Confidence Difference Reflects Various Supervised Signals in Confidence-Difference Classification

Q1. Whether noise is more pronounced with small confidence differences, as large confidence differences can also introduce noise. And it can work well even if no noise is introduced in the experiments.

Thank you for your suggestions. According to the analysis in "General Response: Q1. About the motivation of this paper", $R_{CD}$ encourages $\mathbf{x}$ and $\mathbf{x'}$ to adjust their predictions in opposite directions. This prediction trend holds correctly for pairwise instances with $\left|c(\mathbf{x}, \mathbf{x}')\right| > 0.5$. However, when $\left|c(\mathbf{x}, \mathbf{x}')\right| \le 0.5$, $\mathbf{x}$ and $\mathbf{x'}$ may belong to the same class or different classes. In such cases, the prediction trend may lead to samples from the same class being predicted as belonging to different classes, introducing erroneous supervisory signals. So noise is more pronounced with small confidence differences.

Moreover, our experimental results show that our method outperforms other baselines even in the absence of noise (i.e., noise ratio $m=0$). Furthermore, when the noise ratio is high, our method still demonstrates strong robustness.

 

Q2. The descriptions in Section 4.2 are not so clear, particularly regarding data generation.

Many thanks for your valuable comments. Firstly, in the pretraining phase, we train a probabilistic classifier using standard labeled data through logistic regression. Then, we input randomly pairwise unlabeled data into the classifier to generate class posterior probabilities. Subsequently, we discard misclassified data to ensure that no noise arises from calibration errors of the probabilistic classifier during posterior generation.

Secondly, we apply a Gaussian-kernel-based probability density estimation method to detect outliers in the posterior probability distribution. This step primarily aims to prevent localized density concentration in the posterior distribution during subsequent scaling due to outlier instances.

Thirdly, we retain the original posterior probability values for the outliers identified in the previous step and apply a designed scaling function to the remaining non-outlier instances. This approach ensures that the reconstructed posterior probabilities form a more uniform distribution.

Finally, Gaussian white noise is added to the scaled posterior probabilities, indirectly introducing noise into the confidence difference and yielding the final $\tilde{c}$.

 

Q3. There are some unclear notations and expressions that should be examined.

Thank you for your corrections. The error in lines 89, 235, and 241 will be revised in future versions.

About pointwise information in line 42-43. Pointwise information is also present in weakly supervised learning beyond general supervised learning. In approaches like Soft Label Learning and Confidence Learning, models receive probabilities or confidence scores rather than binary labels. Maximizing the use of such pointwise weakly supervised information is a key direction in weakly supervised learning, driving advancements in techniques like soft labeling, mixup, and others.

About discussion for Theorem 1. Please refer to "General Response: Q2. About the analysis for Theorem 1".

 

Q4. Why do larger confidence differences contain less noise?

Thank you very much for your valuable suggestions. According to the analysis in "General Response: Q1. About the motivation of this paper", the confidence difference is defined as $c(\mathbf{x}\_i,\mathbf{x}'\_i) = p(y'\_i=+1|\mathbf{x}'\_i) - p(y\_i=+1|\mathbf{x}\_i) \in [-1,1]$. If $\left|c(\mathbf{x}, \mathbf{x}')\right| > 0.5$, $\mathbf{x}$ and $\mathbf{x'}$ must belong to different classes, as the posterior difference is must surpass the classification threshold. Additionally, $R_{CD}$ encourages $\mathbf{x}$ and $\mathbf{x'}$ to adjust their predictions in opposite directions. This prediction trend holds correctly for pairwise instances with $\left|c(\mathbf{x}, \mathbf{x}')\right| > 0.5$. Therefore, we can deduce that larger confidence differences contain less noise.

Q1. There is a lack of descriptions of Theorem 1.

Thank you for your comment. Please refer to "General Response: Q2. About the analysis for Theorem 1".

 

Q2. It is recommended that the authors provide a detailed explanation of 'consistency' within the text.

Thank you for your suggestions. In this work, consistency specifically refers to the expectation that, for a given instance pair $(\mathbf{x}, \mathbf{x}') \in D^{C}$, smaller values of $\left|c(\mathbf{x}, \mathbf{x}')\right|$ correspond to more similar model outputs $g(\mathbf{x})$ and $g(\mathbf{x}')$. Based on this, we design a consistency regularization term that aligns the confidence difference with the model outputs, aiming to reduce the ambiguity caused by small confidence differences. Furthermore, the motivation for this consistency regularization term is elaborated in the "General Response: Q1. About the motivation of this paper".

 

Q3. The manuscript needs revision to improve its clarity and academic rigor.

Thank you for your corrections. The error in lines 014, 155, 241, and 387 will be revised in future versions.

About Line 210: Please clarify the meaning of $\theta(D^C)$ and $\theta(D^S)$.** The expression in line 210 may be confusing, as it does not define $\theta(D^C)$ and $\theta(D^S)$ as new forms; rather, $\theta$ and $D^C$ are two separate symbols. This statement aims to clarify that we partition $D$ into two subsets based on the degree of prediction error, such that $D = D^S \cup D^C$: subset $D^S$ contains pairwise data with confidence differences satisfying $|c(\mathbf{x}, \mathbf{x'})| > \theta$, while subset $D^C$ includes pairwise data with confidence differences satisfying $|c(\mathbf{x}, \mathbf{x'})| \leq \theta$.

Q2. It is recommended that the authors provide a detailed explanation of 'consistency' within the text.

Thank you for your suggestions. Building on our previous discussion, we provide a more detailed explanation of 'consistency' from the following aspects.

What is consistency regularization? The consistency regularization term used in this paper is formally defined as $\bigl(\frac{1}{ \log\left(\left| c(\mathbf{x},\mathbf{x}')\right| + \varepsilon \right) } \bigr) \cdot \left \| g(\mathbf{x})-g(\mathbf{x}') \right \|_2$ on $D^{C}$, aiming to encourage the classifier to further reduce output differences when the confidence difference is small. In other words, if the confidence difference between instances $\mathbf{x}$ and $\mathbf{x}'$ is small, $g(\mathbf{x})$ should be similar to $g(\mathbf{x}')$, and vice versa. This strategy helps enhance generalization ability and makes the model more robust to the noise and perturbations.

The motivation for this consistency regularization term. Specifically, given any pairwise instance $(x_i,x'_i)$ we review the definition of its confidence difference $c(x_i,x'_i)$ below: $c(x_i,x'_i)=p(y'_i=+1|x'_i)-p(y_i=+1|x_i)\in[-1,1],$ where $p(y=+1|x)$ denotes the posterior of x belonging to the positive class. Referring to this definition, if $|c(x, x')|>0.5$, $x$ and x' must belong to different classes; and if $|c(x,x')|\le0.5$, x and x' can belong to the same class or different classes, as the posterior difference is insufficient to surpass the classification threshold.

Accordingly, we consider that the pairwise instances whose confidence difference are greater than 0.5 contain more supervised signals, but the other ones may result in noisy signals in the existing ConfDiff method. To explain this, we rearrange the risk of ConfDiff by expanding a prevalent generic form of various loss functions: $R\_{\mathrm{CD}}(g)=\frac{1}{2}\mathbb{E}\_{p(x,x')}\Bigl[\big(\frac{1}{2}-c(x,x')\big)\ell\bigl(g(x),+1\bigr)+\big(\frac{1}{2}+c(x,x')\big)\ell\bigl(g(x'),+1\bigr)\Bigr]+\frac{1}{2}\mathbb{E}\_{p(x,x')}\Bigl[\big(\frac{1}{2}+c(x,x')\big)\ell\bigl(g(x),-1\bigr)+\big(\frac{1}{2}-c(x,x')\big)\ell\bigl(g(x'),-1\bigr)\Bigr] +\frac{1}{2}\mathbb{E}\_{p(x,x')}\Bigl[(1-2\pi)\bigl(g(x)+g(x')\bigr)\Bigr].$ where the first and second terms denote the contrastive losses for positive and negative class predictions, respectively; and the third term serves as a regularization. In the first term, the weights $\frac{1}{2}-c(x,x')$ and $\frac{1}{2}+c(x,x')$ determine the contributions of x and x' to the positive class prediction loss, summing to 1 and making $\frac{1}{2}$ as a boundary for prediction directions. These weights lie on opposite sides of the boundary, encouraging one instance to predict more strongly as the positive class, while the other is encouraged to weaken its positive class tendency (i.e., predict as the negative class). In other words, the first loss term ensures x and x' to adjust their predictions in opposite directions, thereby emphasizing the predictive divergence of pairwise instances in the positive class predictions.

This prediction trend holds correctly for pairwise instances with $|c(x,x')|>0.5$. However, when $|c(x,x')|\le0.5$, x and x' may belong to the same class or different classes. In such cases, the prediction trend may lead to samples from the same class being predicted as belonging to different classes, introducing erroneous supervisory signals. Similarly, the second loss term forces to diverge in their predictions for the negative class, which also introduces incorrect supervision for them.

To address the challenge posed by $|c(x,x')|$ being close to 0, where the guidance information becomes imprecise and prediction becomes difficult, we propose setting a threshold $\theta$ to partition the dataset into two subsets: one with relatively precise predictive information (denoted as $D^S$) and the other with comparatively imprecise predictive information (denoted as $D^C$). For $D^C$, we aim to provide additional information to guide the predictions of pairwise instances toward the correct direction. Specifically, for pairwise instances with small confidence differences, we encourage the model to produce more similar outputs for these pairs. To achieve this, we introduce a consistency regularization term that encourages alignment between the confidence difference and the model’s outputs. Meanwhile, for $D^S$, we retain the original strategy to preserve the accuracy of predictions driven by this strong guidance.

Dear respected reviewer,

Thanks again for your valuable review comments that helped improve the quality of our draft significantly.

Please let us know if our answers resolved your questions/concerns.

Many thanks!

Q1. The paper lacks a clear presentation of the motivation. There seems exist multiple concepts of "noise". At least there are two different noises which are not clearly described.

Thank you very much for your valuable review. First, we provide a detailed discussion of the motivation in the "General Response: Q1. About the motivation of this paper". Second, the paper indeed presents two descriptions of noise: one in Section 3.1 and the other in Section 4.2. We now provide a detailed explanation and distinction of them.

The noise mentioned in Section 3.1 arises from inaccurate predictive direction information within pairwise instances with $|c(\mathbf{x}, \mathbf{x'})| \leq 0.5$. For example, consider an instance $\mathbf{x}$ with a posterior probability of 0.2 and another instance $\mathbf{x}'$ with a posterior probability of 0.4, yielding $c(\mathbf{x}, \mathbf{x'}) = 0.2$. According to the discussion in "General Response: Q1. About the motivation of this paper", the weights of $\mathbf{x}$ and $\mathbf{x'}$ for the positive class prediction loss are 0.3 and 0.7, respectively, while their weights for the negative class prediction loss are 0.7 and 0.3. This ensures that $\mathbf{x}$ and $\mathbf{x'}$ adjust their predictions in opposite direction, implying $\mathbf{x}'$ is more likely to be classified as positive, which contradicts the actual scenario and thereby introduces inherent noise within $R_{CD}$.

The other type of noise, discussed in Section 4.2, refers to the artificially controllable noise we designed introduced in the confidence difference classification. Specifically, Gaussian white noise is added to the posterior probabilities generated by the probabilistic classifier, thereby indirectly injecting noise into the confidence difference.

 

Q2. The consistency regularization term akes a too small part of the whole paper, with unclear motivation, intuition, and behavior of the proposed modification.

Thank you for your suggestions. Overall, since the ConfDiff method introduces inaccurate predictive information when $|c(\mathbf{x}, \mathbf{x'})| \leq 0.5$, we aim to provide precise guidance to encourage the predictions of pairwise instances toward the correct direction. Building on the insight that the consistency regularization term fosters the alignment between confidence differences and model outputs, our method leverages it to encourage more similar outputs for pairwise instances with small confidence differences and reduces the interference of inaccurate guidance on the prediction direction of pairwise instances. Please refer to "General Response: Q1. About the motivation of this paper" for a more detailed analysis and explanation.

 

Q3. The other contribution of consistency risk of $D^S$ is not well motivated.

Thank you for your comment. Please refer to "General Response: Q1. About the motivation of this paper".

 

Q4. Eq.5 introduces the "general form of many commonly used losses" without clarifying which specific loss functions fall under this form or are excluded.

Many thanks for your valuable comments. This function class includes several commonly used loss functions, such as those derived from Generalized Linear Models (GLMs), including the mean squared error (MSE) for linear regression, the logistic loss for logistic regression, the Poisson loss for Poisson regression, the exponential loss for exponential regression, and the cross-entropy loss for neural networks [1].

[1] Linjun Zhang, Zhun Deng, Kenji Kawaguchi, Amirata Ghorbani, and James Zou. How does mixup help with robustness and generalization? In International Conference on Learning Representations, 2021.

Many thanks for your valuable discussion and response.

Q1. There are multiple notations of "noise".

Thank you for your comment. We have provided further clarification on these two issues.. First, we need to clarify that the first type of noise is actually related to the implementation details of estimating $c$ at the experimental level, rather than being the main motivation of this paper. At both the methodological and experimental levels, $c$ is calculated as the difference in posterior probabilities between two instances. The distinction lies in that, at the methodological level, we assume the posterior probabilities used in the analysis are accurate, without involving any estimation. In contrast, at the experimental level, we propose a posterior probability construction method to approximate it. Specifically, we find that the ConfDiff method uses a probabilistic classifier trained on labeled data to generate class posterior probabilities. While this generation method facilitates comprehensive experimental analysis, challenges arise when fitting large-scale real-world datasets and employing sufficiently complex backbones. This often results in the concentration of the posterior probability distribution around certain values, thereby failing to reflect the true distribution influenced by manual annotations in real-world scenarios. Inspired by this, we incorporated a posterior probability construction method based on outlier detection into the probabilistic classifier and calculated confidence differences according to its definition to achieve a more uniform and realistic distribution. The detailed implementation can be found in Section 4.2 in our paper.

The second type of noise is indeed the main motivation of our paper, and a detailed analysis can be found in the description within "General Response: Q1. About the motivation of this paper."

The third type of noise accounts for the fact that in the real world, annotator biases or varying knowledge backgrounds may lead to annotation outcomes that are not entirely precise. To further verify the effectiveness of our method under noisy conditions, we propose adding Gaussian white noise to the posterior probabilities to indirectly introduce noise into the confidence difference. The experimental results strongly demonstrate that our method remains highly competitive even in the presence of noise interference.

Moreover, we acknowledge that certain expressions in the original paper may have been imprecise, potentially leading to confusion regarding the different types of noise and their corresponding motivations. We sincerely thank you for your valuable feedback, which has provided significant guidance for improving our paper. Accordingly, we have revised some descriptions in the manuscript and submitted a new version, which we hope will address your concerns.

 

Q2. For Fig.1, could you explain how to plot a line for different values of $x$.

Thank you for your comment. The construction of this figure is based on an observation that, when attempting to fit the posterior probabilities of a dataset using outputs from sufficiently complex backbones, the output values tend to yield either very high or very low, resulting in instances being classified with high confidence as either positive or negative. In other words, the posterior probabilities are more concentrated near ${0,1}$, disregarding the more uniform distribution typically observed in real-world scenarios. Therefore, when $\pi=0.5$, given two datasets $\mathcal{D}\_{1} = \{\bigl(\mathbf{x}\_i, p(y\_i=+1|\mathbf{x}\_i)\bigr)\}\_{i=1}^n$ and $\mathcal{D}\_{2}=\{\bigl(\mathbf{x}'\_i,p(y'\_i=+1|\mathbf{x}'\_i)\bigr)\}\_{i=1}^n$, we can adjust the ordering of data within $\mathcal{D}\_{1}$ and $\mathcal{D}\_{2}$ to vary the proportion of pairwise instances with $|c| > 0.5$ over the range $[0,1]$. Specifically, a proportion of 0 corresponds to $\mathcal{D}\_{1}$ and $\mathcal{D}\_{2}$ being ordered identically, while a proportion of 1 (equal to $2*\min(\pi,1-\pi)$) corresponds to $\mathcal{D}\_{1}$ and $\mathcal{D}\_{2}$ being ordered in strictly reverse sequences.

Based on this observation, we randomly select two datasets, $\mathcal{D}\_{1}$ and $\mathcal{D}\_{2}$, both satisfying $\pi=0.5$, and sort their data in ascending order according to posterior probabilities. (After sorting under the aforementioned observation, the first 50% of posterior probabilities are concentrated near 0, while the latter 50% are concentrated near 1.) For example, when the x-axis value is 1, we obtain a proportion of 0.5 for pairwise instances with $|c|>0.5$. To achieve this, we exchange the first 25% and last 25% of instances in $\mathcal{D}_{2}$. Then, following the definition $\mathcal{D}\_{\mathrm{CD}}=\{\bigl((\mathbf{x}\_i,\mathbf{x}'\_i),c\_i\bigr)\}\_{i=1}^n,\:c\_i=p(y'\_i=+1|\mathbf{x}'\_i)-p(y_i=+1|\mathbf{x}_i)$, we generate a confidence difference classification dataset for training.

Q1. ConfDiff learning relies on perfect classifier calibration, which is challenging in practice; studying the impact of calibration errors on its performance would be more meaningful.

Thank you for your suggestions. In fact, we discarded the data incorrectly classified by the classifier when reconstructing the confidence differences, and this spirit is detailedly implemented in lines 120-144 of the 'main_strong.py'. Moreover, although the classifier's accuracy during the pretraining phase does not meet the perfect calibration standard, it remains sufficiently high. Therefore, the operation of discarding misclassified data does not significantly affect the scale of the dataset. Detailed information about the accuracy of classifier and the number of misclassified instances is presented in the table below.

 

Q2. The paper presents error bounds without discussing their implications or providing new insights.

Thank you for your comment. Please refer to "General Response: Q2. About the analysis for Theorem 1".

 

Q3. This paper appears to be unclear in many details, why negative risk can lead to overfitting?

Thank you for your suggestions. Risk is typically non-negative, reflecting the deviation between model predictions and ground-truth values. The objective of optimization is to minimize risk; if risk could be negative, the model could be inclined to find an optimization direction that continually reduces risk on the training data, leading to overfitting by learning noise and performing poorly on test data.

Additionally, [1] highlights that negative empirical risk may be a potential cause of overfitting and experimentally demonstrates a strong co-occurrence of negative risk and overfitting across various models and datasets.

[1] Nan Lu, Tianyi Zhang, Gang Niu, and Masashi Sugiyama. Mitigating overfitting in supervised classification from two unlabeled datasets: A consistent risk correction approach. In International Conference on Artificial Intelligence and Statistics, pp. 1115–1125. PMLR, 2020.

 

Q4. Suppose we are given a supervision signal $\tilde{c}(**x**, **x**') < 0.5$ (smaller confidence difference), then by fitting this objective, wouldn’t the $R_{CD}$ inherently encourages $**x**, **x**'$ to be similar? Hence making $R_{CR}$ trivial?

Many thanks for your comment. The motivation for our method arises from the observation that small confidence differences may lead to imprecise guidance within $R_{CD}$, particularly when the confidence difference equals zero, resulting in a complete lack of predictive guidance. Our experiments in Figure 1 further demonstrate that pairwise instances with larger confidence differences dominate the contribution to $R_{CD}$, while those with smaller confidence differences contribute minimally. To address this issue, we propose a consistency regularization term that encourages $\mathbf{x}$ and $\mathbf{x'}$ to produce more similar outputs in the model when $|c(\mathbf{x}, \mathbf{x'})|$ is small. A more detailed explanation and analysis can be found in the "General Response: Q1. About the motivation of this paper".

 

Q5. From line 210-212, shouldn't the $c(\mathbf{x}, \mathbf{x'}) < 0.5$ be $\tilde{c}(\mathbf{x}, \mathbf{x'}) < 0.5$ instead?

Thank you for your correction. We will correct it in future versions.

Q6. How the noise is being defined and formulated in this paper is still a bit puzzling to me.

Thank you very much for your comments. Our paper presents two descriptions of noise, which we now elaborate on in detail below.

The noise mentioned in Section 3.1 arises from inaccurate predictive direction information within pairwise instances with $|c(\mathbf{x}, \mathbf{x'})| \leq 0.5$. For example, consider an instance $\mathbf{x}$ with a posterior probability of 0.2 and another instance $\mathbf{x}'$ with a posterior probability of 0.4, yielding $c(\mathbf{x}, \mathbf{x'}) = 0.2$. According to the discussion in "General Response: Q1. About the motivation of this paper", the weights of $\mathbf{x}$ and $\mathbf{x'}$ for the positive class prediction loss are 0.3 and 0.7, respectively, while their weights for the negative class prediction loss are 0.7 and 0.3. This ensures that $\mathbf{x}$ and $\mathbf{x'}$ adjust their predictions in opposite direction, implying $\mathbf{x}'$ is more likely to be classified as positive, which contradicts the actual scenario and thereby introduces inherent noise within $R_{CD}$.

The other type of noise, discussed in Section 4.2, refers to the artificially controllable noise we designed introduced in the confidence difference classification. Specifically, Gaussian white noise is added to the posterior probabilities generated by the probabilistic classifier, thereby indirectly injecting noise into the confidence difference.

 

Q7. I recommend the authors to improve the writing of the Section 3.1 and add concrete examples to clarify the points.

Many thanks for your valuable comment. Section 3.1 primarily outlines our motivations. We provide a more detailed discussion in "General Response: Q1. About the motivation of this paper", and add concrete examples to clarify key points. We will update Section 3.1 in future revisions.

Thank you for your discussion and response. We have provided further clarification on these two issues.

Q1. I fail to see how this is related to the probability calibration.

Thank you for your comment. I understand that the calculation method of posterior probabilities in the experiments might have caused some confusion. Perhaps you are concerned that using the output of the probabilistic classifier as the posterior probabilities requires the classifier to be highly calibrated. In fact, this method follows the setup presented in [1]. Additionally, we would like to clarify that the confidence difference mentioned in the methodology section is accurately defined as the difference between the posterior probabilities of two instances, specifically $c(\mathbf{x}_i,\mathbf{x}'_i) = p(y'_i=+1|\mathbf{x}'_i) - p(y_i=+1|\mathbf{x}_i)$. In the experimental section, to facilitate comprehensive analysis, we use the output of the probabilistic classifier as an estimate of the posterior probabilities, which is merely an experimental setting. However, we recognize that this estimation may introduce deviations from the true posterior probability distribution. Therefore, in Section 4.2, we propose a posterior probability reconstruction method that detects outliers in the posterior probability distribution using a Gaussian kernel-based discrete posterior probability density estimation. The non-outlier instances are then scaled to achieve a more realistic and uniform posterior probability distribution.

Moreover, we understand your concern regarding the potential noise introduced in the posterior probability calculations if the classifier is not highly calibrated. We have considered this and excluded misclassified instances during the experiments. To further evaluate the impact of noise on our method, we design a method to introduce varying levels of noise to the posterior probabilities and assessed whether our method can achieve stable and accurate results under such artificially induced noise. The experimental results demonstrate that our method maintains increasingly stable and consistent accuracy as the noise ratio increases, with notable improvements in both accuracy and standard deviation, particularly when the noise ratio reaches 100%. This indicates its ability to achieve more competitive results under noise interference.

[1] Wei Wang, Lei Feng, Y uchen Jiang, Gang Niu, Min-Ling Zhang, and Masashi Sugiyama. Binary classification with confidence difference. Advances in Neural Information Processing Systems, 36, 2024.

Q2. About the analysis for Theorem 1

Thank you for your suggestions.

1. How is this related to the noisy supervision signal?

First, as described in the "General Response: Q1. About the motivation of this paper", one of our contributions is the partitioning strategy of the dataset into two subsets based on the threshold $\theta$: one with relatively accurate predictive information (denoted as $D^S$) and the other with relatively inaccurate predictive information (denoted as $D^C$). This suggests that noise signals influence the distribution of $c(x,x')$, which in turn affects the choice of an appropriate $\theta$ to balance the subset partitioning strategy, thereby indirectly impacting the error bound.

Specifically, when the noise ratio is low, the majority of samples are assigned to $D^S$. As the sample size $n_1$ increases, both the Rademacher complexity and the term $\frac{4C_\ell}{n_1}\sqrt{2n\mathrm{In}(2/\delta)}$ decrease, enabling the model to quickly converge when training on low-noise samples.

When the noise ratio is high, a larger proportion of samples are assigned to $D^C$. The consistency regularization term encourages the model to output consistent predictions on samples with small confidence differences, thereby reducing the excessive fluctuations induced by noise. Furthermore, from the perspective of the error bound, we control the number of samples in $D^C$ by setting an appropriate threshold $\theta$, thereby balancing the stability and convergence of the error bound. The logarithmic factor $\left| \frac{1}{\log(\varepsilon)}-\frac{1}{\log(\theta+\varepsilon)}\right|$ also ensures that the model's sensitivity to noise remains within a stable range, preventing noisy supervision signals from excessively affecting the error bound.

2. What properties of your proposed method is related to this error bound?

The two main contributions we propose—the dataset partitioning strategy and the consistency regularization strategy—are both directly related to the error bound.

For the dataset partitioning strategy, its fundamental idea is to partition the dataset into two subsets based on the threshold $\theta$: one with relatively accurate predictive information (denoted as $D^S$) and the other with relatively inaccurate predictive information (denoted as $D^C$). Through this partitioning strategy, The number of pairwise instance in $D^C$ and $D^S$ can be adjusted according to the distribution of $c(x,x')$, thereby influencing the convergence rate of the error bound through $n_1$ and $n_2$. Furthermore, this strategy effectively smooth the influence of noise by applying consistency regularization.

For the consistency regularization strategy, it encourages the model to output consistent predictions on $D^C$, thus preventing overfitting to noisy features. This strategy reduces the Rademacher complexity on $D^C$, accelerating the convergence of the complexity term in the error bound. Additionally, the term $\left| \frac{1}{\log(\varepsilon)}-\frac{1}{\log(\theta+\varepsilon)}\right|$ effectively controls the range of the error bound, ensuring that the influence of noise on the error bound remains within a stable range.

3. About the asymptotic convergence properties and rates of the error bound.

The asymptotic convergence properties. The first and second terms of the error bound involve $\mathfrak{R}\_{n_1}(\mathcal{G})$ and $\mathfrak{R}\_{n_2}(\mathcal{G})$, which represent the Rademacher complexities associated with the sample sizes $n_1$ and $n_2$, respectively. Under standard assumptions, as $n_1,n_2\to\infty$, $\mathfrak{R}\_{n_1}(\mathcal{G})$ and $\mathfrak{R}\_{n_2}(\mathcal{G})$ gradually decrease and converge to zero. The third term, containing $\frac{\sqrt{n}}{n_1}$ and $\frac{\sqrt{n}}{n_2}$, also diminishes as the sample sizes increase. Consequently, as $n\to\infty$, $R(\hat{g}_{\mathrm{CRCR}}) \to R(g^*)$.

The Convergence Rates. The convergence rates of the first and second terms are governed by the Rademacher complexities $\mathfrak{R}\_{n_1}(\mathcal{G})$ and $\mathfrak{R}\_{n_2}(\mathcal{G})$, corresponding to rates of $O(1/\sqrt{n_1})$ and $O(1/\sqrt{n_2})$, respectively. By decomposing the third term into $\frac{4C_\ell}{n_1}\sqrt{2n\mathrm{In}(2/\delta)}$ and $\left| \frac{1}{\log(\varepsilon)}-\frac{1}{\log(\theta+\varepsilon)}\right|\frac{4\alpha C_g}{n_2}\sqrt{2n\mathrm{In}(2/\delta)}$, it can be observed that the convergence rates of these components are dominated by $O(\sqrt{n}/{n_1})$ and $O(\sqrt{n}/{n_2})$, respectively. Consequently, the overall convergence rate is characterized by $O\left(\max(\sqrt{n}/{n_1},\sqrt{n}/{n_2})\right)$. Moreover, the balance between $n_1$ and $n_2$ ensures a faster convergence rate, thereby providing stronger guarantees for the model's performance.

Q1. About the motivation of this paper

Overall, our motivation arises from the fact that $R_{CD}$ encourages $x$ and $x'$ to predict opposite classes, with one inclined toward positive and the other toward negative. However, this prediction direction may be inaccurate when $|c(x,x')|\leq0.5$, as the two instances may belong to the same or different classes in this case. To address this issue, we propose a ConfDiff classification method based on consistency regularization to mitigate the impact of inaccurate predictions when confidence differences are small. To describe our work explicitly, we review the research motivation in detail.

First, given any pairwise instance $(x_i,x'_i)$ we review the definition of its confidence difference $c(x_i,x'_i)$ below: $c(x_i,x'_i)=p(y'_i=+1|x'_i)-p(y_i=+1|x_i)\in[-1,1],$ where $p(y=+1|x)$ denotes the posterior of x belonging to the positive class. Referring to this definition, if $|c(x, x')|>0.5$, $x$ and x' must belong to different classes; and if $|c(x,x')|\le0.5$, x and x' can belong to the same class or different classes, as the posterior difference is insufficient to surpass the classification threshold. For example, possible scenarios may occur when $|c(x,x')|=0.3$: $p(y_i=+1|x_i)=0.1$, $p(y'_i=+1|x'_i)=0.4$ (both negative); $p(y_i=+1|x_i)=0.3$, $p(y'_i=+1|x'_i)=0.6$ (different classes); $p(y_i=+1|x_i)=0.7$, $p(y'_i=+1|x'_i)=1$ (both positive).

Accordingly, we consider that the pairwise instances whose confidence difference are greater than 0.5 contain more supervised signals, but the other ones may result in noisy signals in the existing ConfDiff method. To explain this, we rearrange the risk of ConfDiff by expanding a prevalent generic form of various loss functions: $R\_{\mathrm{CD}}(g)=\frac{1}{2}\mathbb{E}\_{p(x,x')}\Bigl[\big(\frac{1}{2}-c(x,x')\big)\ell\bigl(g(x),+1\bigr)+\big(\frac{1}{2}+c(x,x')\big)\ell\bigl(g(x'),+1\bigr)\Bigr]+\frac{1}{2}\mathbb{E}\_{p(x,x')}\Bigl[\big(\frac{1}{2}+c(x,x')\big)\ell\bigl(g(x),-1\bigr)+\big(\frac{1}{2}-c(x,x')\big)\ell\bigl(g(x'),-1\bigr)\Bigr] +\frac{1}{2}\mathbb{E}\_{p(x,x')}\Bigl[(1-2\pi)\bigl(g(x)+g(x')\bigr)\Bigr].$ where the first and second terms denote the contrastive losses for positive and negative class predictions, respectively; and the third term serves as a regularization. In the first term, the weights $\frac{1}{2}-c(x,x')$ and $\frac{1}{2}+c(x,x')$ determine the contributions of x and x' to the positive class prediction loss, summing to 1 and making $\frac{1}{2}$ as a boundary for prediction directions. These weights lie on opposite sides of the boundary, encouraging one instance to predict more strongly as the positive class, while the other is encouraged to weaken its positive class tendency (i.e., predict as the negative class). In other words, the first loss term ensures x and x' to adjust their predictions in opposite directions, thereby emphasizing the predictive divergence of pairwise instances in the positive class predictions.

This prediction trend holds correctly for pairwise instances with $|c(x,x')|>0.5$. However, when $|c(x,x')|\le0.5$, x and x' may belong to the same class or different classes. In such cases, the prediction trend may lead to samples from the same class being predicted as belonging to different classes, introducing erroneous supervisory signals. Similarly, the second loss term forces to diverge in their predictions for the negative class, which also introduces incorrect supervision for them.

To further validate this perspective, we conduct experiments on the MNIST and CIFAR-10 by varying the proportion of the pairwise instances with $|c(x,x')|>0.5$. The empirical results (in Fig.1) illustrate the accuracy of the binary classifier under different proportions of the pairwise instances with $|c(x,x')|>0.5$. We observe a positive correlation between classification accuracy and the proportion value. These findings demonstrate that the pairwise instances with $|c(x,x')|>0.5$ provide strong guidance and dominate the contribution to $R_{CD}$.

To address the challenge posed by $|c(x,x')|$ being close to 0, where the guidance information becomes imprecise and prediction becomes difficult, we propose setting a threshold $\theta$ to partition the dataset into two subsets: one with relatively precise predictive information (denoted as $D^S$) and the other with comparatively imprecise predictive information (denoted as $D^C$). For $D^C$, we aim to provide additional information to guide the predictions of pairwise instances toward the correct direction. Specifically, for pairwise instances with small confidence differences, we encourage the model to produce more similar outputs for these pairs. To achieve this, we introduce a consistency regularization term that encourages alignment between the confidence difference and the model’s outputs. Meanwhile, for $D^S$, we retain the original strategy to preserve the accuracy of predictions driven by this strong guidance.