Balancing Positive and Negative Classification Error Rates in Positive-Unlabeled Learning
Authors: Ximing Li
Yuanchao Dai
Bing Wang
Changchun Li
Jianfeng Qu
Renchu Guan
Venue: NeurIPS 2025 poster ()
Keywords: positive and unlabeled learning
generalization error bound
Primary area: general_machine_learning

Abstract: Positive and Unlabeled (PU) learning is a special case of binary classification with weak supervision, where only positive labeled and unlabeled data are available. Previous studies suggest several specific risk estimators of PU learning such as non-negative PU (nnPU), which are unbiased and consistent with the expected risk of supervised binary classification. In nnPU, the negative-class empirical risk is estimated by positive labeled and unlabeled data with a non-negativity constraint. However, its negative-class empirical risk estimator approaches 0, so the negative class is over-played, resulting in imbalanced error rates between positive and negative classes. To solve this problem, we suppose that the expected risks of the positive-class and negative-class should be close. Accordingly, we constrain that the negative-class empirical risk estimator is lower bounded by the positive-class empirical risk, instead of 0; and also incorporate an explicit equality constraint between them. we suggest a risk estimator of PU learning that balances positive and negative classification error rates, named $\mathrm{D{\small C-PU} }$, and suggest an efficient training method for $\mathrm{D{\small C-PU} }$ based on the augmented Lagrange multiplier framework. We theoretically analyze the estimation error of $\mathrm{D{\small C-PU} }$ and empirically validate that $\mathrm{D{\small C-PU} }$ achieves higher accuracy and converges more stable than other risk estimators of PU learning. Additionally, $\mathrm{D{\small C-PU} }$ also performs competitive accuracy performance with practical PU learning methods.

Balancing Positive and Negative Classification Error Rates in Positive-Unlabeled Learning
Ximing Lil2,, Yuanchao Dai Biig Wang Changchun 1 Qu Renchu Guan 1.2
'College of Computer Science and Technology, Jilin University, China
Key Laboratory of Symbolic Computation and Knowledge Engineering, Jilin University, China RIKEN Center for Advanced Intelligence Project
Schooo of Computer Science and Technology, Soochow University, China Suzhou Key Lab of Multi-modal Data Fusion and Intelligent Healthcare, Suzhou City University, China fliximing86, yuanchaodai, Shangchun11993)@gmae
Abstract
Positive and Unlabeled (PU) learning is a special case of binary classification with weak supervision, where only positive labeled and unlabeled data are available. Previous studies suggest several specific risk estimators of PU learning such as non-negative PU (nnPU), which are unbiased and consistent with the expected risk of supervised binary classification. In nnPU, the negative-class empirical risk estimated by positive labeled and unlabeled data with a non-negativity constraint. However, its negative-class empirical risk estimator approaches 0, so the negative class is over-played, resulting imbalanced error rates between positive and negative classes. To solve this problem, suppose that the expected risks of the positive-class and negative-class should be close. Accordingly, we constrain that the negative-class empirical risk estimator is lower bounded by the positive-class empirical risk, instead 0; and also incorporate an explicit equality constraint between them. We suggest a risk estimator of PU learning that balances positive and negative classification error rates, named DC-PU, and suggest an efficient training method for DC-PU based on the augmented Lagrange multiplier framework. We theoretically analyze the estimation error of DC-PU and empirically validate that DC-PU achieves higher accuracy and converges more stable than other risk estimators of PU learning. Additionally, DC-PU also performs competitive accuracy performance with practical PU learning methods.
1 Introduction
Positive and Negative (PN) learning refers to conventional supervised binary classification trained with positive and negative labeled data I Positive and Unlabeled (PU) learning is specific case of
PN learning with weak supervision 2 3 w where, as its name suggests, trains a binary classifier with
only positive labeled and unlabeled data but negative labeled data unknown [I]. The paradigm arises various real-world scenarios, such as outlier detection and information retrieval 4 (5 During the past decade, there have been many practical PU learning methods mainly estimating laeuds-labels
for unlabeled data 66
7
8
9] and another research branch of PU lcarning is to formulate specific
risk estimators [10 12 33 while some of them can be unbiased and consistent with the expected
risk of PN learning.
To distinctly discuss the risk estimators, first give the problem assumption of PU learning, and in this work, we concentrate on the two-sample problem setting [14. To be specific, positive data are
Corresponding author
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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drawn separately from unlabeled data. The completely selected at random assumption is typically used, i.e. the positive labeled data are identically distributed as positive unlabeled data, so the positive labeled data are drawn from the positive- classs conditional distribution, and the unlabeled data are drawn from the whole population [1] 12]. Upon this setting, [[]] suggests an unbiased risk estimator of PU learning named uPU, which, specifically, contains positive-class empirical risk with positive labeled data and an unbiased negative-class empirical risk estimated by positive labeled and unlabeled data. However, the subsequent study indicates that uPU may be overfitting because its negative-class empirical risk estimator tends to be less than 0 during model training, so there is non-negativity constraint, upgrading uPU to nnPU [12]. In addition, some variants replace the non-negativity constraint with other tricks such as absolute- value correction [15]
In this paper, we review some intriguing properties of nnPU since it is an efficient benchmark risk estimator in PU learning. We raise a question about imbalance classification error rates in nnPU: the negative-class empirical risk estimator approaches O, whether the negative class is over-play per-played even with the non-negativity constraint, resulting in imbalanced treatment between positive and negative classes. To respond to this question, we evaluate nnPU by designing certain early experiments, and empirical observations suggest that the answer to the question is YES (see details in See 2.2).
To upgrade nnPU, we suggest a constraend risk estimator of PU learning named DC-PU, which applies two straightforward revisions. To maintain balanced eπror rates between the positive and negative classes, we suppose that the expected risks of the positive-class and negative-class should be close. Accordingly, we constrain that the negative-class empirical risk estimator is lower bounded by the positive-class empirical risk, instead of 0, i.e. the non-negativity constraint; and also incorporate an explicit equality constraint between them. We suggest an efficient training method for DC-PU based on the augmented Lagrange multiplier framework. We theoretically analyze the estimation error of DC-PU. We also empirically evaluate DC-PU on benchmark PU learning datasets. The results demonstrate that (1) compared with other risk estimators of PU learning, DC-PU achieves higher accuracy and converges more stably, and (2) compared with practical PU learning methods, DC-PU performs competitive accuracy performance.
We empirically evaluate the problem of balancing positive and negative classification error rates in the risk estimator nnPU.
To solve the problem, propose novel risk estimator of PU learning named DC-PU that balances classification error rates, and theoretically analyze it.
We conduct extensive experiments to indicate the effectiveness of DC-PU.
2
Preliminaries and Analysis
In this section, we briefly review the preliminaries of PU learning: and then empirically investigate the problem of balancing classification error rates in PU learning risk estimator.
2.1 Preliminaries of PU learning
Problem setup
Formally, let x E c Rdd and y E = +1] denote dddiiennsional input feature and binary label, respectively. PN learning is the standard supervised binary classification trained using positive labeled data D, and negative labeled data D,;
{ (xf, +1)
p(x|y +1) Pp
Dp
1=1
D2 (x77 11) x i.
p(xly= -
where np and n,, represent the numbers of the positive and negative labeled samples, respectively, and each sample is drawn independently and identically distributed from its corresponding class conditional distribution.
PU learning is special weakly-supervised binary classification trained using positive labeled data Dp and unlabeled data D,, In this work, we concentrate on the two-sample problem setting [14]. The completely selected at random assumption [16] is applied, D, and D,, are drawn independently

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FPR test
FNR Test
FPR main
FNR train
CIFAR-10-1
CIFAR-10-2
F-MNIST-1
F-MNIST-2
1.0
1.81
0.5
0.3
Rote
Ratee
Rolts
0.2
0.2
0.2
50
100 Epoch
150
200
100
150
200
25
50
25
30
75
100
E
Epoch
Epoch
Epech
Figure 1: False positive rate and false negative rate 4 benchmark PU learning datasets.
and identically distributed from the positive-class conditional distribution and the whole population:
D2 (x², +1) x
iid. p(xly = +1)] "
Du = (x* y,*)
x
i.d.d p(x) imm
where nu denotes the number of unlabeled samples.
Risk estimators The objective is induce classifier g : X - y that can predict the labels for any future samples. Let X' x y + R, be a loss function such as cross-entropy. In PN learning, its expected risk can be formulated follows:
R(g) - n R + (1- (1-))
where π represents the positive class prior ply = +1); R (g) == Ex-p(xl==+) [g(x)) +1)]
and R., (g) := Ex-p(x(===)) [[(g(x), -1)]. Given Dp U DD., the approximation of Eq (I. i.e. the empirical risk, is given below:
RpN(g) n
R (g) + (R,
(2)
where (g) 1
[<(g(x)), n. Eh [((g(()) +1) and ((()))) -1)]
In PU learning, the negative labeled data are unavailable, so its corresponding risk R (g) can be optimized by no means. Thanks to the completely selected at random assumption [66. we have p(x) Tp(x|y = +1) + x)p(x'y y 1)), leading to nR, (g) R, (g) RR (g). where R. (g) := E, Ex-p(xly=+1) [[(g(x). -1)] Accordingly, given D, U Da. the expected risk R(g) of
[ be approximated by unbiased empirical risk for PU learning (uPU) (11):
Rarv(g) n R (g) + Ri (g) -x RT (g)
where R, 1 E. [(() -1]] and (9) = ,, En [e(g(x²): 1)]
Unfortunately, uPU suffers from severe overfitting because the term R (g) - n R, , (g) often continues to decrease and goes negative during model training. To alleviate this problem. [12] proposes non-negativity constraint on this term, leading to a non-negative risk estimator for PU learning (nnPU) formulated follows:
RaMPU (g) = = ( max(0, it (g) - n R (g))
2.2 Empirical Observations of nnPU
We revisit nnPU that the term R. - (g) is used to approximate (1 (1-T1) R,, however, it often continues to rapidly decrease and be truncated by 0 with the non-negativity constraint [12]. Therefore, we raise the question of whether the negative class is over-played, potentially resulting in imbalance between generalization errors of the positive-class R and the megative-class R,, To answer this question, empirically measure R by False Negative Rate (FNR) and R, by False Positive Rate
3

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(FPR) given available labeled data (xx, 31)
R = FNR Di= 1(1 +1 ((x)) = -11
Σx=1 [1 = +1)
== [ -11 - = =11
R FPR
(5)
E Tly: = -1)
where I(()) denotes the indicator function.
We conducted early experiments on benchmark PU learning datasets (see details in Sec 5.1). Specifically, for each dataset, we apply nnPU to train a classifier 9 and report the scores of FNR and FPR measured on both the training and test sets. As depicted in Fig we can observe that the scores of FPR are consistently lower than those of FNR in all the settings, and the gaps between them are significantly large in some cases. For example, in terms of CIFAR-I0- and F-MNIST-2, the scores of FPR are about 0.3 0.4 and 0.2 0.3 lower than those of FNR on the test sets. These empirical results validate that the classifier trained by nnPU tends to predict more accurately for the negative-class than the positive-class, resulting in the imbalanced classification error rates problem. This phenomenon is consistent to the previous finding that the term R, (g) R. (g) of nnPU often approaches 0 12 so as to over-play the negative-class.
3 Balancing Classification Error Rates in PU Learning
In this section, propose PU risk estimator of PU learning named DC-PU, which balances positive and negative classification error rates.
3.1 DC-PU PU Risk Estimator
We empirically validate that nnPU may result in the imbalance problem between R and R because the empirical estimator of R is lower than that of Rt. To solve this problem, in DC-PU suppose that R and RS should be close, so have two equality constraints on their empirical estimators.
Specifically, we revisit the empirical estimators of R and R, in nnPU as follows:
R (g) RR (9))
(6) and have weak and strong equality constraints on them. First, the weak constraint is that (o)) 1-
= R RS
1- π
dynamically lower bounded by the fixed state of Rt. denoted by w, rather than the non-negatyvity constraint unPU. Accordingly, we can reformulate the empirical risk as follows:
PDC.Pulg) = x R)) + max fw, (0) -,) (o)}
where w is dynamically updated during classifier training. Second, the strong constraint is an explicit equality constraint. Upon these ideas, our DC-PU is finally formulated as follows:
s.t. (g) = Ro (g) π RR (g)
min n (g)(()) + max Ru,R,, Rú (g) - nパt (o)
(8)
Training DC-PU is intractable to optimize because it involves an explicit equality constraint. To this end, we suggest efficient training method with the augmented Lagrange multiplier framework. Following [17]. we can transform Eq.(8) into the following augmented Lagrange problem:
RS (g) R (g)
min x xRt (g) + max {w, Rr 9.8
(g) (0) n Tτī (0)
T
RT()))
(9)
where || denotes the l2 norm; θ is the Lagrange parameter; and r is the penalty parameter. Accordingly, we can directly apply any gradient-based method to optimize (q. with respect to {g. θ).

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Algorithm Training of DC-PU
Input: PU learning dataset D U Du; method parameters 8, T, 7, number of iterations T.
Output: A trained classifier g.
1: Initialize g with pre-trained backbone and θ randomly
2: for ........2 do
3: Draw a mini-batch D _ from Dr D
4: Compute R3 (g: (g: RV (g: D("), and Rp (g: )
5: Update w()) by using Eq. TO
6:
if Re (9; D 7 R2 (g; D( 2 w(()
7 then
7:
Compute the stochastic gradient y.
8: else
Compute the gradient vg V (g: - D(
(o:
9:
10:
end if
Update g with vgC and any adaptive learning rate method
12:
Update è by using Eq.
end for
To handle large-scale data, we employ the stochastic gradient decent method. At each iteration
we draw mini-batch D U d from D, D.,, and compute RS (g: D. Rr (g: D!!), and Re (9: )
. In terms of w, update it with exponential moving average trick:
(O) (t-1))) + B) - R) (g: D(,).
(10)
where is the moving parameter. In terms of g. we update with the stochastic gradient formed by DI UD! and solve the max operator by the rollback strategy suggested by [12]. In terms of θ, can be updated as follows:
(o: Ry (g: (()) RR (g: D(/")
@ (e +T +7
(11)
1-
For clarity, we summarize the full training process DC-PU in Algorithm 1.
3.2 Theoretical Analysis
In this section, we analyze the bias, consistency, and estimation error of DC-PU given in Eq. 8 proofs are Appendix AA.
(all
We first clarify some symbols for the following analyses. Let G be a hypothesis space that satisfies the boundedness property, characterized by the existence of constant C, > 0 ensuring sup |lgl| S C, and closure under negation. On this space, we define loss function é R x +1 + R+ that exhibits Le-Lipschitz continuity and is bounded by constant C₃ O, such that E(t, y) < C holds for all |t| < C. and E +1. To ensure the learning feasibility, we impose three additional conditions: a separability condition requiring the existence of a > such that infgeg R, (g) 2 a.
the upper bound condition requiring the existence of B > 0 such that 0 < supgeg R (g) < B, and complexity condition constraining the Rademacher complexity of G to satisfy (9) = 0(1/ and R., (9) = 0(1 /n.). With the DC-PU risk of Eq. we can partition all possible (Dp D.) into D + (D) p. D.)) R, (g) RR, (9) 2 w) and D (DDp. D.)) R. (g) R, (g) w where w is decided by (1 x R),
Lemma 3.1. RDC-ru (g) is positively biased from Rlgg) with a non-zero probability P(D (g)) over repeated sampling of (Dp, Dw), which can be bounded by
2(1 - (1-)))/--333/
(g) &
sexp exp
(12)
T² 22/np + 1/Tru
we can present the exponential decay of the bias and also the consistency for the DC-PU risk of Eq. with the following theorem.
Based Lemma 3.1
5

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Theorem 3.2. Denote the bound of ( in Eq. 12) by A n' max(1,, 1 T) and Xnp.m
2T/ /pp + +1/ // It holds that
0 < E D. [RDc-rr - R(())= S C'S
(13)
For any 8 > O, it has with probability at least 1 -
RiD.. (g) R(g)) Coxapma CC
(14)
and with probability at least 8
Pe R(g)] & CoXayia
(15)
where Cg = Ciyy Cct/nn22/6///
Theorem 3.2 indicates that for fixed g, RDE-Fu (g) R(g) with a convergence rate Ox(T mpp 1/ L/ynul ) which is optimal according the central limit theorem. It means that the proposed DC-PU risk in Eq. (8 is a biased yet optimal estimator to the PN risk.
Theorem 3.3. Let g" = arg ming is the minimizer of the true classification risk in Eq and @Dc-Fu =arg minggen RDc-ee denotes the minimizer of the risk form in Eq P(D (g)) in (1) by A, n' max (TT, π)) and Xnp. 2TT mh 1/ ynu- Then for any
Denote the bound of
5 > 0, it holds with probability at least 1 8:
R( - R(g") <16L,9 p. (9) ML, (9) (9)+ + 2x + CTA
(16)
where =²= Ci///in11/662
Theorem establishes the fundamental generalization bound that explains DC-PU's superior performance through four distinct components: two Rademacher complexity terms (9) and Hp (9), sampling fluctuation term Xng. and distribution discrepancy term A. Under standard assumptions, as Up, Thu + 00, two Rademacher complexity terms gradually decrease and converge to zero, and so does the sampling fluctuation term Xn.. The bias term A is independent R(g").
of sample size and can be treated Additionally, the convergence rates of the first and second terms are governed by the Rademacher complexities (G) and R.,,,, (9), corresponding rates of O(1/ /Ti)) and (11 re- spectively. The fluctuation term Xnp.... contributes additional rate of O(1/ + 1/v
constant. Consequently,
- oo, R(@DC-PU)
Consequently, the overall convergence rate is characterized by (max(1/VTp.. 1/ /m.) It indi- cates that DC-PU achieves minimax optimal convergence while maintaining balanced performance, contrasting with methods that optimize overall accuracy the expense of class balance.
Related Works
Recent years have witnessed significant advancements in PU PU learning, with research on risk estimators playing a pivotal role in its theoretical foundation. uPU [1I] presents solution to PU learning problems by constructing an unbiased risk estimator, which lays the theoretical foundation for future research. However, when models become complex, the unbiased risk estimator produces negative empirical risks, resulting overfitting problems. To address this, nnPU 12 reduces overfitting by adding non-negative constraints on negative risks. Following this, abs-PU [15] introduces simplified method using absolute value correction to handle non-negative constraints. Building on these works, Dist-PU [13 adopt novel perspective by aligning predicted and ground-truth label distributions to address negative prediction bias. A recent contribution, FOPU [88, extend the risk estimator framework by incorporating fairness constraints to address group fairness problems in text classification while maintaining model performance. Handling imperfect annotations has also been
studied in related weakly-supervise supervised settings [9 20 21 22
Beyond risk estimator-based approaches, researchers explore various strategies PU learning. Sample selection methods focus on identifying reliable negative instances from unlabeled data, exem- plified by the probabilistic estimation approach in PUbN
(3 and the dynamic weight adjustment
mechanism in Robust-PU (24). The recent VQ-Encoder [25 learns disentangled representations

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Table 1: Detailed characteristics of datasets.
Dataset
#Input
#Train
#Test
Positive Class
Backbone
F-MNIST-1
28 x 28
60,000
10,000
0.3
1,4,7
5-layer MLP
F-MNIST-2
28 x 28
60,000
10,000
0.7
0,2,3,5,6,8,9
5-layer MLP
CIFAR-10-1
3 x 32 32
50,000
10,000
0,4
0,1,8,9
12-layer CNN
CIFAR-10-2
x 32
50,000
10,000
0.6
2,3,4,5,6,7
12-layer CNN
Alzheimer
3 x 22/ x 224
5,121
1,279
0.499
0,1,3
ResNet-50
from representation learning perspective to achieve clustering separation unlabeled data. An- other category comprises generation-based methods, such as GenPU [26]. which utilizes generative adversarial networks to synthesize negative samples. Recent studies also integrate self-super 27 and predictive trend detection g frameworks enhance feature representations.
In contrast to these approaches, our method specifically addresses the problem of excessive negative class emphasis in PÚ learning, which can result in disparate treatment between positive and negative classes. Accordingly, we propose DC-PU, balanced and robust risk estimator.
5
Experiments
In this section, we compare the performance of DC-PU with existing risk estimators on commonly used benchmarks.
5.1 Settings
To comprehensively evaluate the proposed method, we conduct experiments on two widely-adopted benchmark datasets: Fashion-MNIST (F-MNIST) (8) and CIFAR-10[29 along with a real-world medical dataset Alzheimer. Based on the characteristics of different datasets, we design corresponding model architectures backbone networks following nnPU [[2]. Given that these datasets are originally designed for multi-class classification tasks, we transform them into binary classification problems by partitioning the class labels into positive and negative categories. Table T presents detailed statistics of each dataset.
Datasets
Baseline methods To evaluate the balance of classification error rates in our proposed risk estimator,
we select five classical risk estimator methods for comparison: uPU 111, nnPU [12, abs-PU [15). Dist-PU [3], and FOPU [8]. Given that our focus is on the assessment at the risk estimator level, we simplify the regularization terms in Dist-PU and FOPU to construct their baseline versions, Dist-PU*
and FOPU". The details of baselines presented in the Appendix B All experiments are conducted on a server equipped with two Nvidia RTX4090 GPUs.
Evaluation metrics Here, measure the classification performance by employing Micro-F1 and Macro-F1, where higher scores indicate better performance. Besides, we measure the balance performance by employing the GAP between FNR and FPR, formulated follows:
GAP [FNR FPR]
(17)
For the GAP metric, lower scores imply better performance.
5.2 Comparing with Existing Risk Estimators
From the perspective of enror rates, the GAP metrics delineated in Fig provide crucial insights into the performance of different PU risk estimators balancing classification error rates. The GAP between FPR and FNR demonstrates that DC-PU consistently maintains lower values across all four benchmark datasets (CIFAR-10-1, CIFAR-10-2. F-MNIST-I, and F-MNIST-2), indicating superior performance in balancing classification error rates. For example, in the F-MNIST-1 dataset, DC-PU (denoted by the yellow line) maintains consistently low GAP values throughout the training process, ultimately converging to approximately 0.05, whereas methods such as nnPU exhibit notably higher GAP values. This exceptional performance can be attributed to the explicit equality constraint
R. 19)-nñ (g) which actively promotes
in DC-PU's objective function (specifically, R (g)
7

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Table 2: Results of classification evaluation metrics (mean:istd) four benchmark PU datasets, where "max" represents the maximum value achieved during training and "last" indicates the average performance over the final 5 epochs. The highest scores are indicated in bold.
Micro- Fl
Macro-FI F1
Dataset
Method
max
last
max
last
uPU
86.61+0.00
61.19+0.000
85.91=0.005
40.9440.000
nnPU
88.72+0.00
86.29+0.001
88.06=0.001
85.68=0.002
abs-PU
88.80+0.00 2022
85.86+0.008
88.27=0.003
85.29=0.007
CIFAR-10-1
Dist-PU*
89.14+0.000
86,43+0.007
88.73=0.000
86.00=0.007
FOPU
88.49+0.00
82.53+0.00 002
87.88=0.00
$2.09=0.002
DC-PU
89.37=0.00
88.46+0.000
88.84=0.001
88. 88.11#0.006
uPU
87.53+0.00 007
41. 21.6220.001 0001
86.98=0.007
31.79=0.003
mPUU
88. 10=0.001
85.26+0.01
87,42=0.001
84.51+0.011
abs-PU
88.59+0.000
85.29+0.000 0000
87.97=0.001
84.76+0.004
CIFAR-10-2
Dist-PU"
88.23++00001
85.1550.002
87.58=0.001
84.56+0.0002
FOPU"
87.83+0.00
82.53+0.000 0005
87. 87.41=0.000
82.09+0.005
DC-PU
88.60=0.00
87.03+0.003
88.02#0.00
86.53+0.004
uPU
90.90#0.002
74. 14.1.400000
88.81=0.000
54.85=0.012
mPU
90.92+0.00 000
88 18.0620.001
89.17=0.000
85.7940.002
abs-Pu PU
90.8740.000
87 07.8340.00 0000
88. 88.74=0.00
85.7440.001
F-MNIST-I
Dust-PU"
91.25+#0.00
87 $7.98+0.00 0.005
89.27=0.00
85.68+0.004
FOPU"
90.91
88.50#0.00 .044
89.03=0.001
85.82+0.007
DC-PU
91.48+0.001
89.9720.001
89.8620.001
88.3240.00
uPU
85.5740.011
49. 49.49+0.084
83.09=0.009
48. 48.98+0.092
mPPU
88.5340.00
83.79+0.00
85.78+0.002
81. 11.1090.009
abs-PU
88.71+0.00
83.50#0.001
86.1220.002
81.46+0.003
F-MNIST-2
Dist-PU*
88.40+0.01
84.65+0.00 0007
86.41=0.015
81.68+0.011
FOPU 88. 1430.003
84.41
85.83+0.006
81.86+0.011
DC-PU
88.91=0.003
86,44+0.007
86.72=0.006
83.43+0.002
Table 3: Results of Micro-F1, Macro-F1 and GAP std) Alzheimer, where "min" represents the minimum value achieved during training. The highest scores are indicated in bold.
Micro-Fl
Macro-F1
GAP
Method
max
last
max
last
min
last
uPU
71.02+0.004
63.73+0.096
70.93=0.005
61.94+0.263
0.59=0.000
17.55#1.344
unPU
70.2340.002
67.8740.055
68. 68.98+0.002
66.6140.203
0.4620.000
16.4910.224
abs-PU
71.0940.007
67.9240.000
70.79=0.001
67.00+0.010
0.4420.003
13.4420.699
Dist-PU"
71.25:0.002
68.5620.046
71.14=0.001
66.94+0. 116
0.37=0.001
13.65+0.388
FOPU"
71.56:0.002
64.14=0.053
71.27=0.002
60.96 172
0,26=0.000
12.57+0. 706
DC-PU
71.48+0.002
70.55+0.002
71.03=0.001
69.99:0.009
0.09=0.000
10.79:0.294
balanced prediction between positive and negative classes. This observation is further substantiated by the results presented in Table where DC-PU achieves significantly lower GAP values (0.009 at "min" and 10.79 at "last") compared to nnPU (0.46 at "min" and 16.49 at "last") on the Alzheimer dataset.
From the perspective of the convergence, Fig and Fig present the Micro-F1 and Macro-FI performance trends during the training process across four benchmark datasets, respectively. DC-PU exhibits outstanding convergence stability and final performance, demonstrated by its consistently higher Micor-Fl and Macro-F1 values compared to other PU risk estimator. This pattern is particularly evident in the CIFAR-101 and F-MNIST-1 datasets, where DC-PU maintains stable high F1 level. contrast, uPU shows significant stability issues, characterized by sharp Fl scores decline following initial convergenc. This instability can be attributed to the unconstrained nature of uPU's basic risk formulation, which fails to maintain consistent performance throughout the training process. The improved stability observed in newer methods such as nnPU and abs-PU can be attributed to their

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dPU
RAPU
4bs-PU
DocPU
POPU F-MNIST-1
DCAU
CIFAR-10-1
CIFAR-10-2
0.8
0.6
0.6
0.3
GAP
0.6
3
0.4
0.2
0.1
0.0
0.8
0.0
0.0
400
20
60
1.30
200
Epoch
Epoch
Epoch
Epoch
Figure 2: The curve of GAP during the training process on 4 benchmark PU learning datasets.
DCAUU
CIFAR-I0-1
CIFAR-10-2
F-MNIST-1
E-MNIST-2
0.97
1.00
0.96
0.91
8.91
0.96-
0.91-
Micro-Fi
Micro-F11
0.69
Micro-Fi
0.92
Micro-F11
8.86
0.79
0.58-
B.81
0.73
0.73
0.54
0.76-
1P00
0.67
40
50
120
160
209
80
120
160
200
0.80
20
60
108
0:71
40
60
50
100
Epoch
Epoch
Epoch
Epnch
Figure 3: The curve of Micro-Fl during the training process 4 benchmark PU learning datasets.
naPU
Abs-PU
Dic-PU
POPU F-MNIST-I
CIFAR-10-1
CIFAR-10-2
F-MNIST-2
0.95
0.95
1.00
0.92
0.39
0.96-
0.86
Macro- F1
0.83
Macro-F11
Macrn-FF
0.92
Macro-F1
0.60
0.77
8,77
0.58
0,74
0.71
8.71-
0.84
0.68-
MP
50
120
160
200
80
120
160
200
20
64
100
40
60
50
100
Epoch
Epoch
Epoch
Epoch
Figure 4: The curve of Macro-Fl during the training process on 4 benchmark PU learning datasets.
enhanced risk estimators, which incorporate non-negative constraints or absolute value corrections, effectively preventing model overfitting. Moreover, the results presented in Table further substantiate the convergence advantages, with DC-PU consistently achieving superior Micro-F1 and Macro-F1 scores across all four datasets, both in terms of "last" and "max" performance metrics. Specifically, on the CIFAR-10- 10-1 dataset, DC-PU attains a maximum Micro-F1 of 0.892 and final Micro-F1 of 0.875, surpassing the next best performer (nnPU) by margins of 0.043 and 0.038 respectively. Similarly, for Macro-F1 scores, DC-PU demonstrates exceptional performance on F-MNIST-2, achieving 0.901 at "max" and 0.889 at "last", while competing methods such as uPU and nnPU only reach maximum values of 0.847 and 0.862 respectively. This performance differential is particularly pronounced in the CIFAR- 10-2 dataset, where DC-PU maintains consistently superior metrics with maximum Micro-F1 of 0.903 and Macro-FI of 0.897, representing improvements of approximately 5.2% and 4.8% over the baseline methods.
To further validate the effectiveness of our proposed method, we incorporated the additional regular- ization term inspired by Dist-PU into the aforementioned five risk estimators for comparison well as our risk estimator. Due to the space limitation, the details are presented in the Appendix C
5.3 Parameter Sensitivity
We conduct a comprehensive parameter sensitivity analysis with respect to the parameters 7 and B and the results are presented in Fig 5 For the parameter 7, our experiments show optimal performance x 10 across most datasets, which aligns with our theoretical analysis that moderate penalty parameters achieve balance between constraint strength and optimization stability. For the parameter 3, we find that the range [0.4, 0.5) performs optimally across different datasets, which is significant because 3 controls the update speed of dynamic lower bound. Too small value of 3 leads to overly loose constraints losing the balancing effect, while too large value of causes optimization instability. Our experiments demonstrate that DC-PU exhibits relative robustness to these hyperparameters,

---

Table 4: Results of ablative study (meantstd). The highest scores are indicated in bold.
Variant
(a)
(b)
CIFAR-10-1
CIFAR-10-2
F-MNIST-1
F-MNIST-2
Alzheimer
88.72+0.001
88.10=0.001
90.92=0.000
88 88.53+0.002
70.23+0.002
II
89.2540.001
88.4220.0000
91.43=0.000
88.5620.001
70.9420.001
III
88. 88.90+000001
88.60=0.001
91.21=0.000
88 08.8810.001
71.33+0.006
IV
V
89.37=0.001
88.6020.001
91.48=0.001
88.91=0.003
71.48=0.002
CIFAR-18-1
CIPAR-042
F-MNIST-1
92
92
91
91
90
Micro F1
90
Miero
89
87
87
86
86
10a
10a
10a
10'
10
10
10
0.0
0.1
0.2
0.8 0.9
Figure 5: The parameter sensitivity analysis with respect to the parameters and B on benchmark PU learning datasets.
with parameter variations within reasonable ranges not significantly affecting performance, further validating the practicality of our method.
5.4 Ablation Study
We conduct ablation studies on five datasets to evaluate the impact of two key constraints: the weak constraint (a) and the strong constraint (b). Variants I-IV are designed to investigate the effects of these two constraints. The experimental results demonstrate that employing either the weak or strong constraint independently enhances model performance: Variant with the weak constraint shows consistent improvements across all datasets, while Variant III with the strong constraint exhibits particularly notable performance on CIFAR-10-2 and F-MNIST-2. When both constraints are combined (Variant IV), the model achieves optimal performance. reaching 89.37% 88.6% on CIFAR-10- 10-2, 91.48% on F-MNIST-1, 88.91 8..91% on F-MNIST-2, and 71.48% on Alzheimer, which not only validates the complementary nature of these constraints but also substantiates the effectiveness of our proposed DC-PU.
CIFAR-I0-1,
6 Conclusion
In this paper, we observe issue of balancing classification rates in PU learning where the non-negativity constraint in nnPU leads to over-emphasis of the negative class. To address this, we propose DC-PU, novel risk estimator that balances the error rates between positive and negative classes through two key constraints: a dynamic lower bound constraint and an explicit equality constraint. Through theoretical analysis and extensive experiments benchmark datasets, DC-PU demonstrates improved evaluation metrics and stability across all datasets while balancing error rates between positive and negative classes compared existing methods, establishing itself as an effective approach for achieving both high evaluation metrics and balanced classification error rates in PU learning.
Acknowledgements
We would like to acknowledge support for this project from the National Science and Technology Ma- jor Project (No.2021ZD01 12500), National Natural Science Foundation of China NNo.622761133 and the open research fund of Suzhou Key Lab of Multi-modal Data Fusion and Intelligent Healthcare.
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A Proof of Theorems
A.1 Proof of Lemma 3.1
Proof. Let F(D)) be the cumulative distribution function of Dp and D.., respectively, and F(Dp, D.) F(D,) F(D.) ((D)) be the joint cumulative distribution function of (Dp, D.). Based on the unbiased fact of RuPo (g) and (D)- (g) = 0 on 2 (g), it holds
B[RDe-r( R(gg) BiRDe-rr (g)
1 P.)eDtt) RDC-ro (g) - ( D.)
Ro..l (u D.)
(PP..)ED5()))
(D RuPU (g) D.).
(D).
Due to the fact RDc-ro (g) R. (g) > 0 on (g). B(RDc-ro - RR(g) > O with the probability POP (g)) ((p,...)ED:()) dF (D,, D.). In other words, RDC (9) is positively biased with the
probability (g)) Given the assumptions R, (g) 2 a > and S R (g) s 3, we have
E (g) - n RR (g) = R (g) R
(g) = (1 n) Rn (g) N (1 - -Ta
and
(9) P
R (9) x27 < < (1- x) (8)
P
( - त,, (0) < x) R, (g) (1 - 2) (1- R) (g) [I- (g) (R, (g) π,, (g)) 2 (1 T)a (1 π P2 (g)
p
SP (11 (1-1)) (g) (R (g) RR, (g) 2 (1 T) - -3)
According to the assumption < Ce/np and the change of R,, (g) more than Ce/n,, if replace some (x",, +1) E Dp some
S Ce, the change of R. (g) will be no more than
(xf", yir) E Du. Subsequently, based on McDiarmid's 's inequality, it holds
2)1-7) T) - -3 ? np(CeTpp/1n) n, (Ci/(C//F (Cs/nu)²
P (1 (I- (R, (g) RR (g)) 2 (1 -a B) exp
2(1 TT² /3///2/
exp
n² /np + 1 n
A.2 Proof of Theorem 3.2
Proof. Based on Lemma 3.1 and its proof, we can obtain the exponential decay of the bias in Eq.
13
by
(RD (g) - Rn)r())) /
Ep,.D. D. (DD -Rgg) < sup (.,.D.))2
dF D.)
(D).D))DD
((1 - π) R) R. (g) + n R (g) P((
sup
(DD.
((1 - 7) R (g) + ग (Cr R (g)) R. (g)) ((P_)))
sup
(.).
(aCa + (1 R (g) R, (o) P(DT
sup
< S
20

---

where x' = max(T, - x).
The deviation bound in Eq. 14 is obtained by:
RDD-ru(g) R(g)
- RDC-FU (g) El E[R [g) + [( R(g)
- Pr)) E| nu [g) + T'A.A.
According to the assumption S l(t, +1) S Ct, the change of RDE ru rg(g) will be no more than 2Cr/n, /n or Ce/n,, if we replace some (x. +1) E Dp or some (x",, ut¹ E D.. Subsequently, based on McDiarmid's inequality, holds
PO (g) - E (o)) 5
eexp
m2(20 /T/ /eaTT//))) /n)) + nu (C(nn/) (-m,(2CAm)
or equivalently, with probability at least 1 8
RD (g) ([)D In ln(2/5)C3
(m²
2
1
+
S C6
+
CsXnp,
np
nu
Tip
Vnuu
On the other hand, the deviation bound 15 is given due to
PD R(g) - (g) (ar Rure(g)) R(OL
where Re (g) RuPU (g) 2 O with the probability at most A, and (g) - R(g) shares
the same concentration inequality with (Dc-rc (g) [[(DDc-o()))]
A.3 Proof of Theorem
Proof. We follow the spirit of proving Lemma 5 and Theorem 4 in [12 to give the proof of Theorem 3.3 Specifically, with the assumption inf R, (g) 2 a > 0, Theorem 3.2 and McDiarmid's inequality, have
sup
(g) R(g) gEg
s sup RDD-rL()) E [Roc.rloo] + gEg
(18)
and with probability at least 1 -
sup gEg
(u - -E (DD. [g)
E
sup LgEg
RDD ro(g) -E E RDe rc() S C.Xap.ma (19)
Given a ghost sample (D,, D.), holds
E sup 9gE
(g) E [
||
S (.DDD.DDMDDD.... L
sup
RDC D,)
Ro D', D.)
(20)
The main difference is to decompose given in Eq. we have
Rioc-ull; Dp, Da) - Roc (g: Dr, Day
For the DC-PU risk
Pu Dps Da) - RoC Dy Dp: D'y]
(R (g: Dp) - n RRt (gi D2)) + max w (g: R (g: - max {w', R Rg: D'.) - T RRR (g: D3)
| (g: Dp) - n ~ (g: D2)
+ max f0, Ri (g: D.)) - - nパ. (g: Dp) - ww -max f0, Ri (g: D',)) - n - (g: (D: - w/ + + w'
क R Dp) - R (g: D2) + Ι RT (g: Dp) - Rr (g: D') + RR, (g: Dw) - R D'))
+ 2 - x) 7 (g: R (0: Dy))
(2 x) i7 (g; D,) - R (o: D/) + x
Rí (g: D,) - R7 (g: Dz>)
RR, (g: - Rr (g: D'>|
21

---

where we employed maxx(0, 2} max(0, - 22 2/| R (o: Dp). This decomposition results in
and the fact that w is decided by (1
E sup (D - [ LgEg
<(2 DD. D.D. ||
R (gi D2) - Rs (g: D2)
sup
LoEG
Ry (g: Dp) Rí (o: Dz)1
p. 11 D D/
sup
gEg
R, (g: D.) R. (g: D(o|
+ D..,%
sup LgEg
(21)
Combining Eqs.(T8.
19
21 and the proof of Lemma 5 in 12 for any g > 0. it holds with
probability least
(D)
(g) - R(o) s ap. + 4L, + Ca + C.C. (22)
sup gEg
Then we obtain the estimation bound through
R(īDc-Fo) R(g"))
Pe
(((ōnc.nu) BDc..rl@nn...
(Roc.ull"") R(g"))
< 0 + suu RD (g) R(g)
geg
< 16L (9) + 8L + 20 X0 + C
where PDDere(DDeeeu & RD by the definition @DC-FU-
B Baseline Methods
To evaluate the balance between positive and negative classification error rates of our proposed risk estimator, select five classical risk estimator methods for comparison: uPU [ nnPU [12).
abs-PU [15), Dist-PU
[3], and FOPU [8].
Unbiased
Learning (uPU) [T]
:
An unbiased and convex optimization
framework for PU learning, climinating bias through the strategic application of distinct loss functions to positive and unlabeled samples.
(aru()) (0) (0) R. -R, (g)
R
-uttbvee Learning with Non-Negative Risk Estimator (nnPU) [1] non-negative risk estimator that addresses the overfitting problem PU learning by explicitly constraining empirical risks to be non-negative.
: it introduces
R (g) x > max (0, R, (g) x Rg)
Absolute Positive-Unlabeles Learning (abs-PU) 15 it simplifies the non-negative risk constraint nnPU through absolute-value correction.
Rabs- (g = R (g) + (g) x
Positive-Unlabeled Learning from Label Distribution Perspective (Dist-PU") learns classifier by aligning the predicted and ground-truth -tuth label distributions, mitigating the regative-prediction bias prevalent in conventional PU learning methods.
Rosst-Pug) = 2n Rr (() (g) + R (g)
22

---

Pairmess-Award Online Certive-Unllaeled Learning (FOPU*) [8]
it addresses faimesss
issues in text classification by introducing convex fairness constraint, improving predic- tion fairness across different demographic groups while maintaining model performance. Additionally, we adapt the fairness constraint mechanism from FOPU, originally designed for sensitive attributes, to the framework of class fairness.
RFOPU()) ROMPU + Rrair
Furthermore, to verify the effectiveness of DC-PU, we select three PU additional learning baselines, including MIXPUL (30, PULNS 77. and Mix [6].
Consistency-based Augmentation for Positive and Unlabeled Learning (MIXPUL) 30
it achieves data augmentation in positive-unlabeler learning by combining consistency regularization and reliable negative mining without requiring class prior probabilities. The code provided by its authors is utilized with default parameters, i.e. a = 1, 3 and 1) -==1. 1.
DositieeUUnlabele Learning with Effective Negative Sample Selector (PULNS) 77:
optimizes the negative example selector through reinforcement learning select negative examples from unlabeled data and alternates training with the classifier to effectively handle label noise issues. We take a and {0.4, 0.6, 0.8, 1.0) as suggested in its paper.
Positive and unlabeled learning with Partially Positive Mixup (P*Mix) [6]: it proposes a heuristic mixup technique that selects appropriate positive samples to mix with pseudo- negative samples near the classification boundary, achieving both data augmentation and supervision correction. This approach helps align the learning boundary closer to the supervised boundary. Additionally, two variants are introduced to enhance model robustness: P*Mix-E, which employs a mean teacher to generate auxiliary target vectors for early- learning regularization, and Mix-x-C which directly corrects the labels of high-confidence pseudo-negative samples.
Latent Group- ware Meta Disambiguation (LaGAM) 31 it addresses PU learning by fo- cusing on representation quality. It uses hierarchical contrastive learning module to extract underlying group semanties from PU data, producing more discriminative representations. LaGAM then employs meta-learning to iteratively refine pseudo-labels of unlabeled data.
C Comparing with Practical PU learning Methods
To further validate the effectiveness of our proposed method, we incorporated the additional regular- ization terms inspired by Dist-PU [3 into the aforementioned five risk estimators for comparison as well as our risk estimator. First, to prevent trivial solutions where predictions cluster around ambiguous values, an entropy minimization term Lent is incorporated that encourages more confident predictions on unlabeled data:
Lent
Σ (11- g(x:)) log(1 g(x,)) + g(x;) log (xx:]
(23)
Th
Second, to address the critical issue of confirmation bias, where incorrect early predictions get reinforced during training, the mixup regularization term Lmize is employed:
1 Σ [Nlo (o(x)), g(xi)) + (1 - Xoc (g(x)) g(x)))
Lmiz
(24)
where mixed samples x = + x)x are created using mixing coefficients sampled from a Beta(a, a) distribution, with X' = max(A, A)) A). And (t, () = - -11-6 - log(1 (') log(!").
Additionally, an entropy minimization term L'ni is introduced to the mixed samples themselves:
L'ent =
Ince (g(x))
(25)
Then, the additional regularization function is combined through hyperparameters μl, #2 and μl as:
Lyeg = f1 Lent + u2 Lmmi + H3I
(26)
23

---

Table 5: Results of Miero-F1 (mean:tstd) four benchmark PU datasets after adding the regulariza- tion term. The highest scores among PU learning methods are indicated in bold.
Method
CIFAR-10-1
CIFAR-10-2
F-MNIST-1
F-MNIST-2
uPU+
89.38=0.001
32.7340.106
91.55+00.011
86.33=0.004
nnPU+
89,46=0.001
87.66+0.001
91.66+0.001
88.41#0.001
abs-PU*
89,45=0.001
88. 58.09+0.000
91.68+0.001
88.57+0.009
Dist-PU
89.16=0.002
88.51
91.70+0.000
88.68+0.006
FOPU*
89.51=0.002
87.30+0.000
91.65=0.001
88.39#0.000
MIXPUL
87.00=1 19000
87.00=1.100
87.50#1.500
89.00+0.500
PULNS
87.20=0.600
83.70+2.900
90,70+0.500
87.90=0.500
P*Mix-E
88.20=0.400
84.70+0.500
91.90+0.300
89.50+0.500
M'Mix-C
88.70=0.400
87.00+0.500
92.00=0.400
89.40=0.300
LaGAM
89.91=0.300
87.98
90.15+0.013
80.88+0.084
DC-PU
89.5420.002
88.9940.001
92.73=:0.011
89.41+0.005
The experimental results are presented in Table The methods marked with "+" indicate the incorporation of additional regularization terms. As shown in Table DC-PU + achieves highly competitive performance across all four benchmark datasets. Notably, on the F-MNIST-1 dataset, DC-PU + reaches the highest Micro-F1 of 92.73%, significantly outperforming other methods. The only exception occurs on F-MNIST-2, where DC-PU achieves high Micro-F1 of 89.41% but slightly trails behind 89.50% of Pd M, with a marginal difference of only 0.09 percentage points. Moreover, all methods marked with "+" demonstrate more stable performance compared to their base versions. These results validate that incorporating the regularization term from Dist-PU can indeed enhance model performance, not only improving Micro-F1 but also strengthening model stability.
D Limitations and Broader Impacts
D.1 Limitations
While DC-PU is evaluated on both synthetic and real-world datasets (e.g., Alzheimer), most benchmark datasets used in study (such CIFAR-I0 10 and F-MNIST) originally designed for multi-class classification and are converted into binary classification tasks by partitioning class labels. This transformation simplifies the evaluation setting and may not fully reflect the complexity of real-world scenarios involving inherently multi-label or structured data. In future work, plan to explore the application of DC-PU in more complex PU learning settings beyond binary classification.
Furthermore, our definition of fairness focuses solely equalizing risks between positive and negative classes. This form of fairness does not incorporate demographic or group-level considerations, which essential in socially sensitive applications. Extending DC-PU to address group faireess fairness with respect to sensitive attributes remains an important future direction.
D.2 Broader Impacts
The DC-PU method proposed in this paper addresses fairness issues in positive-unlabeled learning by mitigating the bias of traditional risk estimators toward negative classes. This method enhances the faimess and reliability of PU risk estimators. The method can be widely applied in practical applications (such as medical screening, recommendation systems, fraud detection, etc.), improv- ing model stability and reliability in real-world scenarios where negative samples difficult to obtain. However, it's important to acknowledge that the DC-PU method relies on specific sampling assumptions (i.e., SCAR assumption and one-sample assumption), which may not fully hold in some real-world applications.
24


Reviews:
[{"review_id": "NbLsSLqA23", "reviewer": "Reviewer_kGXk", "summary": "This paper addresses an important and specific problem in Positive-Unlabeled (PU) learning: the imbalanced classification error rates produced by existing risk estimators, particularly the widely-used nnPU framework. The authors empirically demonstrate that classifiers trained with nnPU tend to have a False Positive Rate (FPR) that is significantly lower than the False Negative Rate (FNR), suggesting an over-emphasis on the negative class.\nTo tackle this issue, the paper proposes a novel risk estimator named DC-PU (Dual-Constrained PU). Its core idea is to enforce balance via a dual-constraint mechanism: 1) a \"weak constraint\" that sets a dynamic lower bound for the negative-class empirical risk, tying it to the positive-class empirical risk rather than a fixed value of 0; and 2) a \"strong constraint\" that introduces an explicit equality between the estimated positive- and negative-class risks. To optimize this constrained objective, the authors employ an augmented Lagrange multiplier (ALM) framework.", "questions": "Please correct the legends in Figures 2, 3, and 4, changing the incorrect \"Fair-PU\" label to \"DC-PU.\"", "limitations": "1. Like much of the work in this area, the proposed method relies on the 'selected completely at random' (SCAR) assumption. It would be beneficial to briefly acknowledge in the main body of the paper that this assumption may not hold in all real-world scenarios, thus defining a boundary condition for the method's applicability.\n2. The experiments are conducted on multi-class datasets that have been converted into binary PU tasks. While this is standard practice, these proxy datasets may not fully capture the complexities of native PU problems found in applications like information retrieval or medical diagnosis.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n1. The research is well-targeted, addressing a specific and empirically demonstrated flaw in a strong baseline (nnPU). By first proving the existence of the error rate imbalance, the authors provide a compelling motivation for their work.\n2. The core contribution, the dual-constraint mechanism in DC-PU, is an innovative and direct approach to solving the identified problem. Combining a dynamic lower bound with an explicit equality constraint is an elegant way to enforce balance at the objective function level.\n3.  The experiments show that DC-PU not only excels at its primary goal of balancing error rates (GAP) but also achieves state-of-the-art or competitive performance in standard metrics like F1-scores and stability. This demonstrates a clear and multi-faceted improvement over existing methods.\n\nWeaknesses:\n1. The theoretical exposition is insufficient. The paper is formula-heavy, but the lack of detailed descriptions for these equations makes the content difficult to follow.\n2. The legends in the main experimental figures (Figs 2, 3, 4) incorrectly label the proposed method as \"Fair-PU\" instead of \"DC-PU.\" This is a critical error that is highly confusing for the reader.", "was_revised": true, "final_justification": "The authors have successfully adressed the majority of my concerns as mentioned in the rebuttal.", "rebuttal": "**Q1. The theoretical exposition is insufficient.**\n\nThank you for your valuable suggestion. Our theoretical framework addresses a fundamental challenge in PU learning where existing methods suffer from the imbalanced error rates between positive and negative classes. The risk decomposition in Eq.3 reveals the core mathematical insight: under the SCAR assumption, we can express the negative class risk as $R_n^-(g) = \\frac{R_u^-(g) - \\pi R_p^-(g)}{1-\\pi}$, which allows us to estimate negative class performance without labeled negative samples. However, this unbiased estimator $\\widehat{R}_u^-(g) - \\pi \\widehat{R}_p^-(g) $ frequently becomes negative during training, leading existing methods like nnPU to truncate it at zero, resulting in the negative class being over-played.\n\nOur dual-constraint mechanism in Eq.8 fundamentally transforms this problem through two innovations. The dynamic lower bound $\\omega = (1-\\pi)\\widehat{R}_p^+(g) $ automatically adapts to the current positive class performance, creating a balanced constraint that evolves with the model's learning progress. The explicit equality constraint $\\widehat{R}_p^+(g) = \\frac{\\widehat{R}_u^-(g) - \\pi \\widehat{R}_p^-(g)}{1-\\pi} $ mathematically enforces that the empirical positive and negative class risks remain equivalent throughout training, directly addressing the class imbalance problem at its theoretical root. \nThe mathematical elegance lies in how these constraints work together: while the dynamic bound prevents degenerate solutions, the equality constraint ensures that neither class dominates the optimization landscape. This theoretical foundation explains why DC-PU achieves consistently low GAP values across most experimental settings, as the constraints translate directly into balanced error rates between positive and negative classes.\n\n\n&nbsp;\n\n\n**Q2. The legends are incorrectly labeled.**\n\nWe sincerely apologize for this critical labeling error in figures. We have corrected all figure legends to properly reflect \"DC-PU\" as our method name and will ensure this correction is implemented in the next version.\n\n\n&nbsp;\n\n\n**Q3. The proposed method relies on the SCAR assumption, defining a boundary condition for the method's applicability.**\n\nThank you for your suggestion. This is indeed a fundamental limitation that we should acknowledge more prominently in our main text. As the SCAR assumption can indeed be violated in real-world scenarios, we still believe our method provides robustness beyond strict SCAR assumption. \n\nFirstly, the dual-constraint mechanism inherently provides stability against moderate SCAR violations by preventing extreme class imbalance that could arise from distribution shift, while our dynamic lower bound adapts to the actual observed data distribution rather than relying solely on theoretical distributional assumptions. \n\nFurthermore, our evaluation on the real-world Alzheimer dataset represents a realistic medical diagnosis scenario where SCAR assumptions are likely violated to some degree, yet DC-PU still demonstrates clear improvements in balanced error rates and overall performance. We will expand our limitations discussion in the next version to explicitly address scenarios where SCAR may not hold.\n\n\n&nbsp;\n\n\n**Q4. The experiments are conducted on multi-class datasets that have been converted into binary PU tasks.**\n\nThank you for your comment. In fact, in addition to the artificially converted datasets, our evaluation also includes the real-world Alzheimer dataset, which represents a genuine medical diagnosis scenario with inherent positive-unlabeled structure rather than artificial conversion. The experimental results are presented in Tables 3 and 4 of our paper. Furthermore, we are committed to evaluating on additional real-world PU datasets and will include evaluation on the DIABETES dataset from TableShift as suggested by Reviewer k4vm."}, {"review_id": "rFgF9SCr5i", "reviewer": "Reviewer_YFBq", "summary": "This paper focuses on positive-unlabeled (PU) learning, an important problem in weakly supervised learning. It observes that the loss term with respect to the negative class is overly minimized, causing imbalanced learning of the positive and negative classes. To address this issue, the loss term is modified by raising the minimum value for positive data loss. Extensive theoretical analysis and experimental results validate the effectiveness of the proposed method.", "questions": "Please see \"Weaknesses\". My major question is regarding the main theorem.", "limitations": "The limitations of the paper should be further discussed.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "## Strengths\n- The studied problem is important to the literature.\n\n- The proposed technique is simple yet effective. \n\n- The theoretical analysis is comprehensive and novel, such as the consistency analysis.\n\n- Extensive experiments validate the effectiveness of the proposed method.\n\n## Weaknesses\n- I had a question regarding the proof of Lemma 3.1, which shows that the bias will converge to zero as $n$ goes to infinity. In the last inequality in line 339, it seems that\n\\begin{equation}\n(1-\\pi)\\alpha-(1-\\pi)\\hat{R}_p^+(g)\\leq (1-\\pi)(\\alpha-\\beta).\n\\end{equation}\nTherefore, it seems that the $\\leq$ should be $\\geq$ and the inequality does not hold. It is a very important part since the consistency may not hold.\n\n\n- The proposed technique is related to flooding, but the paper lacks discussion of this relationship. In flooding, a constant value is considered the minimum value of the risk-correction function. This paper proposes using a term proportional to the loss term of the positive class as the minimum value of the risk correction function, which could solve the hyperparameter selection problem. Further discussion would benefit the paper.\n\n- The writing of the paper can be improved; for example, there are several typos. Additionally, the references can be revised for concision. Some minor points: 1) In line 13, it should read \"we.\" 2) The notation \"~\" in line 110 is inaccurate. (3) In line 146, the infinity notation needs to be revised.\n\n- Theorem 3.3 lacks discussion.\n\nReference:\n\n[1] Do We Need Zero Training Loss After Achieving Zero Training Error?, ICML 2020.", "was_revised": true, "final_justification": "Thanks for the rebuttal. Since the error was fixed during the rebuttal, the theoretical analysis looks good to me now. Therefore, I have increase my score. Please include the replies in the rebuttal to the final version of the paper.", "rebuttal": "**Q1. Correction of the proof of Lemma 3.1.**\n\nThank you for pointing out this error. It begins with a mistake in which we wrote the upper bound condition of $\\widehat{R}\\_p^+(g)$, i.e., $0 \\le \\sup_{g \\in \\mathcal{G}} \\widehat{R}\\_p^+(g) \\le \\beta $, as a lower bound condition in the current manuscript. When applying the correct condition $0 \\le \\sup_{g \\in \\mathcal{G}} \\widehat{R}\\_p^+(g) \\le \\beta$, the inequality $(1-\\pi)R\\_n^-(g)-\\bigl(\\widehat{R}\\_u^-(g) - \\pi \\widehat{R}\\_p^-(g)\\bigr)\\ge (1-\\pi)(\\alpha-\\beta)$ is valid.\n\nThis correction actually strengthens our theoretical analysis by demonstrating that the separation between the true negative class risk and our estimator depends on the gap $\\alpha-\\beta$ between the lower bounds of negative and positive class risks. The lemma shows that our bias probability bound $P(D_\\omega^-(g)) \\leq \\exp(-2(1-\\pi)^2(\\alpha-\\beta)^2/C_\\ell^2 \\cdot 1/(\\pi^2/n_p + 1/n_u))$ has an exponential decay rate that explicitly depends on this separation. When $\\alpha > \\beta$, the exponential decay becomes stronger, providing better theoretical guarantees. It explains why DC-PU performs particularly well on datasets with natural class difficulty differences, as our constraint mechanism automatically adapts to these inherent class characteristics while maintaining theoretical consistency.\n\n\n&nbsp;\n\n\n**Q2. The paper lacks discussion of the relationship between flooding.**\n\nThank you for your suggestion. Flooding uses a fixed constant flood level $\\lambda$ where the loss becomes $\\max\\\\{\\ell(f(x), y)-\\lambda,0\\\\}$, creating a uniform lower bound across all samples and classes. It addresses general overfitting by preventing zero loss but lacks class-specific considerations essential for PU learning scenarios. In contrast, our DC-PU employs a dynamic lower bound, creating a class-aware constraint that automatically adapts to the actual positive class risk. The exponential moving average ensures stability while allowing adaptation, contrasting sharply with flooding's static threshold.\n\nMore critically, our explicit equality constraint $\\widehat{R}_p^+(g) = \\frac{\\widehat{R}_u^-(g) - \\pi \\widehat{R}_p^-(g)}{1-\\pi}$ introduces a mathematical relationship between classes that flooding cannot capture.\n\nOverall, DC-PU ensures that class-level risks remain proportionally balanced, addressing the fundamental challenge of PU learning where class imbalance emerges from the absence of labeled negative samples rather than general training instability.\n\n\n&nbsp;\n\n\n**Q3. There are several typos.**\n\nThank you for your corrections. We will revise them in the next version.\n\n\n&nbsp;\n\n\n**Q4. Theorem 3.3 lacks discussion.**\n\nThank you for your comment. Theorem 3.3 establishes the fundamental generalization bound that explains DC-PU's superior performance through four distinct components: two Rademacher complexity terms $\\mathfrak{R}\\_{n_u,p}(\\mathcal{G})$ and $\\mathfrak{R}\\_{n_p,p_p}(\\mathcal{G})$, a sampling fluctuation term $\\chi_{n_p,n_u}$, and a distribution discrepancy term $\\Delta$.\n\n**The asymptotic convergence properties.** Under standard assumptions, as $ n_p, n_u \\to \\infty$, two Rademacher complexity terms gradually decrease and converge to zero, and so does the sampling fluctuation term $\\chi_{n_p,n_u}$. The bias term $\\Delta$ is independent of sample size and can be treated as a constant. Consequently, as $n\\to\\infty$, $R(\\widehat{g}_{\\mathrm{DC-PU}}) \\to R(g^*)$.\n\n**The convergence rates.** The convergence rates of the first and second terms are governed by the Rademacher complexities $\\mathfrak{R}\\_{n_u,p}(\\mathcal{G})$ and $\\mathfrak{R}\\_{n_p,p_p}(\\mathcal{G})$, corresponding to rates of $O(1/\\sqrt{n_p})$ and $O(1/\\sqrt{n_u})$, respectively. The fluctuation term $\\chi_{n_p,n_u}$ contributes an additional rate of $O(1/\\sqrt{n_p}+1/\\sqrt{n_u})$. Consequently, the overall convergence rate is characterized by $ O\\left(\\max(1/\\sqrt{n_p},1/\\sqrt{n_u})\\right)$.  It indicates that DC-PU achieves minimax optimal convergence while maintaining balanced performance, contrasting with methods that optimize overall accuracy at the expense of class balance. \n\nAdditionally, the exponential decay rate of bias probability provides crucial insight into DC-PU's stability advantages. It shows that when classes have natural difficulty separation $\\alpha-\\beta > 0$, our constraint mechanism provides exponentially strong concentration guarantees. This explains why DC-PU performs particularly well on datasets with inherent class differences.\n\n**The sampling fluctuation term.** The sampling fluctuation term $\\chi_{n_p,n_u} = 2\\pi/\\sqrt{n_p} + 1/\\sqrt{n_u}$ provides explicit mathematical guidance for experimental design. This term shows that when positive class prior $\\pi$ is small, positive sample size $n_p$ becomes more critical for maintaining tight bounds, while unlabeled sample size $n_u$ provides consistent benefit regardless of class imbalance.\n\n**The distribution discrepancy term.** The distribution discrepancy term $\\Delta$ represents the controlled bias introduced by our constraint mechanism. Unlike uncontrolled bias in traditional methods, this term decreases exponentially with probability, ensuring that our method maintains consistency while providing stability benefits. It explicitly connects sample sizes, class difficulty separation, and theoretical guarantees:\n$$\\Delta \\leq \\pi' C_\\ell \\exp(-2(1-\\pi)^2(\\alpha-\\beta)^2/C_\\ell^2 \\cdot 1/(\\pi^2/n_p + 1/n_u))$$"}, {"review_id": "XuPyCzTUDe", "reviewer": "Reviewer_cuct", "summary": "This paper addresses the problem of imbalanced classification error rates in positive-unlabeled (PU) learning caused by the fact that the empirical risk estimates of the negative class of existing risk assessors (such as nnPU) approach 0. To solve this problem, the paper proposes a new risk assessor called DC-PU, which aims to balance the classification error rates of positive and negative classes. This method enforces the empirical risk estimate of the negative class to be close to the empirical risk of the positive class through a dynamic lower bound and an explicit equality constraint. Experiments have shown that DC-PU can achieve higher accuracy and more stable convergence than other risk assessors.", "questions": "1. Given that the strong constraint enforces strict equality between positive and negative class risks, is the weak constraint still necessary? Could you clarify the individual contributions and potential redundancy between these two constraints?\n2. Theorems 3.2 and 3.3 present conclusions without adequate analysis. Could you provide more insight into aspects such as convergence rate analysis of the established bounds?\n3. On certain datasets, your method shows inferior \"max\" performance compared to baseline methods, while achieving superior \"last\" performance. How do you interpret this phenomenon?", "limitations": "yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n1. The authors provide a clear empirical demonstration of the imbalance problem between FPR and FNR in nnPU through comprehensive experiments, which effectively motivates their proposed solution with dual constraints.\n2. The authors establish solid theoretical guarantees including bias analysis (Lemma 3.1), consistency analysis (Theorem 3.2), and estimation error bounds (Theorem 3.3), providing a comprehensive theoretical understanding of the proposed method.\n3. Beyond maintaining traditional performance metrics, the authors introduce a novel GAP metric to specifically measure the balance between FPR and FNR.\n\nWeaknesses:\n1. The assumptation that the expected risks of the positive-class and negative-class should be close may not be true in some situations. I think this fundamental assumption is overly idealistic. So the authors should further emphasizing the limitations of this aspect.\n2. Lack of sensitivity analysis of introduced parameters, such as $\\tau$ and $\\beta$.\n3. While detailed proofs are provided in the appendix, the main text lacks sufficient analysis and interpretation of the theoretical results. The authors should provide more intuitive explanations and summarizing conclusions regarding the implications of their theoretical findings.", "was_revised": true, "final_justification": "Consider the response from authors and rebuttal from other reviewers, I keep my original score.", "rebuttal": "**Q1. Lack of sensitivity analysis of introduced parameters.**\n\nThank you for your comment. We conduct comprehensive parameter sensitivity analysis. For the parameter $\\tau$, our experiments show optimal performance at $2\\times10^{-3}$ across most datasets, which aligns with our theoretical analysis that moderate penalty parameters achieve balance between constraint strength and optimization stability. For the parameter $\\beta$, we find that the range $[0.4,0.5]$ performs optimally across different datasets, which is significant because $\\beta$ controls the update speed of dynamic lower bound. Too small value of $\\beta$ leads to overly loose constraints losing the balancing effect, while too large value of $\\beta$ causes optimization instability. Our experiments demonstrate that DC-PU exhibits relative robustness to these hyperparameters, with parameter variations within reasonable ranges not significantly affecting performance, further validating the practicality of our method.\n\n&nbsp;\n\n| $\\tau$ | cifar10-1 | cifar10-2 | fmnist-1 | fmnist-2 |\n|:---------:|:---------:|:---------:|:--------:|:--------:|\n| $2\\times10^{-3}$ | **89.37** | **88.6** | 91.21 | 88.45 |\n| $2\\times10^{-2}$ | 89.31 | 87.8 | 91.23 | 88.06 |\n| $2\\times10^{-1}$ | 89.18 | 88.06 | **91.48** | **88.91** |\n| 0 | 88.54 | 88.36 | 91.28 | 86.97 |\n| $2\\times10^{0}$ | 88.94 | 86.52 | 91.32 | 86.44 |\n| $2\\times10^{1}$ | 88.53 | 69.98 | 90.89 | 66.31 |\n| $2\\times10^{2}$ | 67.82 | 65.07 | 85.3 | 69.61 |\n| $2\\times10^{3}$ | 65.75 | 60.67 | 59.8 | 65.27 |\n\n&nbsp;\n\n| $\\beta$ | cifar10-1 | cifar10-2 | fmnist-1 | fmnist-2 |\n|:-----:|:---------:|:---------:|:--------:|:--------:|\n| 0 | 88.62 | 87.59 | 91.18 | 86.92 |\n| 0.1 | 87.91 | 87.8 | 91.38 | 87.37 |\n| 0.2 | 88.33 | 88.07 | 91.24 | 87.41 |\n| 0.3 | 88.75 | 87.97 | 91.35 | 87.8 |\n| 0.4 | 88.49 | 88.51 | **91.48** | **88.91** |\n| 0.5 | **89.37** | **88.6** | 91.27 | 88.13 |\n| 0.6 | 89.2 | 87.9 | 91.23 | 87.77 |\n| 0.7 | 88.82 | 87.98 | 91.22 | 87.84 |\n| 0.8 | 88.62 | 87.82 | 91.26 | 87.38 |\n| 0.9 | 88.63 | 87.68 | 91.28 | 87.22 |\n| 1 | 88.47 | 86.49 | 91.4 | 87.5 |\n\n&nbsp;\n&nbsp;\n\n**Q2. Lack sufficient analysis of the theoretical results.**\n\nThank you for your valuable suggestion. Theorem 3.3 establishes the fundamental generalization bound that explains DC-PU's superior performance through four distinct components: two Rademacher complexity terms $\\mathfrak{R}\\_{n_u,p}(\\mathcal{G})$ and $\\mathfrak{R}\\_{n_p,p_p}(\\mathcal{G})$, a sampling fluctuation term $\\chi_{n_p,n_u}$, and a distribution discrepancy term $\\Delta$.\n\n**The asymptotic convergence properties.** Under standard assumptions, as $ n_p, n_u \\to \\infty$, two Rademacher complexity terms gradually decrease and converge to zero, and so does the sampling fluctuation term $\\chi_{n_p,n_u}$. The bias term $\\Delta$ is independent of sample size and can be treated as a constant. Consequently, as $n\\to\\infty$, $R(\\widehat{g}_{\\mathrm{DC-PU}}) \\to R(g^*)$.\n\n**The convergence rates.** The convergence rates of the first and second terms are governed by the Rademacher complexities $\\mathfrak{R}\\_{n_u,p}(\\mathcal{G})$ and $\\mathfrak{R}\\_{n_p,p_p}(\\mathcal{G})$, corresponding to rates of $O(1/\\sqrt{n_p})$ and $O(1/\\sqrt{n_u})$, respectively. The fluctuation term $\\chi_{n_p,n_u}$ contributes an additional rate of $O(1/\\sqrt{n_p}+1/\\sqrt{n_u})$. Consequently, the overall convergence rate is characterized by $ O\\left(\\max(1/\\sqrt{n_p},1/\\sqrt{n_u})\\right)$.  It indicates that DC-PU achieves minimax optimal convergence while maintaining balanced performance, contrasting with methods that optimize overall accuracy at the expense of class balance. \n\nAdditionally, the exponential decay rate of bias probability provides crucial insight into DC-PU's stability advantages. It shows that when classes have natural difficulty separation $\\alpha-\\beta > 0$, our constraint mechanism provides exponentially strong concentration guarantees. This explains why DC-PU performs particularly well on datasets with inherent class differences.\n\n**The sampling fluctuation term.** The sampling fluctuation term $\\chi_{n_p,n_u} = 2\\pi/\\sqrt{n_p} + 1/\\sqrt{n_u}$ provides explicit mathematical guidance for experimental design. This term shows that when positive class prior $\\pi$ is small, positive sample size $n_p$ becomes more critical for maintaining tight bounds, while unlabeled sample size $n_u$ provides consistent benefit regardless of class imbalance.\n\n**The distribution discrepancy term.** The distribution discrepancy term $\\Delta$ represents the controlled bias introduced by our constraint mechanism. Unlike uncontrolled bias in traditional methods, this term decreases exponentially with probability, ensuring that our method maintains consistency while providing stability benefits. It explicitly connects sample sizes, class difficulty separation, and theoretical guarantees:\n$$\\Delta \\leq \\pi' C_\\ell \\exp(-2(1-\\pi)^2(\\alpha-\\beta)^2/C_\\ell^2 \\cdot 1/(\\pi^2/n_p + 1/n_u))$$\n\n\n&nbsp;\n\n\n**Q3. Is the weak constraint still necessary?**\n\nThank you for your comments. We argue that the weak constraint cannot be considered as being incorporated within the strong constraint.\n\nFirst, since the expected risk is generally intractable, empirical risk is commonly used as its approximation. Compared to the strong constraint, which enforces strict equality between the empirical risks of the positive and negative classes, and thereby introduces potential estimation error. The weak constraint imposes only a lower bound on the negative class risk, which makes it a more tolerant and robust form of estimation under approximation.\n\nSecond, from a convergence perspective, using the strong constraint alone tends to lead to rapid convergence and overfitting in the early stages of training. In contrast, the weak constraint contributes to a more stable training process and helps mitigate overfitting by introducing a lower-bound formulation.\n\nFinally, ablation studies indicate that the weak constraint can sometimes achieve better performance than the strong constraint, indicating that the two constraints are complementary in practice.\n\n\n&nbsp;\n\n\n**Q4. Why the method outperforms baselines in \"last\" but not in \"max\" on certain datasets?**\n\nThank you for the suggestion. This phenomenon actually demonstrates the strength of our method rather than a limitation, as the \"max\" performance represents peak instantaneous performance that may include overfitting to particular batches, while \"last\" performance reflects stable, converged performance under full constraint enforcement that is more representative of true generalization capability. For example, the results of FOPU* on Alzheimer show that it tends to overfit in later stages while it can find a strong model during training, resulting in lower average performance in the final epochs. This indicates that the model trained by FOPU* is less stable than ours."}, {"review_id": "3JDH9K2YJY", "reviewer": "Reviewer_k4vm", "summary": "This paper studies the Positive-Unlabeled (PU) learning with a dual-constrained risk estimator. Their approach is named by DC-PU. The idea stems from a question about imbalance classification error rates in nnPU. Then, they try to balance the risks of positive-class and negative-class and the theoretical analysis is presented. They train DC-PU on the augmented Lagrange multiplier framework and evaluate it with the Fashion-MNIST and CIFAR-10 datasets.", "questions": "I wonder how their approach applies to the recent dataset DIABETES from [J. Gardner, Z. Popovic, and L. Schmidt. Benchmarking distribution shift in tabular data with tableshift. (NeurIPS 2023)].", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths: \n1. The authors propose a new approach called DC-PU for PU learning. DC-PU is a new risk estimator for PU learning that explicitly addresses the imbalance in classification error rates between positive and negative classes. \n2. The proposed method is motivated by empricial observations of imbalanced error rates in nnPU. This gives a practical solution to real-world applications where such imbalances occur.\n\nWeaknesses: \n1. While the theoretical analysis is rigorous, the paper does not clearly explain how these theoretical results translate into the improved performance observed in experiments. For instance, how do the derived bounds in Section 3 directly influence the training dynamics or generalization of DC-PU? A deeper discussion linking theory to practice would strengthen the paper. I suspect that the theoretical analysis is not necessary to state their main claim.\n2. The ablation study (in Section 5.4) evaluates the impact of two constraints (weak and strong). However, they did not mention the choice of other hyperparameters (such as β in Eq. (10)).", "was_revised": true, "final_justification": "I thank the authors' efforts. \n\nThey presented detailed response to my questions. In particular, I appreciated the performance of their method DC-PU on DIABETES. \n\nI therefore increase the score and the confidence.", "rebuttal": "**Q1. How these theoretical results translate into the improved performance observed in experiments?**\n\nThank you for your suggestion. Our theoretical framework provides direct mathematical explanations for the experimental phenomena.\n\n**The training stability**. Our bias analysis in Lemma 3.1 reveals that DC-PU's positive bias occurs with exponentially decaying probability, which directly explains the improved training stability observed in our experiments. While uPU suffers from severe overfitting because $\\widehat{R}\\_u^-(g)-\\pi \\widehat{R}\\_p^-(g)$ tends to be less than 0 during model training, and non-negativity constraint of nnPU leads to the negative-class risk estimator frequently approaching zero, DC-PU's dual-constraint mechanism prevents both problems with high probability. The asymptotic convergence properties further illuminate this advantage that the generalization error bound in Theorem 3.3 contains four major terms, including two Rademacher complexity terms $\\mathfrak{R}\\_{n_u,p}(\\mathcal{G})$ and $\\mathfrak{R}\\_{n_p,p_p}(\\mathcal{G})$, a sampling fluctuation term $\\chi_{n_p,n_u}$, and a distribution discrepancy term $\\Delta$. Under standard assumptions, as $n_p, n_u \\to \\infty$, the Rademacher complexity terms and sampling fluctuation term converge to zero while the bias term $\\Delta$ remains constant, consequently ensuring $R(\\widehat{g}_{\\mathrm{DC-PU}}) \\to R(g^*)$. The convergence rates analysis shows that the overall rate is characterized by $O(\\max(1/\\sqrt{n_p}, 1/\\sqrt{n_u}))$, which explains the smooth and consistent learning curves we observe in Figures 3 and 4.\n\n**The generalization**. The estimation error bound in Theorem 3.3 provides explicit guidance on how our design choices improve generalization performance. The controlled bias term ensures that DC-PU doesn't suffer from the systematic over-play of negative-class risk. This theoretical framework explains why DC-PU consistently achieves superior \"last\" performance metrics in Tables 2 and 3. As the exponentially small bias probability ensures robust behavior regardless of specific dataset characteristics.\n\n**The GAP**. The explicit equality constraint $\\hat{R}_p^+(g) = \\frac{\\hat{R}_u^-(g) - \\pi\\hat{R}_p^-(g)}{1-\\pi}$ in our DC-PU theoretically guarantees that the empirical risks of positive and negative classes remain close throughout training. This mathematical constraint directly translates into the consistently low GAP values shown in Figure 2, where DC-PU maintains balanced error rates across all datasets.\n\n\n&nbsp;\n\n\n**Q2. Lack the choice of other hyperparameters.**\n\nThank you for your comment. We conduct comprehensive parameter sensitivity analysis. For the parameter $\\tau$, our experiments show optimal performance at $2\\times10^{-3}$ across most datasets, which aligns with our theoretical analysis that moderate penalty parameters achieve balance between constraint strength and optimization stability. For the parameter $\\beta$, we find that the range $[0.4,0.5]$ performs optimally across different datasets, which is significant because $\\beta$ controls the update speed of dynamic lower bound. Too small value of $\\beta$ leads to overly loose constraints losing the balancing effect, while too large value of $\\beta$ causes optimization instability. Our experiments demonstrate that DC-PU exhibits relative robustness to these hyperparameters, with parameter variations within reasonable ranges not significantly affecting performance, further validating the practicality of our method.\n\n| $\\tau$ | cifar10-1 | cifar10-2 | fmnist-1 | fmnist-2 |\n|:---------:|:---------:|:---------:|:--------:|:--------:|\n| $2\\times10^{-3}$ | **89.37** | **88.6** | 91.21 | 88.45 |\n| $2\\times10^{-2}$ | 89.31 | 87.8 | 91.23 | 88.06 |\n| $2\\times10^{-1}$ | 89.18 | 88.06 | **91.48** | **88.91** |\n| 0 | 88.54 | 88.36 | 91.28 | 86.97 |\n| $2\\times10^{0}$ | 88.94 | 86.52 | 91.32 | 86.44 |\n| $2\\times10^{1}$ | 88.53 | 69.98 | 90.89 | 66.31 |\n| $2\\times10^{2}$ | 67.82 | 65.07 | 85.3 | 69.61 |\n| $2\\times10^{3}$ | 65.75 | 60.67 | 59.8 | 65.27 |\n\n&nbsp;\n\n| $\\beta$ | cifar10-1 | cifar10-2 | fmnist-1 | fmnist-2 |\n|:-----:|:---------:|:---------:|:--------:|:--------:|\n| 0 | 88.62 | 87.59 | 91.18 | 86.92 |\n| 0.1 | 87.91 | 87.8 | 91.38 | 87.37 |\n| 0.2 | 88.33 | 88.07 | 91.24 | 87.41 |\n| 0.3 | 88.75 | 87.97 | 91.35 | 87.8 |\n| 0.4 | 88.49 | 88.51 | **91.48** | **88.91** |\n| 0.5 | **89.37** | **88.6** | 91.27 | 88.13 |\n| 0.6 | 89.2 | 87.9 | 91.23 | 87.77 |\n| 0.7 | 88.82 | 87.98 | 91.22 | 87.84 |\n| 0.8 | 88.62 | 87.82 | 91.26 | 87.38 |\n| 0.9 | 88.63 | 87.68 | 91.28 | 87.22 |\n| 1 | 88.47 | 86.49 | 91.4 | 87.5 |\n\n&nbsp;\n&nbsp;\n\n\n**Q3. How DC-PU applies to the recent dataset DIABETES?**\n\nThank you for your comment. We add the experimental results on the DIABETES dataset, as shown below. The results on DIABETES dataset provide compelling evidence for the effectiveness of our method in real-world tabular data scenarios. It demonstrates consistent performance throughout training and indicates that DC-PU maintains balanced performance across both classes, which is crucial for real-world medical applications. The most significant finding lies in the GAP metrics: 4.78±0.009 (min) and 10.12±0.009 (last), substantially lower than competing methods like uPU and nnPU. This dramatic reduction in classification error imbalance demonstrates that our dual-constraint mechanism effectively addresses the fundamental challenge of imbalanced error rates even in complex tabular data with intricate feature interactions. The consistent low GAP values across all metrics confirm that our mathematical constraints translate effectively to practical performance improvements in scenarios.\n\n| Method     | Micro-F1 (max) | Micro-F1 (last) | Macro-F1 (max) | Macro-F1 (last) | GAP (min)  | GAP (last)  |\n|------------|----------------|-----------------|----------------|-----------------|------------|-------------|\n| uPU        | 59.09±0.004    | **57.59±0.001** | 51.23±0.028    | 38.37±0.006     | 10.46±0.017 | 40.78±0.006 |\n| nnPU       | 59.08±0.004    | 55.41±0.006     | 55.82±0.006    | 54.33±0.006     | 5.71±0.011  | 10.76±0.008 |\n| abs-PU     | 59.08±0.004    | 55.61±0.004     | 55.70±0.009    | 54.39±0.006     | 5.95±0.013  | 11.39±0.012 |\n| Dist-PU*   | 58.34±0.002    | 55.33±0.004     | 55.80±0.004    | 54.15±0.005     | 5.22±0.007  | 10.69±0.010 |\n| FOPU*      | **59.14±0.004**| 55.22±0.003     | 54.86±0.009    | 53.08±0.005     | 7.18±0.007  | 11.94±0.008 |\n| DC-PU      | 59.10±0.004    | 56.17±0.004     | **56.16±0.008**| **54.79±0.013** | **4.78±0.009** | **10.12±0.009** |"}]

Author rebuttal:

Meta-review: This paper presents a well-motivated approach to addressing a long-standing issue in PU learning — imbalanced classification error rates between positive and negative classes.  The proposed method DC-PU introduces a dual-constraint mechanism that combines a dynamic lower bound and an equality constraint to maintain balanced empirical risks.  

The reviewers generally agree that the paper is solid, with a clear motivation drawn from empirical observations, well-grounded theoretical analysis, and strong experimental results. A concern raised by multiple reviewers is that the theoretical section lacks sufficient intuitive explanation. The authors provided clarifications in the rebuttal. To improve the clarity and accessibility of the work, the revised version should include a more reader-friendly interpretation of the theoretical results within the main text. Overall, I find the paper to make a meaningful contribution to PU learning with both theoretical and practical impact, and I recommend acceptance.
Decision: Accept (poster)