}(T)$:\n$$\n\\mathcal{L}_{\\text{LLP}}(T) := \\sum_j \\sum_{i \\in B_j} \\sum_k T_{k,i} (1 - p(k \\mid B_j)).\n$$\nMinimizing this term is equivalent to maximizing the alignment between the transport plan's implied label proportions and the true bag proportions. By defining the transport plan's label distribution in bag $B_j$ as $P_T(k \\mid B_j)$, minimizing $\\mathcal{L}_{\\text{LLP}}(T)$ becomes equivalent to maximizing the sum of dot products between the predicted and true label distributions within each bag:\n$$\n\\max_{T} \\sum_j \\langle P_T(\\cdot \\mid B_j), p(\\cdot \\mid B_j) \\rangle.\n$$\n**Maximizing this dot product encourages the distribution $P_T$ to align with the true prior $p$**, which is analogous to minimizing statistical divergences like the KL divergence. Therefore, by introducing the LLP-Proportion Penalty into the cost matrix, we directly enforce that the optimal transport plan $T^*$—the unique solution to the optimization problem—favors assignments that are consistent with the known bag-level supervision.\n\n\n\nReference:\n\n[1] Byrne, Charles L. Alternating minimization and alternating projection algorithms: A tutorial.\n\n# Weakness：more experiments are suggested.\n\n We appreciate the feedback on the experimental scope. To address this, we have significantly expanded our evaluation:\n- **Beyond image datasets**: we have conducted additional experiments on tabular datasets to demonstrate the applicability of our method beyond image-based modalities. Specifically, we performed experiments on the UCI Adult dataset, a commonly used benchmark in tabular data. We also reproduced the results of two existing methods, LLP-AHIL and ROT, on this dataset for comparison. The experimental results are summarized in the table below:\n| Model | RLPL(ours) | LLP-AHIL | ROT |\n|:-----:|:----------:|:--------:|:---:|\n| UCI Adult | 77.57 | 75.99 | 72.82 |\n\nFrom the table, we observe that our method (RLPL) outperforms both LLP-AHIL and ROT on the UCI Adult dataset, achieving an accuracy of 77.57%. This indicates that our method is effective in the tabular data domain, achieving competitive performance compared to state-of-the-art methods. These results reinforce the robustness of our method across different non-vision modalities. We believe these experiments provide strong evidence that our approach generalizes well beyond image data and performs effectively in diverse domains such as tabular and structured data. In the final version of the paper, we will include these additional experiments and the corresponding analysis to further demonstrate the broad applicability of our method.\n\n\n# Question： Multi-class vs. Multi-label Setting \n\nThank you for your comments. We give the following explanations. \n\n**1. Clarification of the Problem Setting:** Our work addresses the standard **multi-class** Learning from Label Proportions (LLP) problem, where each instance belongs to exactly one class from a set of $K$ mutually exclusive classes. Multiple samples form a bag, and we have the label distribution of the bag as supervision information. The problem is formally defined in Section 3, where the proportion for class $k$ in a bag $B_j$ is given by $p_{j,k} = \\frac{1}{n_j} \\sum_{i \\in B_j} I(y_i = c_k)$, with $\\sum_{k=1}^K p_{j,k} = 1$. This formulation inherently assumes a multi-class ground truth. Consequently, our entire framework, including the pseudo-labeling step, is designed for this multi-class setting. The use of `argmax` to generate a single pseudo-label per instance is the standard and appropriate procedure for deriving hard labels from a multi-class classifier's probability distribution.\n\n**2. Potential Source of Misunderstanding:** We believe the term \"label proportions\" itself, while standard in the LLP literature, might be interpreted in other machine learning contexts as relating to multi-label problems. We will ensure this distinction is made clearer in the final version of our paper to prevent future ambiguity.\n\n**3. On the Extension to a \"Truly Multi-label\" Setting:** We agree that extending this work to the **multi-label LLP** setting is a fascinating and important future research direction. Such an extension would require some significant modifications to our framework:\n\n- The classifier head would need to be changed from a `softmax` to multiple independent `sigmoid` activations.\n- The bag proportion loss would need to be redefined, as the sum of proportions would no longer be 1.\n- The OT formulation would be more complex, as the goal is no longer to assign an instance to a single class prototype. One might need to explore alternative matching frameworks."}, {"review_id": "F5tj4eivrT", "reviewer": "Reviewer_z6fq", "summary": "The paper presents RLPL, an innovative and robust framework for Learning from Label Proportions (LLP), which utilizes optimal transport and semi-supervised learning strategies to improve the accuracy of instance-level classifiers. The proposed approach addresses the problem of noisy pseudo-labeling effectively and demonstrates strong empirical performance across multiple benchmark datasets.", "questions": "1. How does the Optimal Transport (OT) process scale with larger datasets? Have you considered methods to reduce the computational cost of OT, especially in high-dimensional spaces?\n\n2. How does the model perform when bag-level proportions are severely noisy or ambiguous? Would the performance drop significantly, or does the LLP-OTD mechanism sufficiently handle extreme noise scenarios?\n\n3. Are there any theoretical guarantees for the LLP-OTD method? Specifically, how do you ensure convergence and robustness to outliers in highly noisy datasets?\n\n4. How do the two weighting coefficients in Eq. (13) affect the performance? How do you tune them for different datasets?", "limitations": "1.\tThe Optimal Transport step in LLP-OTD is computationally expensive. This could limit its application in real-world scenarios where large datasets or high-dimensional data are common. An evaluation of its performance in terms of time complexity would be beneficial.\n\n2.\tThe method relies on accurate bag-level proportions, which may not always be available in practice. If the bag proportions themselves are imprecise, this could significantly impact model performance, and further exploration of such edge cases is needed.\n\n3.\tThe experiments primarily use standard benchmark datasets (CIFAR-10, CIFAR-100, etc.) with relatively controlled conditions. An evaluation of RLPL on real-world datasets, or more complex data modalities (e.g., textual or medical data), would provide better insights into its generalization abilities.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 2, "strengths_and_weaknesses": "**Strengths**:\n- The paper introduces the RLPL framework, a two-stage approach that integrates contrastive learning and optimal transport (OT) techniques to improve Learning from Label Proportions (LLP). This dual approach of pretraining and refinement via pseudo-labeling is highly innovative and shows potential to significantly advance the weakly-supervised learning paradigm.\n\n- One of the key innovations in this work is the LLP-OTD mechanism, which utilizes optimal transport to refine pseudo-labels by distinguishing high-confidence labels from low-confidence ones. This is a critical issue in LLP tasks where noisy pseudo-labels degrade classifier performance. The paper convincingly shows that LLP-OTD significantly reduces noise, providing a more reliable set of labels for subsequent training stages.\n\n- The LLPMix component, inspired by MixMatch, is a valuable extension to this framework. By combining the high-confidence pseudo-labels with bag-level constraints, the method benefits from both instance-level supervision and bag-level consistency. This not only improves the classifier's performance but also bridges the gap between supervised and unsupervised learning within the LLP framework.\n\n**Weaknesses**:\n\n- While the paper provides strong empirical evidence for RLPL’s effectiveness, it lacks a thorough theoretical analysis of the proposed methods, especially the LLP-OTD and LLPMix mechanisms. Formal results, such as convergence guarantees or robustness under varying bag sizes, would provide a deeper understanding of the model’s properties and its scalability across different domains.\n\n- Although the LLP-OTD mechanism improves the quality of pseudo-labels, pseudo-labeling remains an inherent challenge in weakly-supervised learning. There is no mention of how the model performs when faced with highly ambiguous bag-level proportions, which could limit the generalization capabilities in more complex or imbalanced datasets. The framework would benefit from a discussion on extreme cases of noisy data.\n\n- The framework heavily relies on the Optimal Transport (OT) process, which can be computationally expensive, especially for large datasets or high-dimensional feature spaces. The authors do not address the potential computational bottlenecks of the OT process, nor do they provide any discussion on how the method scales with larger datasets. An evaluation of the trade-offs between performance improvement and computational complexity would be valuable.", "was_revised": true, "final_justification": "The author's rebuttal has addressed my concerns.", "rebuttal": "### Theoretical Analysis of LLP-OTD (W1 & Q3)\nThank you for your valuable suggestion. We provide a three-step theoretical proof for our method. \n1. **Existence and Uniqueness of the Optimal Solution:** We first establish that the core optimization problem in our LLP-OTD mechanism has a unique global minimizer. The optimization problem is defined as:\n   $$\n   \\min_{T \\in U(a, b)} \\langle C, T \\rangle - \\gamma H(T),\n   $$\n   where $\\langle C, T \\rangle $ is the transport cost, $ H(T) $ is the entropy term, and $ U(a, b)$ is the transport polytope. The proof of existence and uniqueness is based on the strict convexity of the objective function and the compactness of the feasible set $U(a, b) $, guaranteeing a unique global minimizer.\n\n2. **Convergence of the Sinkhorn-Knopp Algorithm:** We show that the Sinkhorn-Knopp algorithm, used to solve the OT problem, converges to the unique optimal solution. The optimal transport plan $T^* $ is known to have a specific structure, and the convergence follows from interpreting the algorithm as an alternating projection procedure with Kullback-Leibler (KL) divergence. The process converges to the unique optimal solution $ T^* $ due to the convexity of the feasible sets.\n\n3. **Role of the LLP-Proportion Penalty:** Finally, we explain the intuition behind our innovative LLP-Proportion Penalty, which helps align the pseudo-label distribution with the true LLP prior. This penalty term effectively shifts the position of the optimal solution, ensuring that the resulting label assignments are more consistent with the given bag-level proportions.\n\n**Note:** Due to space constraints, we refer to the detailed theoretical analysis provided in our response to **Reviewer oaZQ**. We will include the full discussion in the final version of the paper.\n\n### Performance on Noisy and Imbalanced Data (W2)\nThis is a very valid point. To evaluate our model's robustness, we conducted new experiments under more challenging conditions, including datasets with **imbalanced class distributions** and **noisy label proportions**.\n**Evaluations on Imbalanced Datasets:**  \nFor the imbalanced data experiments, we used the CIFAR-10 dataset and constructed a long-tailed distribution by setting the imbalance ratio as the ratio of the maximum number of samples in one class to the minimum number of samples in another class. The imbalance ratios tested are 5, 10, 15, 50, and 100. The accuracy results are as follows:\n\n| Imbalance Ratio | 5 | 10 | 15 | 50 | 100 | \n| ------ | ------ | ------ | ------ | ------ | ------ | \n| Accuracy | 93.55 | 91 | 89.72 | 81.61 | 70.28 | \n\nAs shown in the table, our method maintains strong performance even under high levels of imbalance. The accuracy decreases as the imbalance ratio increases, which is expected due to the growing difficulty of learning from such skewed data. However, our method outperforms the baseline, demonstrating its robustness in handling class imbalance.\n**Evaluations on Noisy Datasets:**  \nFor noisy label experiments, we used the CIFAR-10 dataset and generated LLP data with clean labels, then introduced Gaussian noise and uniform noise with different intensities. The noisy labels were generated by the following formulas:\n- **Gaussian noise**: $ p' = \\text{clip}(p + N(0, \\sigma^2)) $\n- **Uniform noise**: $ p' = \\text{clip}(p + U(-r, r)) $\nThe evaluation results are as follows:\n\n| Noise Type | gaussian moderate | gaussian_heavy  | uniform_moderate | uniform_heavy |\n| ------ | ------ | ------ | ------ | ------ |\n| Accuracy | 94.05 | 93.84 | 93.95 | 93.94 |\n\nAs can be seen from the table, our method remains resilient under both moderate and heavy noise conditions. Even with significant label noise, the accuracy only slightly decreases, which highlights the robustness of our LLP-OTD mechanism in handling imperfect and ambiguous supervision signals.\nThe results from both the imbalanced and noisy datasets demonstrate that our method consistently maintains high accuracy and robustness, making it highly effective in settings with imperfect supervision. This analysis will be included in the final version of the paper.\n### Computational Cost and Scalability of OT (W3 & Q1) \nWe thank the reviewer for raising this critical concern. We have analyzed the computational cost of OT from three perspectives.\n**1. Computational Complexity:** Our LLP-OTD module, solved via the **Sinkhorn-Knopp algorithm**, has a time complexity of $O(KN(d + L))$, where $ K$ is the number of classes, $N$ is the number of instances, $d$ is the dimensionality of each sample, and $ L$ is the number of iterations required by the Sinkhorn-Knopp algorithm. This complexity is **linear with respect to the number of instances $N$**, which provides a significant advantage over classical Optimal Transport (OT) solvers that typically have a time complexity of $O(N^3)$. As a result, our approach is highly scalable and efficient for large datasets.\n**2. Empirical Runtime:** To empirically verify the cost, we profiled our model's runtime on CIFAR-10 (bag size=256). The results show that the entire LLP-OTD module accounts for only **9.6%** of the total training time in the second stage. This directly demonstrates that the OT refinement is not a computational bottleneck in practice.\n**3. Impact of Sinkhorn Iterations (L):** The complexity depends on the number of Sinkhorn iterations, $L$. We tested its impact on accuracy.\n**Table: Accuracy(at epoch 50, on CIFAR10, bag size=256) vs. Sinkhorn Iterations** \n\n| Sinkhorn Iterations (L) | 1 | 2 | 3 | 5 | 10 | 20 | 50 |\n| ------ | ------ | ------ | ------ | ------ | ------ | ------ | ------ |\n| Accuracy (%) | 89.24 | 89.48 | 89.96 | 90.00 | 89.61 | 89.85 | 89.68 |\n\nAs shown, the accuracy quickly saturates with a small number of iterations (e.g., L=5-10). This confirms that a small, constant L is sufficient, keeping the practical computational cost low.\n### Hyperparameter Tuning (Q4)\nThis is an important aspect. We have performed a detailed sensitivity analysis for the two weighting coefficients, namely $ \\lambda_{\\text{OTD}} $ and $ \\omega_{\\text{LLP}} $, on the CIFAR-10 dataset with a bag size of 256. The results, summarized in the table below, show that our model is robust across a reasonable range of hyperparameter settings. The accuracy at 100 epochs for different combinations of $\\lambda_{\\text{OTD}} $ and $\\omega_{\\text{LLP}} $ is as follows:\n**Table: Accuracy (at 100 epochs) on different combinations of $\\lambda_{\\text{OTD}} $ and $ \\omega_{\\text{LLP}} $**\n\n| $\\lambda_{\\text{OTD}} \\backslash \\omega_{\\text{LLP}} $ | 0.05   | 0.1    | 0.2    | 0.5    | 1.0    |\n|----------------------------------------------------------|--------|--------|--------|--------|--------|\n| 0.05                                                     | 91.06% | 90.83% | 91.44% | 90.92% | 90.82% |\n| 0.1                                                      | 90.94% | 90.95% | 91.33% | 91.17% | 91.32% |\n| 0.2                                                      | 90.97% | 91.28% | 90.92% | 90.94% | 91.20% |\n| 0.5                                                      | 91.15% | 90.98% | 91.06% | 91.12% | 91.02% |\n\nAs shown in the table, the model achieves high and stable accuracy across a range of values for both $ \\lambda_{\\text{OTD}} $ and $\\omega_{\\text{LLP}} $, with accuracy values ranging from 90.82% to 91.44%. The results demonstrate that our model is robust to variations in these hyperparameters, with no significant degradation in performance even for extreme values. This confirms that our method is not overly sensitive to specific hyperparameter choices, allowing for flexibility in tuning.\nWe will include this analysis in Section 5.3 in the final version of the paper.\n\n### Evaluation on Other Modalities\nWe agree that demonstrating broader applicability is important. We have conducted **new experiments on tabular data**, including the UCI Adult dataset, and observed strong performance. Specifically, we performed experiments on the UCI Adult dataset, a widely used benchmark in tabular data. We also reproduced the results of two existing methods, LLP-AHIL and ROT, on this dataset for comparison. The experimental results are summarized in the table below:\n\n| Model        | RLPL (ours) | LLP-AHIL | ROT    |\n|--------------|-------------|----------|--------|\n| UCI Adult    | 77.57       | 75.99    | 72.82  |\n\nFrom the table, we observe that our method (RLPL) outperforms both LLP-AHIL and ROT on the UCI Adult dataset, achieving an accuracy of 77.57%. This indicates that our method is effective in the tabular data domain, achieving competitive performance compared to state-of-the-art methods. These results reinforce the robustness of our method across different non-vision modalities.\nWe believe these experiments provide strong evidence that our approach generalizes well beyond image data and performs effectively in diverse domains such as tabular and structured data. In the final version of the paper, we will include these additional experiments and the corresponding analysis to further demonstrate the broad applicability of our method."}, {"review_id": "CTHXanbgQg", "reviewer": "Reviewer_3KP4", "summary": "This work proposes a method for learning with label proportions that involves first employing bag-level proportions to generate pseudo-labels. In the next stage, the LLP-OTD mechanism refines them via an optimal transport process. A final LLPMix inspired semi-supervised pipeline has an explicit bag-level consistency loss and trains the classifier on these sets.", "questions": "1. Have you tried using Count Loss instead of averaging the predicted instance probabilities? [1]. \n2. What is the performance against Count Loss, GLWS, and etc.\n3. Is there any way to observe how well the final classifier obeys the bag proportions?\n4. Please provide details and comparisons with baselines w.r.t. the timing for training\n5. Why are the other baselines astonishingly bad on ImageNet?\n6. I would like to see some ablations on the hyperparmaters used (loss weight and $\\lambda_{OTD}$).\n\n[1] Shukla, Vinay, et al. \"A Unified Approach to Count-Based Weakly Supervised Learning.\" Advances in Neural Information Processing Systems 36 (2023): 38709-38722.\n\n[2] Chen, Hao, et al. \"A general framework for learning from weak supervision.\" arXiv preprint arXiv:2402.01922 (2024).", "limitations": "Yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 4, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\n- Clear and well-written particularly when explaining the method.\n- Method is novel to my knowledge and leverages a neat way to refine pseudo labels.\n\nWeakness:\n- High computation overhead bc of the optimal transport refinement (limited analysis of runtime, complexity, and etc.) (Especially with larger datasets with more instances, making it less practical.)\n- The approach relies on contrastive features that cleanly separate the classes (SimCLR).\n-The training mix is indeed new although its principle components (OT for LLP, consistency loss) are not.\n- Few ablations -- although the existing ones are helpful.\n- Baselines can be more comprehensive\n- The method loses on 2 out of the 4 datasets.\n\nIn line with the criteria for borderline reject, I think that there is a need for further analysis/experiments. The current ones do show some promise.\n\nTypo: Line 81", "was_revised": true, "final_justification": "All my concerns were answered. See strengths for further details.", "rebuttal": "### Concerns about the computational complexity aof the Optimal Transport module (W1) \nWe thank the reviewer for raising this important point. We provide a detailed complexity analysis and empirical runtime measurements to demonstrate the practicality of our approach.\n\n**Complexity Analysis:**  \nIn our implementation, we solve the entropy-regularized OT problem using the **highly efficient Sinkhorn-Knopp algorithm**. The complexity is determined by two main steps:\n1. Constructing the $K \\times N$ cost matrix, which takes $O(KNd) $ time, where $K$ is the number of classes, $N$ is the number of instances, and $d$ is the feature dimension.\n2. Running the Sinkhorn-Knopp algorithm for $ L$ iterations, which takes $O(LKN) $ time.\n\nThis results in a total complexity of **$O(KN(d + L)) $**. As this is linear with respect to the number of instances $N $, our method is scalable and efficient for large-scale LLP problems, especially compared to traditional OT solvers with polynomial complexity (e.g., $ O(N^3)$). For extreme-scale scenarios, our framework can also incorporate more advanced solvers like Greedy Sinkhorn, and we will add a discussion of these options in the appendix of our revised paper.\n\n**Empirical Training Time:**  \nTo empirically verify the cost, we profiled our method on CIFAR-10 with a bag size of 256. The LLP-OTD module accounts for only **9.6%** of the total training time in the second stage, demonstrating that the OT refinement does not pose a computational bottleneck in practice.\n\n**Impact of Sinkhorn Iterations (L):**  \nWe also evaluated the impact of the number of Sinkhorn iterations ($L$) on accuracy. The results from the CIFAR-10 dataset (bag size=256) at epoch 50 are shown in the table below:\n\n| Sinkhorn Iterations (L) | 1    | 2    | 3    | 5    | 10   | 20   | 50   |\n|-------------------------|------|------|------|------|------|------|------|\n| Accuracy (%)            | 89.24 | 89.48 | 89.96 | 90.00 | 89.61 | 89.85 | 89.68 |\n\nAs shown, the accuracy quickly saturates with a small number of iterations (e.g., $ L = 5 $ to $ 10 $). This confirms that a small, constant $L$ is sufficient, keeping the practical computational cost low.\n\nWe will include this detailed analysis and runtime measurement in the appendix of our revised paper.\n\n### On the novelty of using established components like Optimal Transport (W2)\nWe agree that OT and consistency loss are established tools. Our core novelty lies not in using these tools, but in **how we fundamentally adapt OT for the LLP problem.**\nWhile prior works typically modify the overall OT objective function, our key innovation is to redesign the **internal cost matrix** for each sample-prototype pair. By embedding the LLP-Proportion Penalty directly into the cost function, $C_{k,i} = \\|f_{\\text{main}}(x_i) - \\mu_k\\|_2^2 + \\lambda_{\\text{OTD}}(1-p_{j(i),k})$, we reshape the cost landscape (refer to response to reviewer z6fq). This modification forces the optimal transport plan to a new equilibrium that is inherently more consistent with the true label proportions. This micro-level intervention, which alters the matching cost for each instance, is the central contribution that distinguishes our method from previous OT-based approaches.\n\n### Request for comparison against other baselines like Count Loss and GLWS (W4, Q2)\nFollowing the reviewer's suggestion, we have broadened our experimental comparison to include Count Loss [1] and GLWS [2].\nWe have run new experiments on CIFAR-10 to compare with these baselines. Our method consistently outperforms them across various bag sizes. We will add the following table into section 5.3 in the revised paper:\n\n| | | | |\n|:-:|:-:|:-:|:-:|\n|Bag Size|Count Loss|GLWS|RLPL|\n|16|87.5%|85.46%|92.54%|\n|32|83.61%|81.11%|94.02%|\n|64|68.35%|64.64%|93.50%\n\n### On Count Loss replacement(Q1)\nThank you for your valuable suggestion to explore potential improvements for our work.\nFollowing your advice, we implemented Count Loss, referencing the official implementation provided in [1], to replace the KL divergence loss used for averaging predicted instance probabilities in our two-stage method.\nHowever, our experiments revealed that this approach introduces a prohibitive computational overhead. The Count Loss algorithm requires a dynamic programming step to compute the loss for each instance within a bag, leading to a complexity of O(N²), where N is the bag size. This computational cost proved to be impractically expensive for our setting.\nTo substantiate this, we ran a comparative experiment on an RTX 4090 GPU to measure the average training time per epoch. The results are presented below:\n\n| bag_size |16 |32| 64|128|\n| ------ | ------ | ------ | ------ | ------ |\n|Count Loss (s/epoch）|2245| 5012| 9215| 19964|\n| KL Divergence Loss (s/epoch) | 17| 9| 6| 6 |\n\nAs the table demonstrates, the training time per epoch for Count Loss increases substantially with the bag size, exceeding 5 hours for a bag size of 128. This computational demand made it infeasible for us to obtain complete training results within a reasonable timeframe.\nIn contrast, the epoch time for our method using KL divergence loss remains highly efficient. It even slightly decreases with larger bag sizes due to the corresponding reduction in the total number of bags, demonstrating that it introduces no extra computational overhead. This highlights a significant efficiency advantage of our proposed approach.\nGiven this substantial computational bottleneck, we maintain that our original choice of KL divergence loss represents a more practical and scalable solution.\n\n### On Observing Adherence to Bag Proportions (Q3)\nTo directly address your question, we conducted comprehensive bag proportions prediction experiments that quantitatively measure how well our final classifier adheres to the original bag-level constraints. We evaluate this through ****Mean Absolute Error (MAE)**** between predicted and true bag proportions, where MAE represents the average absolute deviation of predicted class proportions from ground-truth proportions across all bags. Our method demonstrates superior adherence:\n\n\n****Experimental Results (MAE $\\downarrow$):****\n| Method | Bag Size 16 | Bag Size 32 | Bag Size 64 | Bag Size 128 | Bag Size 256 |\n|--------|-------------|-------------|-------------|--------------|--------------|\n| ****RLPL (Ours)**** | ****0.010600**** | ****0.009505**** | ****0.008061**** | ****0.006725**** | ****0.005273**** |\n| LLP-AHIL | 0.015280 | 0.013498 | 0.010868 | 0.008544 | 0.006758 |\n\n****Key Observations:****\n1. ****Superior Bag Constraint Adherence****: Our RLPL method consistently achieves lower MAE across all bag sizes, demonstrating that our final classifier better respects the original bag proportion constraints compared to the strong baseline LLP-AHIL.\n2. ****Robust Scale Performance****: The performance gap widens as bag size decreases (more challenging scenarios), with RLPL achieving 30.6% lower MAE at bag size 16 compared to LLP-AHIL, indicating our method's robustness in preserving bag-level supervision signals.\n3. ****Practical Case Study****: For bag size 256, we present a detailed example of one bag below, which demonstrates remarkable precision in proportion prediction:\n\n| CIFAR-10 Classes | airplane | automobile | bird | cat | deer | dog | frog | horse | ship | truck |\n|------------------|----------|------------|------|-----|------|-----|------|-------|------|-------|\n| ****True Proportions**** | 0.125 | 0.102 | 0.078 | 0.070 | 0.137 | 0.113 | 0.078 | 0.098 | 0.090 | 0.109 |\n| ****Pred Proportions**** | 0.129 | 0.105 | 0.082 | 0.070 | 0.133 | 0.102 | 0.078 | 0.102 | 0.090 | 0.109 |\n| ****Individual MAE**** | 0.004 | 0.003 | 0.004 | 0.000 | 0.004 | 0.011 | 0.000 | 0.004 | 0.000 | 0.000 |\n\nThis demonstrates that our LLP-OTD mechanism successfully preserves bag-level constraints while enabling accurate instance-level classification. The integration of LLP consistency loss in our LLPMix framework (Eq. 9 in our paper) ensures that the final classifier's predictions, when aggregated at the bag level, closely match the ground-truth proportions, directly answering your concern about bag proportion adherence.\n\n### On Training Time Details(Q4)\nThank you for raising the training-time concern. Here are the single-epoch numbers on CIFAR-10 (bag size = 256):\n\n| Model | LLP-AHIL | ROT | LLP-VAT | RLPL (ours) |\n|------|----------|-----|---------|-------------|\n| Time (s) | 10.607 | 10.894 | 16.087 | 16.11 |\n\nAs the table shows, RLPL is a touch slower than some baselines. The extra cost comes almost entirely from the **LLP-OTD module**—the Sinkhorn iterations and the on-the-fly cost-matrix updates. Three quick points to put this in perspective:\n1. **The jump in accuracy is well worth it.** RLPL outperforms LLP-VAT by **47.02 %**, ROT by **30.85 %**, and even beats LLP-AHIL by **1.07 pp**—a large margin in the LLP world.\n2. **Overhead is modest and tunable.** The OT piece accounts for only 9.6 % of a training step (see our reply to Reviewer z6fq). Dialing the Sinkhorn iterations down to L ≤ 5 cuts that slice further, with < 0.5 % accuracy loss.\n3. **Real-world trade-offs favor accuracy.** In domains like healthcare or finance, a single-point absolute gain often outweighs a few extra seconds per epoch. The complexity is O(KN(d + L)), so scaling is linear; on UCI Adult the added cost is only 2.1 s per epoch.\n\nWe’ll fold this discussion into the camera-ready version and release the code so users can freely trade speed for precision."}, {"review_id": "1VKSYdHkzF", "reviewer": "Reviewer_3xeg", "summary": "The paper introduces a novel multi-step method for Learning from Label Proportions. The method named RLPL has the following steps:\n1. Unsupervised learning to learn a good representation using SimCLR.\n2. Learning a weak classifier by minimizing the KL divergence between the true and predicted label proportion of each bag.\n3. Using the weak classifier to pseudo-label each instance.\n4. Iteratively solving an Optimal Transport problem to refine the pseudo-labels.\n5. Partitioning the set of pseudo-labels into High Confidence PLs and Low Confidence PLs based on whether the refined PLs agree with the initial PLs.\n6. Optimize a combined objective of supervised loss on High Confidence PLs, unsupervised loss on Low Confidence PLs and a newly introduced loss enforcing label proportion constraint.\n\nThe paper demonstrates the effectiveness of the method on Vision datasets with randomly sampled bags. The paper also performs ablation study to quantify the importance of each step of the method.", "questions": "[Q.1] Can the authors perform and report the performance of their methods on the LLP-Bench [1] tabular dataset? If the focus of this paper is on vision datasets then the authors can keep this in mind for future work.\n\n[Q.2] Do the authors have any hypothesis about why their method outperforms baselines when the bag sizes are large but fail to do so with smaller bag sizes?\n\n[Q.3] Can the authors explain why they only use a two-step process for updating their pseudo-labels and their barycenters? This could be a hyperparameter to the method.\n\n[Q.3] It would be nice if the authors could include some of the theoretical justification mentioned in the appendix in the main paper while explaining the method.\n\nSome corrections: Line 81 – reference to MixUp is missing, Line 99 – “Beginning by” -> “Beginning with”\n\n\n[1] Anand Brahmbhatt, Mohith Pokala, Rishi Saket, and Aravindan Raghuveer. 2024. LLP-Bench: A Large Scale Tabular Benchmark for Learning from Label Proportions. In Proceedings of the 33rd ACM International Conference on Information and Knowledge Management (CIKM '24), 4374–4381. https://doi.org/10.1145/3627673.3680032", "limitations": "The authors mention the limitations of their work in the appendix.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "**Strengths**\n\n[S.1] The paper is able to leverage ideas from LLP methods like PLOT (Pseudo-Labelling with Optimal Transport) and ideas from semi-supervised learning like MixUp to come up with a method which outperforms the baselines on most of the datasets used in the paper, particularly when the bag sizes are large.\n\n[S.2] The paper performs good analysis to demonstrate that their method behaves as it is expected to wherever such evaluation is possible. The ablation study performed in the paper showcases the importance of each component of the proposed method. Figure 2 demonstrates that the pseudo-labeling component is performing as expected.\n\n[S.3] The paper is well written and puts across its point clearly.\n\n**Weaknesses**\n\n[W.1] Although the method performs well empirically and its individual steps are conceptually sound, it lacks rigorous theoretical justification, leaving its performance in broader settings uncertain.\n\n[W.2] The paper focuses solely on the vision modality and does not include experiments on tabular data, which is crucial for many real-world applications of LLP. Additionally, the experimental setup is restricted to scenarios where bags are constructed via random sampling, which may not reflect the data generation process in practical settings. The authors acknowledge these limitations in their discussion.", "was_revised": true, "final_justification": "The authors address my concerns regarding performance on Tabular datasets. Their results in the rebuttal prove the robustness of their method on a different modality, as well as in the setting when bags are not sampled uniformly at random. The authors also address my minor concerns about the performance on Large Bags vs Small Bags and Number of OT passes effectively. I am unsatisfied with their response on the theory, but I feel the focus of the paper is empirical and the authors have managed to perform extensive empirical evaluation and ablation studies. Hence, I am raising my score to 5.", "rebuttal": "In the response, we provde three theoretical guarantees for the prosoed method. We will add them to the final version of our paper. First, we prove the existence and uniqueness of the optimal solution for our entropy-regularized optimal transport (OT) problem. Second, we demonstrate that the Sinkhorn-Knopp algorithm, which we employ, is guaranteed to converge to this unique solution. Finally, we explain how our innovative LLP-Proportion Penalty, incorporated into the cost function, guides the pseudo-label distribution to align with the true LLP prior. \n\n1. Existence and Uniqueness of the Optimal Solution\nWe first establish that the core optimization problem in our LLP-OTD mechanism has a unique global minimizer. The problem is defined as\n$$\nmin_{T in U(a, b)} langle C, T rangle - gamma H(T),\n$$\nwhere $ langle C, T rangle = sum_{k,i} C_{k,i} T_{k,i} $ is the transport cost, $H(T) = -sum_{k,i} T_{k,i} (log T_{k,i} - 1) $ is the entropy term, $gamma  0$ is the regularization strength, and $ U(a, b) $ is the transport polytope defined by marginal constraints.\nProof \nLet $F(T) = langle C, T rangle - gamma H(T)$. Then we will prove the strict convexity of the object function.  For the term $langle C, T rangle$, it is linear in $T$, hence convex. For the entropy term $-H(T) = sum_{k,i} T_{k,i} (log T_{k,i} - 1)$, we can compute out its Hessian matrix\n$$\nfrac{partial^2 (-H)}{partial T_{k,i}^2} = frac{1}{T_{k,i}}  0 quad text{since } T_{k,i}  0,\n$$\n implying strict convexity of $-H(T) $. Therefore, $F(T) = langle C, T rangle - gamma H(T)$ is the sum of a convex and strictly convex function, and thus strictly convex. Meanwhile, for $U(a,b)= left{ T in mathbb{R}_{geq 0}^{K times N} , middle , sum_i T_{k,i} = a_k, sum_k T_{k,i} = b_i right }$, we will prove it to be convex, closed and bounded. As $U(a,b)$ is defined by linear equalities and nonnegativity constraints, it is a convex polytope. Because all constraints are equation or non-strict non-equation, $U(a,b)$ is closed. Since $sum a_k = sum b_i = 1$, we have$0 le T_{k,i} le 1$, and $U(a,b)$ is a compact convex set. Finally, by standard results in convex optimization, a strictly convex function over a non-empty compact convex set has a unique global minimizer $T^$.\n\n2. Convergence of the Sinkhorn-Knopp Algorithm\nNext, we prove that the Sinkhorn-Knopp algorithm, used to solve the OT problem, converges to the unique optimal solution $T^$. The optimal transport plan T∗ is known to have a specific structure\n$$\nT^_{k,i} = u_k cdot K_{k,i} cdot v_i, quad text{where } K_{k,i} = expleft(-frac{C_{k,i}}{gamma}right)\n$$\nThe Sinkhorn-Knopp algorithm is an iterative procedure to find the scaling vectors $u in mathbb{R}^K$ and $v in mathbb{R}^N$ that satisfy the marginal constraints.\nProof：\nThe convergence can be shown by interpreting the algorithm as an alternating projection procedure using the Kullback-Leibler (KL) divergence. Let the sets of matrices satisfying the row and column constraints be\n- $mathcal{C}_1 = { T in mathbb{R}_{geq 0}^{K times N} mid sum_i T_{k,i} = a_k }$  (Row constraints)\n- $mathcal{C}_2 = { T in mathbb{R}_{geq 0}^{K times N} mid sum_k T_{k,i} = b_i$ (Column constraints)\nEach iteration of the Sinkhorn-Knopp algorithm can be viewed as performing the following alternating KL projections\n$$\nbegin{aligned}\nT^{(l+frac{1}{2})} &= argmin_{T in mathcal{C}_1} mathrm{KL}(T parallel T^{(l)}), \nT^{(l+1)} &= argmin_{T in mathcal{C}_2} mathrm{KL}(T parallel T^{(l+frac{1}{2})}),\nend{aligned}\n$$\nwhere the KL divergence is a specific type of Bregman divergence. Since $ mathcal{C}_1$ and $ mathcal{C}_2$ are convex sets, the theory of alternating Bregman projections guarantees that this iterative process converges [1]. The intersection $ mathcal{C}_1 cap mathcal{C}_2 =U(a,b)$ is non-empty and contains the unique optimal solution T∗. Therefore, the sequence of iterates T(l) produced by the Sinkhorn-Knopp algorithm is guaranteed to converge to the unique optimal solution $T^$.\n\n3. Role of the LLP-Proportion Penalty\nFinally, we provide the intuition for why our innovative LLP-Proportion Penalty helps align the pseudo-label distribution with the true LLP prior. Our effective cost function is\n$$\nC_{text{eff}}(c_k, z_i) = d(c_k, z_i) + lambda_{text{LLP}}(1 - p(k mid B_j(i)))\n$$\nThe key component is the penalty term. To understand its effect, we analyze the part of the objective function corresponding to this penalty, which we denote $mathcal{L}_{text{LLP}}(T)$\n$$\nmathcal{L}_{text{LLP}}(T) = sum_j sum_{i in B_j} sum_k T_{k,i} (1 - p(k mid B_j)).\n$$\nMinimizing this term is equivalent to maximizing the alignment between the transport plan's implied label proportions and the true bag proportions. By defining the transport plan's label distribution in bag $B_j$ as $P_T(k mid B_j)$, minimizing $mathcal{L}_{text{LLP}}(T)$ becomes equivalent to maximizing the sum of dot products between the predicted and true label distributions within each bag\n$$\nmax_{T} sum_j langle P_T(cdot mid B_j), p(cdot mid B_j) rangle.\n$$\nMaximizing this dot product encourages the distribution $P_T$ to align with the true prior $p$, which is analogous to minimizing statistical divergences like the KL divergence. Therefore, by introducing the LLP-Proportion Penalty into the cost matrix, we directly enforce that the optimal transport plan $T^$—the unique solution to the optimization problem—favors assignments that are consistent with the known bag-level supervision.\n\nReference\n[1] Byrne, Charles L. Alternating minimization and alternating projection algorithms A tutorial.\n\n\n### Limited experimental scope on modality (W2 & Q1)\nThis is a very valid point. To address this limitation and demonstrate the broader applicability of our method, we have conducted **new experiments on tabular data**. Specifically, we evaluated RLPL on the **UCI Adult** dataset and the **LLP-Bench [1]** dataset as suggested. Furthermore, to explore bag generation beyond random sampling, we constructed **feature-based bags** on the UCI Adult dataset, where bags are formed by clustering instances in the feature space. Our method consistently achieves strong performance in these new settings, showcasing its versatility.\n\nWe performed experiments on the UCI Adult dataset, a commonly used benchmark in tabular data, and compared our method with two existing methods, LLP-AHIL and ROT. The experimental results are summarized in the table below:\n\n| Model        | RLPL (ours) | LLP-AHIL | ROT    |\n|--------------|-------------|----------|--------|\n| UCI Adult    | 77.57       | 75.99    | 72.82  |\n\nFrom the table, we observe that our method (RLPL) outperforms both LLP-AHIL and ROT on the UCI Adult dataset, achieving an accuracy of 77.57%. This indicates that our method is effective in the tabular data domain, achieving competitive performance compared to state-of-the-art methods.\n\nFurthermore, we also tested our method on the **LLP-Bench [1]** dataset, where we achieved an accuracy of **66.85%** on the Criteo dataset. This result further confirms that our method is capable of handling tabular data effectively, demonstrating its broad applicability beyond image-based modalities.\n\nThese results reinforce the robustness of our method across different non-vision modalities. We believe these experiments provide strong evidence that our approach generalizes well beyond image data and performs effectively in diverse domains such as tabular and structured data. In the final version of the paper, we will include these additional experiments and the corresponding analysis to further demonstrate the broad applicability of our method.\n\n### Performance on large vs. small bag sizes (Q2)\nThis is a very insightful question. Our hypothesis is that the performance gap correlates with the **degree of label ambiguity** inherent in different bag sizes.\n- **With smaller bags**, the label proportions provide a relatively strong and less ambiguous supervisory signal. In this setting, even simpler methods can perform reasonably well, leading to a smaller performance gap between different approaches.\n- **With larger bags**, the label ambiguity increases significantly, as a single proportion vector can correspond to a vast number of instance-level label configurations. This is where the robustness of our pipeline, particularly the **LLP-OTD refinement process**, becomes critical. Our method's ability to effectively denoise pseudo-labels from a highly ambiguous signal allows it to excel and outperform baselines more significantly in these challenging scenarios.\nWe will include this analysis in the final version of the paper to further explain the relationship between bag size and performance in different settings.\n\n### Number of OT passes in LLP-OTD (Q3)\nThank you for this excellent question. We investigated the effect of the number of OT passes, and the results on CIFAR-10 (bag size=256) are as follows:\n| Number of OT Passes | 1 | 2 | 3 | 5 |\n| ------ | ------ | ------ | ------ | ------ |\n| Accuracy (%) | 94.00 | 94.55 | 94.16 | 94.15 |\nOur rationale for choosing two passes is based on these results. A single pass is insufficient as it relies on barycenters computed from the initial, noisy pseudo-labels. A second pass is beneficial because it re-computes barycenters from the refined labels of the first pass, leading to a more accurate refinement. However, further passes did not yield additional gains, which we hypothesize is due to diminishing returns and a potential risk of overfitting to the refined (but still imperfect) pseudo-labels. Based on this, two passes offer the best trade-off between performance and efficiency. However, we agree this is a natural hyperparameter. **Following your great suggestion, we will make the number of OT passes a user-configurable parameter in our public code release**, defaulting to our empirically optimal choice of 2."}]

Author rebuttal:

Meta-review: The paper concerns the problem of learning from label proportions (LLP). The Authors introduce a two-stage approach. In the first stage, they employ unsupervised contrastive learning and train an auxiliary classifier with bag-level supervision. In the second stage, they refine the obtained labels using an optimal-transport-based method. Finally, inspired by the MixMatch framework, they train the main classifier within a semi-supervised pipeline.

This is primarily an experimental paper. Nevertheless, the Reviewers appreciate the idea, which combines various concepts (contrastive learning, optimal transport, MixMatch) into a natural and "worth trying" approach for LLP problems. Most of the main concerns raised by the Reviewers were addressed in a satisfactory manner. While the paper lacks theoretical results, this weakness is outweighed by its other strengths.

Since NeurIPS is a very competitive conference, the final positive evaluation may not be sufficient for the paper to be accepted at this venue.
Decision: Accept (poster)