PiCO: Contrastive Label Disambiguation for Partial Label Learning
Authors: Haobo Wang
Ruixuan Xiao
Yixuan Li
Lei Feng
Gang Niu
Gang Chen
Junbo Zhao
Venue: ICLR 2022 Oral (2022)
Keywords: Partial Label Learning
Contrastive Learning
Prototype-based Disambiguation

Abstract: Partial label learning (PLL) is an important problem that allows each training example to be labeled with a coarse candidate set, which well suits many real-world data annotation scenarios with label ambiguity.  Despite the promise, the performance of PLL often lags behind the supervised counterpart. In this work, we bridge the gap by addressing two key research challenges in PLL---representation learning and label disambiguation---in one coherent framework. Specifically, our proposed framework PiCO consists of a contrastive learning module along with a novel class prototype-based label disambiguation algorithm. PiCO produces closely aligned representations for examples from the same classes and facilitates label disambiguation. Theoretically, we show that these two components are mutually beneficial, and can be rigorously justified from an expectation-maximization (EM) algorithm perspective. Extensive experiments demonstrate that PiCO significantly outperforms the current state-of-the-art approaches in PLL and even achieves comparable results to fully supervised learning. Code and data available: https://github.com/hbzju/PiCO.

Published as a conference paper at ICLR 2022
PICO: CONTRASTIVE LABEL DISAMBIGUATION FOR PARTIAL LABEL LEARNING
Haobo Wang'
Ruixuan Xiao
Yixuan Li Junbo Zhao? 1
Lei Feng
Gang -
Gang Chen'
'Zhejiang University
University of Wisconsin-Madison
Chongging University
'RIKEN
ABSTRACT
Partial label learning (PLL) is an important problem that allows each training ex- ample to be labeled with coarse candidate set, which well suits many real-world data annotation scenarios with label ambiguity. Despite the promise, the perfor- mance of PLL often lags behind the supervised counterpart. In this work, we bridge the gap by addressing two key research challenges in PLL PLL-representattion learning and label disambiguation-in one coherent framework. Specifically, our proposed framework PiCO consists of a contrastive learning module along with novel class prototype- based label disambiguation algorithm. PiCO pro- duces closely aligned representations for examples from the same classes and facilitates label disambiguation. Theoretically, we show that these two compo- nents mutually beneficial, and can be rigorously justified from expectation- maximization (EM) algorithm perspective. Extensive experiments demonstrate that PiCO significantly outperforms the current state-of-the-art approaches in PL.L and even achieves comparable results to fully supervised learning. Code and data available: htttps://gi
cwm/hb// /u/PiCO
INTRODUCTION
The training of modern deep neural networks typically requires massive labeled data, which imposes formidable obstacles in data collection. Of particular challenge, data annotation in the real-world can naturally be subject to inherent label ambiguity and noise. For example, as shown in Figure identifying an Alaskan Malamute from a Siberian Husky can be difficult for human annotator. The issue of labeling ambiguity is prevalent yet often overlooked in many applications, such as web mining (Luo & Orabonal 2010) and automatic image annotation (Chen al. 2018). This gives rise to the importance of partial label learning (PLL) (Hüllermeier & Beringer [006) Cour et al.) (001)), where each training example equipped with a set of candidate labels instead of the exact ground- truth label. This stands in contrast to its supervised counterpart where one label must be chosen as the "gold". Arguably, the PLL problem is deemed more common and practical in various situations due to its relatively lower cost to annotations.
Despite the promise, a core challenge in PLL label disambiguation, i.c., identifying the ground- truth label from the candidate label set. Existing methods typically require a good feature representa- tion (Liu & Dietterich, 2012 Zhang et al. 2016 Lyu et al. 2021). and operate under the assumption that data points closer in the feature space are more likely to share the same ground-truth label. However, the reliance on representations has led to a non-trivial dilemma-the inherent label uncer- tainty can undesirably manifest in the representation learning process-the quality of which may, in turn, prevent effective label disambiguation. To date, few efforts have been made to resolve this.
This paper bridges the gap by reconciling the intrinsic tension between the two highly depen- dent problems--repreeentttion learning and label disambiguation-in one coherent and synergistic framework. Our framework, Partial label learning with COntrastive label disambiguation (dubbed PiCO), produces closely aligned representations for examples from the same classes and facilitates label disambiguation. Specifically, PiCO encapsulates two key components. First, we leverage con- trastive learning (CL) (Khosla et al. 2020) to partial label learning, which is unexplored in previous
"Correspondence to j.zhao@ @ wwwu.cdu.en.

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Published as a conference paper at ICLR 2022
PLL literature. To mitigate the key challenge of constructing positive pairs, we employ the classi- fier's 's output and generate pseudo positive pairs for contrastive comparison (Section 3.1 Second, based on the learned embeddings, we propose novel prototype-basee label disambiguation strategy (Section 3.2). Key to our method, we gradually update the pseudo target for classification, based on the closest class prototype. By alternating the two steps above, PiCO converges to solution with a highly distinguishable representation for accurate classification. Empirically, PiCO estab- lishes state-of-the-ar performance on three benchmark datasets, outperforming the baselines by significant margin (Section D and obtains results that are competitive withfulll supervised learning.
Theoretically, we demonstrate that our contrastive representation learning and prototype- based la- bel disambiguation are mutually beneficial, and can be rigorously interpreted from an Expectation- Maximization (EM) algorithm perspective (Section 5). First, the refined pseudo labeling improves contrastive learning by selecting pseudo positive examples accurately. This can be analogous to the E-step, where we utilize the classifier's output to assign each data example to one label-specific cluster. Second, better contrastive performance in turn improves the quality of representations and thus the effectiveness of label disambiguation. This can be reasoned from an M-step perspective, where the contrastive loss partially maximizes the likelihood by clustering similar data examples. Finally, the training data will be mapped to a mixture of von Mises-Fisher distributions on the unit hypersphere, which facilitates label disambiguation by using the component-specifie label.
Our main contributions are summarized as follows:
1 (Methodology). To the best of our knowledge, our paper pioneers the exploration of contrastive learning for partial label learning and proposes a novel framework termed PiCO. As an integral part of our algorithm, we also introduce a new prototype-based label disanbiguation mechanism, that leverages the contrastively learned embeddings.
2 (Experiments). Empirically, our proposed PiCO framework establishes the stale-of-the-art per- formance on three PLL tasks. Moreover, we make the first attempt to conduct experiments on fine-grained classification datasets, where show classification performance improvement by up to 9.61% compared with the best baseline on the CUB-200 dataset.
3 (Theory). We theoretically interpret our framework from the expectation-maxtt maximization perspec- tive. Our derivation is also generalizable to other CL methods and shows the alignment property in CL (Wang & Isola 2020) mathematically equals the M-step in center-based clustering algorithms.
BACKGROUND
The problem of partial label learning (PLL. is defined using the following setup. Let X' be the input space, and y = {1, 2, C} be the output label space. We consider a training dataset D {(x,, Y:), where each tuple comprises of an image x; E X and a candidate label set Y, y. Identical to the supervised learning setup, the goal of PLL is to obtain functional mapping that predicts the one true label associated with the input. Yet differently, the PLL setup bears significantly more uncertainty in the label space. A basic assumption of PLL is that the ground- truth label y is concealed in its candidate set, i... V/ E Y, and is invisible to the learner. For this reason, the learning process can suffer from inherent ambiguity, compared with the supervised learning task with explicit ground-truth.
A dog Image x; with (Husky, Malamnte,, Samoyed]
Figure I: An input image with three candidate labels, where the ground- truth Ma Malute..
The key challenge of PLL is identify the ground-truth label from the candidate label set. During training, we assign each image æ; normalized vector 8; E [0, 1] as the pseudo larget, whose entries denote the probability of labels being the ground-truth. The total probability mass of is allocated among candidate labels in Y,. Note that 8; will be updated during the training procedure. Ideally, s; should put more probability mass on the (unknown) ground-truth label y; over the course of training. We train a classifier f X' + [0, 1 using cross-entropy loss, with s; being the target prediction. The per-sample loss is given by:
(as 24, Y,) E. S,, (og(f((() s.t.
Σ $i.j = 1 and s.j O. Vj $ Y. (1)
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Dkambiguard Jabel
Cross Entropy Loss
Classifier
Augmenied
Guary Veew
RE Senteereneet Serdel
E Enooder
MI
Augmented
Protetypes
Momentum
Contrastive
MI
Simillarity
Encoder
loss
Mommetiim konteddngge
Figure 2: Illustration of PiCO. The classifier's 's output is used to determine the positive peers for contrastive
learning. The contrastive prototypes are then used to gradually update the pscudo target. The momentum embeddings are maintained by queue structure. "//" means stop gradient.
where j denotes the indices of labels. Kt, denotes the j-th pseudo target of Øy. Here I' is the softmax output of the networks and we denote f as its j-th entry. the remainder of this paper, we omit the sample index when the context is clear. We proceed by describing our proposed framework.
3 METHOD
In this section, we describe our novel Partial label learning with COntrastive label disambigua- tion (PiCO) framework in detail. In a nutshell, PiCO comprises two key components tackling the representation quality (Section 3.1) and label ambiguity respectively (Section 3.2). The two com- ponents systematically work as a whole and reciprocate each other. We further rigorously provide theoretical interpretation of PiCO from an EM perspective in Section
3.1 CONTRASTIVE REPRESENTATION LEARNING FOR PLL
The uncertainty in the label space posits a unique obstacle for learning effective representations. In PiCO, we couple the classification loss in Eq. with contrastive term that facilitates clustering effect in the embedding space. While contrastive learning has been extensively studied in recent literature, it remains untapped in the domain of PLL. The main challenge lies in the construction of a positive sample set. In conventional supervised CL frameworks, the positive sample pairs can be easily drawn according the ground-truth labels (Khosla et al. 2020). However, this is not straightforward in the setting of PLL.
Training Objective. To begin with, we describe the standard contrastive loss term. We adopt the most popular setups by closely following MoCo (He et al., 2020) and SupCon (Khosla et al.) 2020). Given each sample (x,, Y), we generate two views-a query view and a key view view-by by way of randomized data augmentation Aug(z). The two images then fed into query network g(-) and a key network g(()) yielding a pair of Ln-normalized embeddings q = g(Aug, (z)) and = g (x)). In implementations, the query network shares the same convolutional blocks as the classifier, followed by prediction head (see Figure 2). Following MoCo, the key network uses a momentum update with the query network. We additionally maintain a queue storing the most current key embeddings k., and we update the queue chronologically. To this end, we have the following contrastive embedding pool:
A By U B₃ Qqueue
where B, and By are vectorial embeddings corresponding to the query and key views of the current mini-batch. Given an example æ, the per-sample contrastive loss is defined by contrasting its query embedding with the remainder of the pool A.
1
Σ
log cxp(qTk+/7)
Lcon (g: x, A,
[P(x)
(P(m))
EvEA(E exp(e ( k'/T)'
(3)
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where P(z) is the positive set and A(x) A\1q). is the temperature.
Positive Set Selection. As mentioned carlier, the crucial challenge is how to construct the positive set P(x). We propose utilizing the predicted label y arg maxzeY f (Aug, (x)) from the classi-
fier. Note that we restrict the predicted label to be in the candidate set Y. The positive examples are then selected follows,
P(x) = (k'K'' E A(x). 0)
(4)
where y' is the predicted label for the corresponding training example of k'. For computational efficiency, we also maintain label queue to store past predictions. In other words. we define the positive set of æ to be those examples carrying the same approximated label prediction j. Despite its simplicity, we show that our selection strategy can be theoretically justified (Section [ and also lead to superior empirical results (Section 4. Note that more sophisticated selection strategies can be
explored, for which we discuss in Appendix B.4] Putting it all together, we jointly train the classifier as well as the contrastive network. The overall loss function is:
L Lats Xo
(5)
Still, our goal of learning high-quality representation by CL relies on accurate classifier prediction for positive set selection, which remains unsolved in the presence of label ambiguity. To this end, we further propose a novel label disambiguation mechanism based on contrastive embeddings and show that these two components are mutually beneficial.
3.2 PROTOTYPE-BASED LABEL DISAMBIGUATION
As we mentioned (and later theoretically prove in Section Sp. the contrastive loss poses a clustering effect in the embedding space. As a collaborative algorithm, we introduce our novel prototype- basedd label disambiguation strategy. Importantly, we keep a prototype embedding vector f4c corresponding to each class c {1, 2, ...: C). which can be deemed as a set of representative embedding vectors. Categorically, a naive version of the pscudo target assignment is to find the nearest prototype of the current embedding vector. Notably this primitive resembles a clustering step. We additionally soften this hard label assignment version by using posit intuitively that the employment of the prototype builds a connection with the clustering effect in the embedding space brought by the contrastive term (Section 3.1). We provide more rigorous justification in Section 5
moving-averagg style formula. To this end, we may
Pseudo Target Updating. We propose softened and moving-average style strategy update the pseudo targets. Specifically, we first initialize the pseudo targets with a uniform distribution, 8j |(j E Y). We then iteratively update it by the following moving-average mechanism,
[1 c arg maxjey q Pj: 8 = 08 + (1 -)z, 2e = {0
(6)
else
where ó 10.1) is positive constant, and μ4 is prototype corresponding to the j-th class. The intuition is that fitting uniform pseudo targets results in a good initialization for the classifier since the contrastive embeddings are less distinguishable at the beginning. The moving-average style strategy then smoothly updates the pseudo targets towards the correct ones, and meanwhile ensures stable dynamics of training; see Appendix [B.2] With more rigorous validation provided later in [ we provide an explanation for the prototype as follows: (i)-for a given input æ, the closest prototype is indicative of its ground-truth class label. At each step, has the tendency slightly move toward the one-hot distribution defined by based on Eq. (6); (ii)-if an example consistently points to one prototype, the pseudo target can converge (almost) a one-hot vector with the least ambiguity.
Section
Prototype Updating. The most canonical way to update the prototype embeddings is to compute it in every iteration of training. However, this would extract heavy computational toll and in turn cause unbearable training latency. As a result, we update the class-conditional prototype vector similarly in moving-average style:
ae Normalize(714: + 7))), if arg maxjey I (x)n)((
(7)
where the momentum prototype μ4, of class c is defined by the moving-average of the normalized query embeddings q whose predicted class conforms to c. y is tunable hyperparameter.
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Table I:
Accuracy comparisons on benchmark datasets. Bold indicates superior results. Notably, PiCO
achieves comparable results to the fully supervised learning (less than 1% in accuracy with & 1 false candidate).
Dataset
Method
0.1
9 0.3
0.5
PiCO (ours)
94.39 0.18%
94.18 + 0.12%
93.38 0.06%
LWS
90.30 1: 0.60%
88.99 + 1.43%
86.16 0.85%
PRODEN
90.24 +: 0.32%
89.38 + 0.31%
87.78 0.07%
CIFAR-10
CC
82.30 0.21%
79.08 + 0.07%
74.05 0.35%
MSE
79.97 0.45%
75.64 + 0.28%
67.09 0.66%
EXP
79.23 0.10%
75.79 + 0.21%
70.34 1.32%
Fully Supervised
0.07% 94.91
Dataset
Method
g= 000.01 0.01
q= 0.05
= 0.1
PICO (ours)
73.09 + 0.34%
72.74 + 0.30%
69.91 = 0.24%
LWS
65.78 + 0.02%
59.56 + 0.33%
53.53 = 0.08%
PRODEN
62.61) 0.02%
60.73 + 0.03%
56.80 + 0.29%
CIFAR-100
CC
49.76 + 0.45%
47.62 + 0.08%
35.72 + 0.47%
MSE
49.17 0.05%
46.02 + 1.82%
43.81 + 0.49%
EXP
44.45 1.50%
41.05 + 1.40%
29.27 2.81%
Fully Supervised
73.56 + 0.
3.3 SYNERGY BETWEEN CONTRASTIVE LEARNING AND LABEL DISAMBIGUATION
While seemingly separated from each other, the two key components of PiCO work in collaborative fashion. First, as the contrastive term favorably manifests a clustering effect in the embedding space, the label disambiguation module further leverages via setting more precise prototypes. Second, set of well-polished label disambiguation results may, in turn, reciprocate the positive set construction which serves as a crucial part in the contrastive learning stage. The entire training process converges when the two components perform satisfactorily. We further rigorously draw a resemblance of PiCO with classical EM-style clustering algorithm in Section [S] Our experiments, particularly the ablation study displayed in Section 4.3 further justify the mutual dependency of the synergy between the two components. The pseudo-code of our complete algorithm is shown in Appendix C
4 EXPERIMENTS
4.1 SETUP
Datasets and Baselines. First, evaluate PiCO commonly used benchmarks CIFAR-10
and CIFAR-10W (Krizhessky et al. 2009). Adopting the identical experimental settings in previous work (Lv al. 2020 Wen al. (021), generate partially labeled datasets by flipping negative labels 7 / to false positive labels with a probability
P(ğ E Y # y). In other words, all C- 1 negative labels have uniform probability be false positive and we aggregate the flipped ones with the ground-truth to form the candidate set. We consider E {0.1, 0.3, 0.5} for CIFAR- 10 and q (0.01, 0.05, 0.1) for CIFAR-100. Section 4.4, we further evaluate our method on fine-grained classification tasks, where label disambiguation can be more challenging.
We choose the five best-performed partial label learning algorithms date: 1) LWS (Wen et al. 2021] weights the risk function by means of trade-off between losses on candidate labels and the remaining; 2) PRODEN (Lv et 2020) iteratively updates the latent label distribution in a self- training style; 3) CC (Feng et al.) 2020b) is classifier-conssste
method that assumes set-level uniform data generation process; 4) MSE and EXP (Feng et al. 2020a) are two simple baselines that adopt mean square eπror and exponential loss the risks. The hyperparameters are tuned according to the original methods. The detailed implementation of our method PiCO is presented in Ap- pendix B.T For all experiments, report the mean and standard deviation based on 5 independent runs (with different random seeds).
4.2 MAIN EMPIRICAL RESULTS
PiCO achieves SOTA results. As shown in Table I PiCO significantly outperforms all the rivals by significant margin on all datasets. Specifically, on CIFAR-10 10 dataset, we improve upon the best baseline by 4.09%, 4.80%, and 5.80% where q is set to 0.1, 0.3, 0.5 respectively. Moreover, PiCO consistently achieves superior results as the size of the candidate set increases, while the baselines demonstrate significant performance drop. Besides, it is worth pointing out that previous
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1055
(a) Uniform features
(b) PRODEN features
(c) PiCO features (ours)
Figure 3: T-SNE visualization of the image representation on CIFAR-10 with
0.5. Different colors
represent the corresponding classes.
works (Lv et al. 2020 Wen et al. 202T) are typically evaluated on datasets with a small label space (C 10). We challenge this by showing additional results CIFAR-100. When & = 0.1, all the baselines fail to obtain satisfactory performance, whereas PiCO remains competitive. Finally, we observe that PiCO achieves results that are comparable to the fully supervised contrastive learning model, showing that disambiguation is sufficiently accomplished. The comparison highlights the superiority of our label disambiguation strategy.
PiCO learns more distinguishable representations. We visualize the image representation pro- duced by the feature encoder using t-SNE (Van der Maaten & Hinton 2008) Figure 3 Different colors represent different ground-truth class labels. We use the CIFAR-I T0 dataset with = 0.5. We contrast the t-SNE embeddings of three approaches: (a) model trained with uniform pseudo targets, i.e., 5j 1 (i E Y), (b) the best baseline PRODEN, and (c) our method PiCO. We can observe that the representation of the uniform model indistinguishable since its supervision signals suffer from high uncertainty. The features of PRODEN are improved, yet with some class overlapping (e.g.. blue and purple). In contrast, PiCO produces well-separated clusters and more distinguishable representations, which validates its effectiveness in learning high-quality representation.
4.3 ABLATION STUDIES
In this section, we present part of our ablation results to show the effectiveness of PiCO. We refer readers to Appendix B.2 for more ablation experiments.
Effect of fcont and label disambiguation. We ablate the contributions of two key components of PiCO: contrastive learning and prototype-bases label disambiguation. In par- ticular, compare PiCO with two variants: 1) PiCO w/o
70
Accuracy
0
disambiguation which keeps the pseudo target as uniform 1/IY|; and 2) PiCO w/o Looms, which further removes the contrastive learning and only trains classifier with uniform pseudo targets. From Table [ we can observe that vari- ant 1 substantially outperforms variant 2 (e.g., +8.04% on CIFAR-10), which signifies the importance of contrastive learning for producing better representations. Moreover, with label disambiguation. PiCO obtains results close fully supervised setting, which verifies the ability of PiCO in identifying the ground-truth.
55
$00.9
1.0
43
100
500 Epoch
400
800
Figure 4: Performance of PiCO with
varying & on CIFAR- (q = 0.05).
Different disambiguation strategy. Based on the contrastive prototypes, various strategies can be used to disambiguate the labels, which corresponds to the E-step in our theoretical analysis. We choose the following variants: 1) One-hot Prototype always assigns one-hot pseudo target 8 =
by using the nearest prototype (Ø = 0): 2) Soft Prototype Probs follows (Li et al. 2021a) and uses exp(y" pt/T)
soft class probability s; Σ,er rxplq ps/T] the pseudo target (ó = 0): 3) MA Soft Prototype Probs gradually updates pseudo target from uniform by using the soft probabilities in moving- average style. From Table 2 we can see that directly using either soft or hard prototype-based label assignment leads to competitive results. This corroborates our theoretical analysis in Section 5, since center center-based class probability estimation is common in clustering algorithms. However, MA Soft Prototype Probs displays degenerated performance, suggesting soft label assignment is less reliable identifying the ground-trutt Finally, PiCO outperforms the best variant by xe 2% in accuracy both datasets, showing the superiority of our label disambiguation strategy.

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Table 2: Ablation study CIFAR-10 with q = 0.5 and CIFAR- with 7 = 0.05.
CIFAR-10
CIFAR-100
Ablation
Louit
Label Disambiguation
(q 0.5)
(g 0.05)
PICO
Ours
93.58
72.74
PICO w/o Disambiguation
V Uniform Pseudo Target
84.50
64.11
PiCO w/o Lcom PICO with e PiCO with 6 PICO
X
Uniform Pscudo Target
76.46
56.87
V
Soft Prototype Probs
91.60
71.07
/ One-hot Prototype
91.41
70.10
/ MA Soft Prototype Probs
81.67
63.75
Effeet of moving-average factor o. We then explore the effect of pseudo target updating factor on PiCO performance. Figure shows the learning curves PiCO on CIFAR-100 (q 0.05). We can see that the best result is achieved at 6 = 0.9 and the performance drops when e takes smaller value, particularly on the early stage. When e = O, PiCO obtains a competitive result but much lower than 0.9. This confirms that trusting the uniform pseudo targets at the early stage is crucial in obtaining superior performance. At the other extreme value Φ = 1, uniform pseudo targets used, and PiCO demonstrates a degenerated performance and severe overfitting phenomena. In general, PiCO performs well when á Rs 0.9.
4.4 FURTHER EXTENSION:
PARTIAL LABEL LEARNING
Recall the dog example highlighted Section where semantically similar classes more likely to cause label ambiguity. It begs the question of whether PiCO is effective the challenging fine- grained image classification tasks. To verify this, conduct experiments on two datasets: CUB- 200 dataset (Welinder et al. 2010)) contains 200
Table 3: Accuracy comparisons on fine-grained
classification datasets.
CUB-200
CIFAR-100-H
Method
(g 0.05)
(g 0.5)
PiCO
72.17 + 0.72%
72.04 + 0.31%
LWS
39.74 + 0,47%
57.25 + 0.02%
PRODEN
62.56 + 0.10%
60.89 + 0.03%
bird species: 2) CIFAR-I00 with hierarchical labels CCFAR-110-HH where we generate candidate labels that belong the same superclass) We set q = 0.05 for CUB-200 and 0.5 for CIFAR- with hi- erarchical labels. In Table 3 we compare PiCO with baselines, where PiCO outperforms the best method PRODEN by large margin (+9.61% on CUB-200 and +11.15% CIFAR-100-HD Our results validate the effectiveness of our framework, even in the presence of strong label ambiguity.
CC
55.61 + 0.02%
42.60 0.11%
MSE
22.07 1: 2.36%
39.52 + 0.28%
EXP
9.44 1: 2.32%
35.08 1 1.71%
5 WHY PICO IMPROVES PARTIAL LABEL LEARNING?
In this section, we provide theoretical justification on why the contrastive prototypes help disam- biguate the ground-truth label. We show that the alignment property in contrastive learning (Wang & Isolaa 2020) intrinsically minimizes the intraclass covariance in the embedding space, which com- cides with the objective of classical clustering algorithms. It motivates to interpret PiCO through the lens of the inpectation-maxmmizzue each training step, all data examples are accessible and the augmentation copies are also included in the training set, i.e., A = D. Then, the contrastive loss is calculated as,
algorithm. To see this, we consider an ideal setting: in
log exp(q k+/T)
Lcom (0: D)
Σ
PP(x)
ENEA(E) exp(q' K(/T)
#ED
k+EP(x)
(8)
Σ
(q k+/T)
log Σ expla (xplaT(//T) AeeA(a)
1
[P(x)
72
RCD
(b)
We focus analyzing the first term (a), which is often dubbed as the alignment term (Wang & this term is to optimize the tightness of the clusters in the
Isola 2020). The main functionality embedding space. In this work, connect it with classical clustering algorithms. We first split the dataset to C' subsets S, E Dc (1 I & C), where each subset contains examples possessing
CIFAR-100 dataset consists of 20 superclasses, with 5 classes in each superclass.

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the same predicted labels. In effect, our selection strategy in Eq. selecting examples from the same subset. Therefore, we have,
4
constructs the positive by
P(x)) En (1lq k- 2)) 1 Σ RED
(a)
1
Σ S ED |sy]
1
Σ Eawes, 11((()) g( )1( + K
2rn
Th Σ E S,EDo Eues, 11((z) 2 K,
where is a constant and j4j is the mean center of Sj. Here we approximate [5g] since is usually large. We omitted the augmentation operation for simplicity. The miforriity term (b) can benefit informatiionpreserviinn
15 [P(=]]
and has been analyzed in (Wang & Isola 2020).
We now ready to interpret the PiCO algorithm maximizes the likelihood a generative model. At the E-step. the classifier assigns each data example to one specific cluster. At the M-step, the contrastive loss concentrates the embeddings to their cluster mean direction, which is achieved by minimizing Eq. (9. Finally, the training data will be mapped mixture of von Mises-Fisher distributions the unit hypersphere.
pnpeectationmmxximizzitie
algorithm that
EM Perspective. Recall that the candidate label set is a noisy version of the ground- To estimate the likelihood P(Y,, æ;:). we need establish the relationship between the candidate and the ground-truth label. Following (Liu & Dietterich 2012), we make mild assumption,
Assumption 1. All labels ys in the candidate label set have the same probability of generating Y, but no label outside of Y, can generate Y,, iee. P(Yilys) = h(Y,) if y: E Y; else O. Here h(-) is some function making it valid probability distribution.
Then, we show that the PiCO implicitly maximizes the likelihood as follows,
E-Step. First, we introduce some distributions over examples and the candidates &े 2 0 s i S n, 1 < j C) such that n/ = 0 if & Y; and Ejen ... = 1. Let be the parameters of g. Our goal is to maximize the likelihood below,
argmax Σ log P(Y, argmax
log Σ
P( 1= Σ i==
log(h
WE
BEEY ! P(æ,, y:10)
log Σ
argmax
0
i=1
m Ui
(10)
Σ MEY T log P( v)
argmax
T. The last step of the derivation uses Jensen's 's inequality. By using the fact that log(-) function is concave, the inequality holds with equality when P(m..vl0)) is some constant. Therefore,
x
P(z,, 1.
P(a,.3100
P y-l0) P(x,10)
4: Ey.ev, P( 34 0)
= P(y k 0),
(11)
E P(x,, 9/)
which is the posterior class probability. In PiCO, it is estimated by using the classifier's 's output.
To estimate P 0), classical unsupervised clustering methods intuitively assign the data exam- ples to the cluster centers, c.g. k-means. As in the supervised learning setting, we can directly use the ground-truth. However, under the setting of PLL., the supervision signals situated between the supervised and unsupervised setups. Based on empirical findings, the candidate labels are more reliable for posterior estimation the beginning; yet alongside the training process, the prototypes tend to become more trustful. This empirical observation has motivated us to update the pseudo tar- style. Thereby, have good initialization in estimating class posterior, and it will be smoothly refined during the training procedure. This is verified in our empirical stud- ies; see Section 4.3 and Appendix B.2 Finally, we take one-hot prediction ÿ arg maxjeY f (z,) since each example inherently belongs to exactly one label and hence, we have π? = I[ū = j).
gets in moving-averan
M-Step. At this step, aim maximizing the likelihood under the assumption that the poste- rior class probability is known. We show that under mild assumptions, minimizing Eq. p also maximizes lower bound of likelihood in Eq. (T0).
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Theorem 1. Assume data from the same class in the contrastive output space follow d- variate von Slises-Fisher (vMF) distribution whose probabilistic density is given by f(zlp;, k) = ca(K)e g(m), where μr /../llulll is the mean direction, is the concentration parameter, and Ca(K) is the normalization factor. We further assume a uniform class prior P(y = = 1/C. Let n = [Syl. Then, optimizing Eq. and Eq. (10) equal maximize R1 and R2 below, respectively.
R1 = R==5s S,EDo n
[1j || E
|4 R2-
(12)
S,EDo
The proof can be found in Appendix A Theorem Sonoreeeniniicttsss that minimizing Eq. 9 also maximizes a lower bound of the likelihood in Eq. (10). The lower bound is tight when || is close to 1, which in effect means a strong intraclass concentration on the hypersphere. Intuitively, when the hypothesis space is rich enough, it is possible to achieve low intraclass covariance in the Euclidean space, resulting in a large norm of the mean vector ||ر Then, normalized embeddings in the hypersphere also have intraclass concentration in a strong sense, because a large || || also results in a large x. (Banerjee et al. 2005). Regarding the visualized representation in Figure 3 PiCO is indeed able to learn compact clusters. Therefore, loss also partially maximizes the likelihood defined in Eq. (1O).
we note that have that minimizing the contrastive
6 RELATED WORKS
Partial Label Learning (PLL) allows each training example to be annotated with candidate label set, in which the ground-truti is guaranteed to be included. The most intuitive solution is average- based methods (Hüllermeier & Beringer [2006] Cour et al., 2011] ZZang & Yu 2015), which treat all candidates equally. However, the key and obvious drawback is that the predictions can be severely misled by false positive labels. To disambiguate the ground-truth from the candidates, identification- based methods (Jin & Ghahramani 2002). which regard the ground-truth as latent variable, have recently attracted increasing attention; representative approaches include maximum margin- based methods (Nguyen & Caruana, 2008 Wang et al.) 2020), graph-based methods (Zhang et al., 2016 Wang et al. 2019 Xu et al.j 2019] et al.) 2021). and clustering- -hased approaches (Liu & Diet- terich] (011)) Recently, self-training methods (Feng al., [0200] [v al., 2020 Wen et al.) 2021) have achieved state-of-the-arr results on various benchmark datasets, which disambiguate the candi- date label sets by means of the model outputs themselves. But, few efforts have been made to learn high-quality representations to reciprocate label disambiguation.
Contrastive Learning (CL) (van den Oord et al., 2018 He al. 2020) is framework that learns A plethora of works has explored the effectiveness of CL in unsupervised representation learning (van den Oord et al. 2018, Chen et al. 2020 He et al., 2020). Recently, Khosla et al. (2020) propose supervised contrastive learning (SCL), an approach that aggregates data from the same class as the positive set and obtains improved performance
discriminative representations through the use of instance similarit/dissimiin
various supervised learning tasks. The success of SCL has motivated a series of works to apply CL to a number of weakly supervised learning tasks, including noisy label learning (Li et al. 2021a Wu al. 2021), semi-supervised learning (Li et al. 2020 Zhang et al. 2021), etc. Despite promising empirical results, however, these works, lack theoretical understandings. Wann & Isola (2020) theoretically show that the CL favors alignment and unifor- mity, and thoroughly analyzed the properties of uniformity. But, to date, the terminology alignment remains confusing; we show it inherently maps data points to mixture of vMF distributions.
7 CONCLUSION
In this work, we propose a novel partial label learning framework PiCO. The key idea is to identify the ground-truth from the candidate set by using contrastively learned embedding prototypes. Em- pirically, we conducted extensive experiments and show that PiCO establishes state-of-the-as per- formance. Our results are competitive with the fully supervised setting, where the ground-truth label is given explicitly. Theoretical analysis shows that PiCO can be interpreted from EM-algorithm perspective. Applications of multi-class classification with ambiguous labeling can benefit from our method, and we anticipate further research in PLL to extend this framework to tasks beyond image classification. We hope our work will draw more attention from the community toward a broader view of using contrastive prototypes for partial label learning.
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ACKNOWLEDGMENT
HW, RX, GC and JZ are supported by the Key R&D Program of Zhejiang Province (Grant No. 2020C01024). JZ would also like to thank the Fundamental Research Funds for the Central Uni- versities. YL supported by Wisconsin Alumni Research Foundation (WARF), Facebook Research Award, and funding from Google Research. FL is supported by the National Natural Science Foun- dation of China (Grant No. 62106028).
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A THEORETICAL ANALYSIS
Derivation of Eq.
We provide the derivation of the second equality in Eq. ( It suffices to
show that LHS
2ng Σ x ES, ((9(x) - g(x')12 Emes, 119(#) - |4 1 RHS. We have,
LHS
Σ
(1 -()(()))))(()) g((²) + (l0(
2n
#ES, x'ES,
1
Σ
(nj g(x)T( g(x'))
x'E8,
(13)
1
Σ (ng g( (mzt4a)))
xES;
n2 g(x))) T 12)) ny (1 |1,,)11))
xES,
On the other hand,
RHS Σ (11g(=)112 2g(xx) pej + |||m
s
(ng - 2( Σ g(t)) T Mg + ny (|
(14)
SES)
n2 - u (2) = LHS.
Proof of Theorem from Eq. IOP..
By regarding गं as constants w.r.t 6 we can get the following derivation
P( 0:
arg max
m; log
arg max
त log P(xxlyr, P(y)))
n
1=1 V.EY
W.EYY
arg max
I(ū = y:) log ((x;
weY:
arg max
Σ
log P(xly = j,0)
(15)
S,EDo #ES,
arg max Σ
Σ
(Kp g(x) + log(ca(#)))
SyEDo #ES;
Σ
n2j
||m
arg
n
S
Σn Eyer,, o. log n." w.r.t. 0 the first equality. In the last equality, use the fact that μ, ny Exes, g(x) and μ, is the unit directional
where n, = |Syl- Here we ignore the constant factor
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vector of Mj. From Eq.
( have that,
arg min Σ Σ
(| 13ll2 = arg min
E (I1g(x))1 2g(xx) (ooT) Mj (111)113))
Se zES,
S,EDO WES,
arg min
(ny - nz [uy (1²)
0
SD
Σ
||ml|
arg max
n
SyEDe
(16)
Note that the contrastive embeddings are distributed the hypersphere s and thus ||ي||| [0, 1]. It can be directly derived that,
R1 = Σ | S,EDe
Σ S,EDe
|lyl = R2.
(17)
Therefore, maximizing the intraclass covariance Eq. 99 equivalent maximizing a lower bound of the likelihood in Eq. (10). It can also be shown that (R2)² < R1 followed by the convexity of the squared function. Since arg max R2 = arg max R2)), we have that the contrastive loss also maximizes an upper bound of R2.
Unfortunately, there is no guarantee that the lower bound tight without further assumptions. To see this, assume that we have two classes y E {1,2} with equal-sized samples and their mean vectors have the norm of | || = 0.5 and || || = 1. We have that R = 0.625 and R2 = 0.75, which demonstrates large discrepancy. It is interesting to see that when the norm of the mean vectors are the same, i.c. | || = || for all 1 S j I < C. we have (R₂)² = R₁ by the Jensen's inequality, making the upper bound tight. But, it not trivial condition.
To obtain a tight lower bound, what we need is a rich enough hypothesis space to achieve a low intraclass covariance in Eq. (9). and hence large R1. We show that it inherently produces compact vMF distributions. To see this, should be noted that the concentration parameter x of a vMF distribution is given by the inverse of the ratio of Bessel functions of mean vector píj. Though is not possible to obtain an analytic solution of K, we have the following well-known approximation Banerjee et al. 2005).
d - 1
2(1 [1yl1)) K
valid for large ||
(18)
dd 8) (d (d+ 2) + | |1m))1*):
K d |||l|||
1 d+ 14
valid for small ||a
(19)
The above approximations show that larger norm of Euclidean's mean H4 typically leads stronger concentration on the hypersphere. By Eq. (16), we know that contrastive loss encourages a large norm of faj. and thus also tightly clusters the embeddings on the hypersphere; see Figure
We further note that we do not include a k-means process in our PiCO method. PiCO related center-based clustering algorithms in our theoretical analysis. Since we restrict the gold label to be included in the candidate set, we believe that this piece of information could largely help avoid the bad optimum problem that occurs in a pure unsupervised setup. For the convergence properties of our PiCO algorithm, we did not empirically find any issues with PiCO converging to (perhaps locally) optimal point. However, we want to refer the readers to the proof of k-means clustering in (Bottou & Bengio 1994) for the convergence analysis.
Discussion. Regarding our empirical results where PiCO does indeed learn compact representa- tions for each class, we can conclude that PiCO implicitly clusters data points in the contrastive embeddings space as a mixture vMF distributions. In each iteration, our algorithm can be viewed as alternating the two steps until convergence, though different in detail. First, it is intractable to handle the whole training dataset, and thus we accelerate via a MoCo-style dictionary and MA- updated prototypes. Second, the contrastive loss also encourages the uniformity of the embeddings to maximally preserve information, which serves as a regularizer and typically leads to better rep- resentation. Finally, we use copies of data examples such that data also aligned in its local
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100
072
025
4.35
-015
-2.23
-2.0
-0.23
Figure 5: A large norm of Euclidean's 's mean vector also leads to strong concentration of unit vectors to its mean direction.
region. Moreover, our theoretical result also answers why merely taking the pseudo-labels to select positive examples also leads to superior results, since the selected positive set will be refined as the training procedure proceeds.
Our theoretical results are also generalizable to other contrastive learning methods. For example, the classical unsupervised contrastive learning (He al. 2020) actually assumes n-prototypes to cluster all locally augmented data; prototypical contrastive learning (Li et al. (2221)) directly assigns the data examples to one cluster get the posterior, since there are no supervision signals; supervised contrastive learning (Khosla et al. 2020) chooses the known ground-truth as the posterior. In our problem, we follow two extreme settings to progressively obtain an accurate posterior estimation. Finally, it is also noteworthy that the objective in Eq. has a close connection to the intraclass deviation (Saunshi et al., 2019), minimizing which is proven to be beneficial obtaining tighter generalization error bound downstream tasks. It should further be noted that work differs from existing clustering- baasering-based CL methods (Caron et al.) 2020): Li et al.. (2021b), which explicitly involves clustering aggregate the embeddings; instead, results derived from the loss itself.
B ADDITIONAL EXPERIMENTAL SETUPS AND RESULTS
B. IMPLEMENTATION DETAILS
Following the standard experimental setup in PLL (Feng et al. 20206 Wen et al. 2021), we split a clean validation (10% of training data) from the training that, we transform the validation set back to its original PI.I. form and incorporate into the training set to accomplish the final model training. We use 18-layer ResNet as the backbone for feature extraction. Most of experimental setups for the contrastive network follow previous works (Hc) al., 2020 Khosla et 2020). The projection head of the contrastive network a 2-layer MLP that outputs 128- dineenmnns embeddings. We use two data augmentation modules SimAugment (Khosla et al. 2020) and RandAugment (Cubuk et al., 2019) for query and key data augmentation respectively. Empirically, we find that even weak augmentation for key embeddings also leads to good results. The size of the queue that stores key embeddings fixed to be 8192. The momentum coefficients are set as 0.999 for contrastive network updating and 0.99 for prototype calculation. For pseudo target updating, we linearly ramp down á from 0.95 to 0.8. The temperature parameter set 0.07. The loss weighting factor is SGD optimizer with a momentum of 0.9 and the batch size is 256. We train the model for 800 epochs with cosine learning rate scheduling. We also empirically find that classifier warm-up leads to better performance when there are many candidates. Hence, we disable contrastive learning in the first 100 epoch for CIFAR-100 with q = 0.1 and I epoch for the remaining experiments.
select the hyperparameters. After
À 0.5. The model is trained by a standard
B.2 ABLATION STUDIES
Moving-average updating factor d. We first present more ablation results about the effect of pseudo target updating factor PiCO performance. (ag shows the results two datasets
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92.5
90
90.0
87.5
85
Accuracy
85..0
80
CIFAR-100
82.5
CIAR-10 CIAAR 100
80.0
75
77.5
70
75.0
65
72.5
0.0
0.2
0.4
0.6
0.8
1.0
10
10 1
10c A (log scale)
10a
$
(a) Ablation
(b) Ablation À
Figure 6: More ablation results CIFAR-10 10 (q = 0.5) and CIFAR-100 (q 0.05). (a) Performance of PICO
with varying o. (b) Performance of PiCO with varying A.
CIFAR-10 (g 0.5) and CIFAR 100 (g 0.05). The overall trends on both datasets follow Specifically, performance on CIFAR-100 achieves the best result when = 0.7, and slight drops when ø 0.9. Therefore, in practice, we may achieve better result with careful fine-tuning on á value. In contrast, PiCO works well in a wide range of & values on CIFAR-10. The reason might be that CIFAR-IO is simpler version of CIFAR-100, and thus the prototypes can be high-qualing quickly. But, setting ø to either 0 or leads to worse result, which has been discussed earlier.
arguments in section 4.3
Loss weight A. F reports the performance of PiCO with varying À values that trade-off the classification and contrastive losses. selected from (0.01, 0.1, 0.5, 5, 50). We can observe that on CIFAR-10, 10, the performance is stable, but CIFAR-100. the hest performance is obtained 50, PiCO shows inferior results on both two datasets.
= 5. When small À (< 10) usually leads to good performance than much larger value. When À is large, the contrastive network tends to fit noisy labels at the early stage of training.
general, relatively
Table 4: Performance of PiCO with varying on CIFAR-I0 and CIFAR-100.
Dataset
? 0.1
7
0.5
7
0.9
? = 0.99
? 0.999
CIFAR-10 (q = 0.5)
93.61 72.87
93.51
93.52 72.54
93.58 72.74
93.66
CIFAR-100 = 0.05)
73.09
67.33
Prototype updating factor y. Finally, we show the effect of that controls the speed of prototype updating and the results listed in Table On the CIFAR-10 dataset, the performance is stable with varying 7. But, on CIFAR- 100, can be seen that too large À leads to significant performance drop, which may be caused by insufficient label disambiguation.
Table 5: Training accuracy of pseudo targets on CIFAR-10 10 and CIFAR-100.
CIFAR-10
CIFAR-100
Dataset
g = 0.1
= 0.3
q ==00 0.5
= 0.01
= 0.05
= 0.1
Accuracy
98.28
98.26
96.79
99.06
96.27
90.58
The disambiguation ability of PiCO. Finally, we evaluate the disambiguation ability of the pro- posed PiCO. To see this, we calculate the max confidence to represent the uncertainty of an exam- ple, which has been widely used in recent works (Hendrycks & Gimpell 2017). one example uncertain about its ground-truth, then it typically associates with low max confidence max, 8j- To represent the uncertainty (MMC) score. In Figure 7 plot the MMC scores of different label disambiguation strategies in different training epochs. First, we can observe the MMC score of PiCO smoothly increases and finally achieves near results, which means most of the labels are well disambiguated. In contrast,
the whole training dataset, we calculate the mean max confidence
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1.0
1.0
Mean Max Confidence
Mean Max Confidence
0.8
0.8
0.7
PICO
0.7
PICO
One-Hoo Prototype
One-Hot Prototype
0.6
Seft Prototype Probs
0.6
Saft Protstype Probs
MA Soft Prototype Probs
0.5
MA Soft Prototype Probs
0.4
0.4
0.3
0.3
0.2
0.2
100
200
300
400 500 Epoch
600
700
100
200
400 500 600 Epoch
700
(a) CIFAR-IO (q 0.5)
(b) CIFAR-100 (g 0.05)
Figure 7: The mean max confidence curves of different label disambiguation strategies.
the Soft Prototype Probs strategy oscillates at the beginning, and then also increases to a certain value, which means that directly adopting the soft class probability also helps disambiguation. But, it is worth noting that it ends with a smaller MMC score compared with PiCO. The reason might be that the cosine distances to non-ground-truth prototypes are still at a scale. Hence, the Soft Prototype Probs strategy always holds a certain degree of ambiguity. Finally, we can see that the MA Soft Prototype Probs strategy fails to achieve great disambiguation ability. Compared with the non-moving-averidn version, it fails to get rid of severe label ambiguity and will finally converge to uniform pseudo targets again.
Furthermore, we evaluated the accuracy of the pseudo targets over training examples. From Table 5) we can find that the pseudo targets achieve high training accuracy. Combined with the fact that the mean confidence score of the pseudo target is close to 1, the training examples finally become near supervised ones. Thus, the proposed PiCO is able to achieve near-supervised learning performance, especially when the label ambiguity is not too high. These results verify that PiCO has a strong label disambiguation ability to handle the PLL problem.
B.3 ADDITIONAL RESULTS ON FINE-GRAINED CLASSIFICATION
In the sequel, we show the full setups and experimental results on fine- classification datasets. In particular, on CUB-200, we set the length of the momentum queue as 4192. For CUB-200, we set the input image resolution as 224 x 224 and select E (0.01, 0.05, 0.1}. When q = 0.05/0.1, warm up for 20/100 100 epochs and train the model for 200/ 200/300 epochs, respectively. For CIFAR- 100-H, we select g E {0.1, 0.5, 0.8) and warm up the model for 100 epochs when = 0.8. Other hyperparameters are the same as default setting. The baselines also fine-tuned to achieve their best results.
From Table 6 we can observe that PiCO significantly outperforms all the baselines mentioned in section on all the datasets. Moreover, as the size of candidate sets grows larger, PiCO con- sistently leads by best bascline, performance improvement reaches 9.50%, 11.15% and 22.53% in accuracy when q 0.1, 0.5, 0.8, respectively. The comparison emerges the dominance of our label disambiguation strategy among semantically similar classes.
wider margin. For example,
CIFAR- 100-H, compared with the
B.4 STRATEGIES FOR POSITIVE SELECTION
While our positive set selection strategy is simple and effective, one may still explore more compli-
cated strategies to boost performance. We have empirically tested two strategies: 1) Filter-based: we set filter [YnY] | S p (p 0.5) to remove example pairs who have dissimilar candidate sets at the early stage. 2) Threshold-basee
set threshold max f (Aug, (x)) < (6 0.95) to move those uncertain examples at the end of training, which has been widely used in -upervise learning (Sohn al. 2020). Our basic principle is that contrastive learning is robust noisy nega-
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Table 6: Accuracy comparisons fine-grained datasets. Bold indicates superior results.
Dataset
Method
q 0.1
g 0.5
4 0.8
PICO (ours)
73.41 + 0.27%
72.04 + 0.31%
66.17 + 0123%
LWS
62.41 + 0.03%
57.25 + 0.02%
20.64 + 0.48%
PRODEN
62.91 +: 0.01%
60.89 : 0.03%
43.64 + 1.82%
CIFAR-100-F
CC
50.40 = 0.20%
42.60 = 0.11%
37.80 = 0.09%
MSE
46.05 + 0.17%
39.52 + 0.28%
15.18
+ 0.73%
EXP
45.73 + 0.22%
35.08 + 1.71%
22.31
+ 0.39%
Dataset
Method
g 0.01
9 0.05
9 0.1
PICO (ours)
74.14 1: 0.24%
72.17 1 0.72%
62.02 1: 1.16%
LWS
73.74 + 0.23%
39.74 + 0.47%
12.30 + 0.77%
PRODEN
72.34 +: 0.04%
62.56 + 0.10%
35.89 + 0.05%
CUB-200
CC
56.63 +: 0.01%
55.61 + 0.02%
17.01 + 1.44%
MSE
61.12 +: 0.51%
22.07 + 2.36%
11.40 + 2.42%
EXP
55.62 + 2.25%
9.44 2.32%
7.3 + 0.99%
Fully Supervised
76.02 + 0.19%
tive pairs and thus, we can flip those less reliable positive pairs to negative. Unfortunately. we did observe statistically significant improvement our vamilla strategy in experiments. These nega- tive results suggest that the proposed PiCO has strong error correction ability, which corroborates our theoretical analysis.
B.5 THE INFLUENCE OF DATA GENERATION
In practice, some labels may be more analogous the true label than others, which makes their prob- ability of label flipping q larger than others. In other words, the data generation procedure non- uniform. In Section 4.4, have shown one such case on CIFAR- labels have a larger probability of being a false posi- tive. Moreover, we follow (Wen et al.] 2021) to con- duct empirical comparisons on data with alternative generation processes. In particular, we test two com- monly used cases on CIFAR-10 10 with the following flipping matrix, respectively:
Table Accuracy comparisons with different data generation processes on CIFAR-10.
Method
Case (1)
Case (2)
PiCO
94.49 1 0.08%
94.11 1:2 0.25%
LWS
90.78 + 0.01 %
68.37 1: 0.04%
where semantically similar
PRODEN
90.53 + 0.01% %
87.02 1 0.02%
CC
75.81 = 0.13%
66.51 0.11%
MSE
68.11 = 0.23%
39.49 + 0.41%
EXP
71.62 = 0.79%
48.87 + 2.32%
0.5
0
0.9
0.7
0.5
0.3
0.1
0
1
0.5
...
0
0
1
0.9
0.7
0.5
0.3
0.1
(1)
(2)
10.5
L..9
0.7
0.5
0.3
0.1
where each entry denotes the probability of label being candidate. As shown in Table 7 PiCO outperforms other basclines in both cases. It is worth noting that Case (2), each ground-truth label has maximum probability of 0.9 of being coupled with the same false positive label. In such a challenging setup, PiCO still achieves promising results that are competitive with the supervised performance, which further verifies its strong disambiguation ability.
B.6 THE INFLUENCE OF PROTOTYPE CALCULATION
There are several ways to calculate the prototypes and hence, we further test a variant PiCO that re-computes the prototypes by averaging embed- dings of all training examples at the end each epoch. We train the models using one Quadro P5000 GPU respectively and evaluate the average training time per epoch. From Table we ob- serve that the Re-Compute variant achieves com- petitive results, but is much slower than PiCO.
Table 8: Training time (min/epoch) and accuracy of different prototype calculation methods.
Dataset
Method
Time
Accuracy
CIFAR-10
PiCO Re-Compute
0.94
93.58
(q 0.5)
1.39
93.55
CIFAR- 100
PiCO
0.96
72.74
(g = 0.05)
Re Re-Compute
1.40
72.35
17

---

Published as a conference paper at ICLR 2022
Algorithm 1: Pseudo-code of PiCO (one epoch).
Input:
Training dataset D, classifier f. query network 9, key network g', momentum queue, uniform pscudo-labels 8; associated with æ; in D, class prototypes μ4 (1 S C).
for iler = ........ ......... do sample mini-batch B from D
/ query and key embeddings generation
Bg = {q: = (Aug, (z,)), E B}
B+ (k: 9 (Aug, (a:)) le, B}
= Bg U B5 L queue
for x; & B do
// classifier prediction
arg maxjev; 8 (z.))
// momentum prototype updating
Normalize(yHe: + (1-7)))), if =
// positive set generation
10
P(w,) = (k'W'' E A(x.). }
end
// bratetype-based label disambiguation
12
for q: E B, do
OneHot maxyer, q. M4))
14
S. Φs; + z)
15
end
// contrastive loss calculation
16
Lou(giiT, |
[Bql ENER, { (P())))
log EnEALe expla, W w/T)
explal kt/T)
// classiffication loss calculation
La(J:B)= T& EzEB E sis log( (Aug,
17
/1 network updating
minimize loss C= Cas + XCoo
// update
key network and momentum queue
19 momentum update g' by using
enqueue B3 and classifier predictions and dequeue
21 end
C PSEUDO-CODE OF PICO
We summarize the pseudo-code of our PiCO method in Algorithm
D THE LITERATURE OF PROTOTYPE LEARNING
Prototype learning (PL) aims to learn a metric space where examples are enforced to be closer to its class prototype. PL typically more robust handling few-shot learning (Snell et al. 2017). zero- shot learning (Xu et al. 2020). and
samples (Arik & Pfister 2019). Recently, PL
has demonstrated promising results in weakly-supervised learning, such as semi-supervised learn- ing (Han et al. 2020), noisy-label learning (Li et al.) (221a),, etc. For example, USADTM (Han) jet al., (0200) shows that informative class prototypes usually lead to better pscudo-labels for semi- supervised learning than classical pseudo-labeling algorithms (Sohn al. 2020) which reuse clas-
sifier outputs. Motivated by this, we also employ contrastive prototypes for label disambiguation.
18


Reviews:
[{"review_id": "lBwdz18uwtF", "reviewer": "Reviewer_k7zr", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "## Strengths \n\nThe problem that the paper tackles, namely partial label learning, is important in the real life where labelling is difficult due to semantic ambiguity (e.g. husky vs malamute). The problem has several connections especially with weakly supervised learning. \n\nThe approach proposed in the paper is very well motivated. The idea of making contrastive learning and prototype learning is sound as they can work collaboratively in the EM fashion (proved in the paper). \n\nThe approach is backed by very impressive empirical results. The presented analyses support the motivation of the approach, e.g. cluster visualisation shows that the clusters are well formed with few classification errors. \n\nThe paper is well written with a clear structure. Reproduction doesn't seem difficult. \n\n\n## Weaknesses \n\nAs the approach is EM-like, it also inherits some cons from EM. This is however not presented in the paper. For instance, in which case the learning will get stuck in a bad optimum? What is the consequence? Is there any way to avoid that?\n\nIt is unclear how the models were fine-tuned. Did the authors use clean dev sets which don't have any false positive labels? If this is the case, would it be more realistic to fine-tune the models on noisy dev sets instead? One concern is that clean dev sets can provide lots of information for removing labelling ambiguity. For instance, one can achieve reasonably good prototype representations just from a few clean labelled instances. \n\nIt seems that the paper mentions too little about prototype learning literature. \n\n", "summary_of_the_paper": "The paper proposes an innovative approach to partial label learning, where an instance is assigned with some false positive labels besides its true label (due to difficulty in labelling). The approach blends contrastive learning with prototype learning: (i) the former helps to form good clusters for the latter to learn prototype representations (ii) in return the latter helps to select positive samples for the former. Theoretically, the authors prove that the two of them work collaboratively in the EM fashion. \n\nThe paper demonstrates the effectiveness of the proposed approach using a typical setting for the task. Specifically, CIFAR 10 and CIFAR 100 were used and each instance was randomly assigned false labels with some probabilities. The proposed approach achieved impressive results, substantially outperforming five recent models in the literature. Moreover, its performance strongly approaches the one of supervised learning. \n\nThe paper also presents a much stricter setting when false positive labels are semantically correlated with true labels. The proposed approach also gained impressive results on CUB-200 and CIFAR-100-H datasets. Moreover, the paper shows several in-depth analyses.\n\n", "summary_of_the_review": "This is a strong paper tackling an important problem. The approach is interesting and the technique is sound. The claims are well backed by impressive empirical result and theory. Therefore, overall, the contributions of the paper will have a certain impact to the community.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works.", "empirical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works."}, "extra_text": {}}, {"review_id": "MsJN5PBk1Z_", "reviewer": "Reviewer_fAwc", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "I have the following comments.\nPros:\n1.\tThe structure of this paper is well-written and easy to follow. The motivation is clear, and the solution is simple but effective.\n2.\tI think the theoretical justification of the proposed method from the expectation-maximization perspective is very interesting. It is a generic result and potentially help the community understand the property of contrastive learning.\n3.\tThe experimental results are quite strong and the ablation study also looks good. I especially appreciate the results on new fine-grained datasets to test the performance of PLL methods.\nCons:\n1.\tExperiments are mainly conducted on datasets with uniformly generated candidates, it is not clear how PiCO performs on non-uniform datasets, such as in the label-dependent setting where the probability of label flipping depends on its ground-truth label.\n2.\tWhy does the second equality in Eq. (9) hold? It would be appreciated if a clear explanation is provided.\n3.\tThere is a redundant space notation in Table 1 (the method CC with q=0.5 on CIFAR10).\n", "summary_of_the_paper": "This work approaches the partial label learning problem where each training example is annotated with multiple candidate labels, in contrast to the conventional supervised learning setup that the ground-truth label is provided. The proposed framework comprises two key components: (1) a contrastive learning module that uses the classifier output to select positive pairs and (2) a label disambiguation method that uses the contrastive prototypes to update the pseudo targets in a moving-average style. The experimental results look quite strong, where the performance of PiCO nearly approaches the fully-supervised results.", "summary_of_the_review": "This paper is clear and technically solid. The theoretical justification is interesting.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "CRpXMhxuyZ0", "reviewer": "Reviewer_7qAy", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Strengths\n- Incredibly strong PLL performance close to supervised learning and substantially stronger than baselines.\n- Helpful ablations and results on various datasets.\n- Provides theoretical connection of the proposed method to EM.\n\nWeaknesses\n- More should be said about the closeness of PLL performance and supervised learning performance. This comment may seem counterintuitive, but I do not think this closeness can be so easily glossed over, especially when it is one of the main results.\n- (minor) Treatment of clustering is simplistic. K-means is a very specific type of clustering algorithm, and only one of multiple classic clustering techniques.\n\nQuestions and comments:\n\n- How is the “novel” prototype-based label disambiguation so different from using a softmax layer? A softmax layer will also have a prototype vector for each class.\n- I can see the convenience of momentum style updating of prototype vectors, but re-computing them every iteration (or every N iterations) is not so expensive and at most increases training by 2x.\n", "summary_of_the_paper": "The authors present a new technique for partial label learning (PLL). PLL is the task where the labels for each instance include both the ground truth label and a randomly sampled set of distractor labels, and during training the model learns a latent decision for which among this set is the ground truth. The technique presented by the authors uses a combination of momentum (in the representation) and contrastive learning (to augment the label set) that leads to improved PLL results, reaching nearly fully supervised performance.", "summary_of_the_review": "This seems like a particularly strong result. More could be said about the implications of their results, but otherwise seems like a good paper.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works."}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper presents PiCO, a novel approach for partial label learning, which achieves very strong performance close to that of fully supervised learning and outperforms PPL baselines. The experiments are extensive with very impressive results and the analysis are thorough.
Decision: Accept (Oral)