Paper: Learning the Latent Noisy Data Generative Process for Label-Noise Learning (ICLR-2024)

Reviewer comment (Novelty):
To the best of my knowledge, BLTM (one of the baselines in the experiment part), which trains a instance-dependent transition matrix network using the distilled dataset, has no assumption for the similarity between noise transition for different instances. I think this study contradicts to author's proposal, which says, there are assumptions of similarity in the previous researches.

Author response:
**A1**: BLTM can learn the noise transition of some examples using a neural network. The authors of BLTM hope the network can generate the noise transition for unseen instances. However, the author does not discuss how to generalize the learned noise transition to other examples, and they do not provide a theoretical guarantee that the noise transition can be generalized. Some hidden assumptions must exist to build the connection. Otherwise, the noise transition can not be generalized to other examples. Our claim focuses on the methods that have theoretical guarantees on the generalization of noise transitions, e.g., the noise transition is class-dependent, and the noise transition for the instances in the same manifold is the same. To avoid confusion, we have modified the paper.

We are the first to explore when the learned noise transition in some instances can be generalized to other instances in learning with noisy labels. Our aim is to enable deep neural networks to capture the connections of noise transitions among different instances automatically. Since once the connections are captured, the noise transition learned in some examples can be generalized to other examples. To achieve this, we need to model the joint distribution of all variables $p(\mathbf{X},\tilde{Y},Y,\mathbf{Z},\mathbf{N})$. The noise transition can be obtained through
$$
p(\tilde{Y}|\mathbf{X},Y)=\frac{\int_{\mathbf{Z},\mathbf{N}}p(\mathbf{X},\tilde{Y},Y,\mathbf{Z},\mathbf{N})d\mathbf{Z}d\mathbf{N}}{\int_{\tilde{Y},\mathbf{Z},\mathbf{N}}p(\mathbf{X},\tilde{Y},Y,\mathbf{Z},\mathbf{N})d\tilde{Y}d\mathbf{Z}d\mathbf{N}}.
$$
To obtain this joint distribution, we propose a principled way to model the whole data generative process. Specifically, we not only model the generative process from the latent variable $\mathbf{Z}$ to the instance $\mathbf{X}$ but also model and learn causal structure among different latent variables $\{Z_1,Z_2,\dots,Z_m\}$ from observed noisy data. 

To achieve this, the generation of the variable $\mathbf{N}$ is modeled by a Gaussian distribution. The generation of the latent variable $\mathbf{Z}$ is modeled by a Structure Causal Model (SCM): $Z_i:=\mathbf{W}^T_i \mathbf{Z}+N_i$, where $\mathbf{W}$ is a matrix used to represent the association among the latent variables. Two generators are used to model the generation of the instance $\mathbf{X}$ and the noisy label $\tilde{Y}$, respectively.

Moreover, our paper has analyzed the identifiability of the proposed method theoretically in Appendix C. We discuss the conditions required for identifiability. Specifically, when the number of distinct classes is large (more than the sufficient statistics of the latent variable $\mathbf{Z}$) and we have a confident example in each class, the latent variable $\mathbf{Z}$ can be identified up to permutation.

Reviewer comment (Theory):
Since Y is latent, there is no way to prove that Y is true (so that the authors relied on distillation). Therefore, it means the authors actually makes an another assumption saying that the distilled dataset is clean.

Author response:
**A2**: We assume that the selected dataset is likely clean. This assumption is reasonable because the accuracy of the selected dataset is high empirically. For example, when the dataset is CIFAR-10, and the noise type is "worst", the accuracy of the selected dataset is 0.9568.

It is also worth mentioning that without clean labels, recovering the causal factor $\mathbf{Z}$ is an ambitious goal and generally impossible. This is because the nonlinear mixture function $f(\cdot)$ used to generate observed variable $\mathbf{X}$ is unknown. A variable $\mathbf{Z}'$ can be constructed via a nonlinear function $\mathbf{Z}'=g(\mathbf{Z})$. Another nonlinear mixture function $f'(\cdot)$ can generate the same observation, i.e., $\mathbf{X}=f'(\mathbf{Z}')$. Previous studies also prove that it is ill-posed to recover the latent causal factors with only the observed variables [1,2]. Existing methods in causal representation learning also have to use the auxiliary variables to recover the causal factors [3,4]. In the context of learning with noisy labels, auxiliary variables necessitate clean labels.

Reviewer comment (Evaluation):
Since it should generate X (usually an image), time complexity and memory will be large. Why should we generate an image to classify? Also want to see the exact time and memory comparison over several baselines.

Author response:
**A3**: The proposed method does not increase the time complexity and memory because the encoder and the decoder introduced by our method are lightweight. The training time of the proposed method on CIFAR-10 is 13.68 hours, and the training time of DivideMix is 9.08 hours. The memory used for the proposed method is 7.06 GB, and the memory used for DivideMix is 6.99 GB.

Moreover, the encoder and decoder can be discarded during inference. The inference time is the same as the time of baselines.

Reviewer comment (Evaluation):
Lack of any analysis. In the experiment part, nothing but the test accuracy are reported. Please show analytical empirical results for whether the training are processed as authors proposed. For example, at least whether the error of the inferred transition is small or not should be expressed.

Author response:
**A4**: We have conducted experiments on the dataset CIFAR-10 to calculate the noise transition estimation error. The labels are corrupted manually using the instance-dependent noisy label generation method proposed in [5]. The baselines used to compare the noise transition estimation error are BLTM, MEIDTM and VolMinNet. The experiment results are shown as follows.

|           | IDN-10%               | IDN-20%               | IDN-30%               | IDN-40%               | IDN-50%               |
| --------- | --------------------- | --------------------- | --------------------- | --------------------- | --------------------- |
| BLTM      | 0.275 $\pm$ 0.022     | 0.339 $\pm$ 0.020     | 0.467 $\pm$ 0.008     | 0.670 $\pm$ 0.106     | 0.816 $\pm$ 0.056     |
| MEIDTM    | 0.240 $\pm$ 0.002     | 0.374 $\pm$ 0.001     | 0.563 $\pm$ 0.001     | 0.775 $\pm$ 0.002     | 0.985 $\pm$ 0.001     |
| VolMinNet | 0.477 $\pm$ 0.005     | 0.590 $\pm$ 0.000     | 0.680 $\pm$ 0.000     | 0.720 $\pm$ 0.000     | 0.783 $\pm$ 0.005     |
| GenP      | **0.224 $\pm$ 0.005** | **0.302 $\pm$ 0.016** | **0.374 $\pm$ 0.008** | **0.434 $\pm$ 0.021** | **0.548 $\pm$ 0.041** |

Empirical results show that our method can reduce the error of inferred noise transition.

Reviewer comment (Experiments):
Can authors show (at least, the empirical) evidences of the proposal saying that "noise transitions can be inferred of a special group of instances"?

Author response:
**A6**: The evidence is the noise transition estimation methods in previous work. For example, the special group of instances can be the anchor points in the dataset, which is stated in Section 2 in [6] and Theorem 3 in [7].

Reviewer comment (Theory):
Why should we use graph structure? Can't this structure be changed to other structure?

Author response:
**A5**: To model the causal structure among latent variable $\mathbf{Z}$, the graph structure has to be used. This is because the causal structure among $\mathbf{Z}$ is a part of the noisy data generative process. As mentioned in A1, modeling the noisy data generative process can obtain the joint distribution $p(\mathbf{X},\tilde{Y},Y,\mathbf{Z},\mathbf{N})$, then the connections of noise transitions among different instances can be captured automatically. The more accurately the generation process is modeled, the more accurate the joint distribution obtained will be.

Reviewer comment (Experiments):
How can we be sure that the process the authors proposed improve the model performance?

Author response:
**A8**: Our method is conceptually better than other methods. As discussed in the Introduction, to generalize the noise transition learned in some instances to other instances, existing work without theoretical guarantee manually defines similarity for noise transition across various instances. However, the human-defined similarity is hard to verify in the real world. If the similarity is not truthful, the estimation error of the noise transition could be large. Thus, the performance of the classifier would decrease. 

To avoid human-defined similarity, we propose a principled way to capture the connections of noise transitions among different instances automatically. Specifically, our method models the data generative process, and then the joint distribution $p(\mathbf{X},\tilde{Y},Y,\mathbf{Z},\mathbf{N})$ can be obtained. Thus, the noise transition can be captured  (please refer to the answer in A1). Once the model can capture the connections, the noise transition estimated in some examples can be generalized to other examples. The information contained in other examples can be inferred and learned by the model. Thus, the performance of the model will be improved. The experiment results also have verified the effectiveness of our method.

Reviewer comment (Novelty):
Why using the estimated clean labels from distillation is not enough? As the authors said, distillation methods are well studied. By pointing out that the model performance of the method the authors proposed should depend on the quality of the estimated clean labels, the distilled labels should be clean enough. If the estimated labels are clean enough, why should we do more process fundamentally?

Author response:
**A7:** The size of selected examples is much smaller than the size of the whole dataset, and the distribution of selected examples is not the same as the distribution of examples in the clean domain. As a result, training classifiers on selected examples with different distributions will lead to the performance degradation of the classifiers because the parameters of the learned classifier are not statistically consistent with the parameters of the optimal classifier defined on the clean domain. 

To guarantee the statistical consistency of classifiers, the information contained in the remaining examples has to be used. Modeling the noise transition for the whole dataset is a way to employ the information contained in the training examples. The reason is that the clean class posterior $p(\tilde{Y}|\mathbf{X}=\mathbf{x})$ for an instance $\mathbf{x}$ can be inferred through the noise transition $p(\tilde{Y}|Y,\mathbf{X}=\mathbf{x})$ and the noisy class posterior $p(\tilde{Y}|\mathbf{X}=\mathbf{x})$, i.e.,  $p(\tilde{Y}|\mathbf{X}=\mathbf{x})=p(\tilde{Y}|Y,\mathbf{X}=\mathbf{x})p(Y|\mathbf{X}=\mathbf{x})$.

Reviewer comment (Presentation):
Figure 2 should be polished up. For example, in Figure 2, the difference between the blue arrow and black arrow should be explained. Meanwhile, the generative process related to Figure 2 in Section 3, the core of this paper, should be given more details about.

Author response:
**A2**: Thanks for your suggestion. The black arrow indicates the causal direction in the structural causal model, while the blue arrow indicates the weights are changed across the clean label $Y$. More details about the generative process are included in Sec 3.

Reviewer comment (Theory):
The assumption that the generation process of causal factors Z is linear seems a bit unreasonable, which should be given more details on.

Author response:
**A3**: Since causal factor $\mathbf{Z}$ is latent, discovering the causal structure is a challenging problem. If $\mathbf{Z}$ is observed, nonlinear causal mechanisms can be discovered easily. However, we do not assume the causal factors are observed. To provide a theoretical guarantee, we assume the causal mechanism among causal factors is linear. We empirically find that the performance of the proposed method is good, which indicates that this simplicity does not have negative effects.

Reviewer comment (Theory):
Eq.(1) is very confusing. It seems that $Z_i$ is the element in $\mathbf{Z}$ but also calculated by $\mathbf{Z}$, which provides an implicit constraint.

Author response:
**A4**: $\mathbf{W}$ is an upper triangular matrix with zero-value diagonal elements. When calculating $Z_i$, only the parents nodes of $Z_i$ are used, i.e., $Z_i=\sum_{j\in pa_i}W_{i,j}Z_{j}$.

Reviewer comment (Novelty):
Some related works such as [2, 3] are missing.

Author response:
**A5**: Thanks for your advice. We have added these works to the Related Work.

Reviewer comment (Writing):
What is the meaning of $N_i$? In Section 3, it mentions that $N_i$ is the corresponding latent noise variable. Could you take Figure 1 as an example to illustrate it?

Author response:
**A6**: $N_i$ is an unexplained random variable that determines $Z$ in the structural causal model, i.e., $Z_i:=f_Z(pa(Z_i),N_i)$. This variable is referred as the noise variable in previous work [1], which is different from the noise in the label noise.

Reviewer comment (Novelty):
In the field of noisy label classification, using deep generative models (DGMs) to infer latent true labels is not a novel approach. Various studies, including [1], have explored this concept by leveraging the capacity of DGMs to model complex data distributions and underlying noise patterns.

Author response:
**Q1: Various studies have explored leveraging deep generative models to model complex data distributions and underlying noise patterns, thus using deep generative models (DGMs) to infer latent true labels is not a novel approach.**

**A1**: Though some previous studies also use generative models, our work is different from previous work in several points. 

- The aim of previous work CausalNL and InstanceGM is to exploit the information contained in the distribution of instances to help the learning of classifiers. To achieve this, the generative model is introduced. However, **existing methods do not model the causal structure among latent variables. They only “partially” model the latent noisy data generative process without identifiablity analysis.**

- Our aim is to enable deep neural networks to capture the connections of noise transitions among different instances automatically. Since once the connections are captured, the noise transition learned in some examples can be generalized to other examples. To achieve this, we propose a principled way to "fully" model the data generative process. **Our method also models the causal structure among latent variables. We also analyzed sufficient conditions required for identifiability.**

Here we would like to include more details. We have also included below in Appendix D to avoid confusion.

> Previous work helps the learning of classifiers by exploiting the information contained in the distribution of instances. To be specific, when the latent clean label $Y$ is a cause of the instance $\mathbf{X}$, the distribution of instance $p(\mathbf{X})$ will generally contain some information about the distribution of clean class posterior $p(Y|\mathbf{X})$. To exploit the information contained in the instances, CausalNL and InstanceGM use the generative model to model the generative relationship between the latent variable $\mathbf{Z}$ and the observed instance $\mathbf{X}$. However, they do not model the causal structure among the latent variable $\mathbf{Z}$. Therefore, their methods only “partially” model the noisy data generative process. These methods do not analyze the identifiability of the generative process.

> Our aim is to enable deep neural networks to capture the connections of noise transitions among different instances automatically. Since once the connections are captured, the noise transition learned in some examples can be generalized to other examples. To achieve this, we need to model the joint distribution of all variables $p(\mathbf{X},\tilde{Y},Y,\mathbf{Z},\mathbf{N})$. The noise transition can be obtained through

$$
p(\tilde{Y}|\mathbf{X},Y)=\frac{\int_{\mathbf{Z},\mathbf{N}}p(\mathbf{X},\tilde{Y},Y,\mathbf{Z},\mathbf{N})d\mathbf{Z}d\mathbf{N}}{\int_{\tilde{Y},\mathbf{Z},\mathbf{N}}p(\mathbf{X},\tilde{Y},Y,\mathbf{Z},\mathbf{N})d\tilde{Y}d\mathbf{Z}d\mathbf{N}}.
$$

> To obtain this joint distribution, we propose a principled way to model the whole data generative process. Specifically, we not only model the generative process from the latent variable $\mathbf{Z}$ to the instance $\mathbf{X}$ but also model and learn causal structure among different latent variables $\{Z_1,Z_2,\dots,Z_m\}$ from observed noisy data. 

To achieve this, the generation of the variable $\mathbf{N}$ is modeled by a Gaussian distribution. The generation of the latent variable $\mathbf{Z}$ is modeled by a Structure Causal Model (SCM): $Z_i:=\mathbf{W}^T_i \mathbf{Z}+N_i$, where $\mathbf{W}$ is a matrix used to represent the association among the latent variables. Two generators are used to model the generation of the instance $\mathbf{X}$ and the noisy label $\tilde{Y}$, respectively.

Moreover, our paper has analyzed the identifiability of the proposed method theoretically in Appendix C. We discuss the conditions required for identifiability. Specifically, when the number of distinct classes is large (more than the sufficient statistics of the latent variable $\mathbf{Z}$) and we have a confident example in each class, the latent variable $\mathbf{Z}$ can be identified up to permutation.

Reviewer comment (Experiments):
The primary focus of this study on low-resolution datasets such as MNIST and CIFAR-10 in experimental evaluations is possibly due to the inherent difficulties of generating high-resolution inputs. Although results are reported for the higher-resolution Clothing1M dataset, the lack of specific mention of standard deviations casts doubt on the reliability of these findings. Furthermore, the performance improvement over DivideMix is marginal, raising questions about the necessity of employing a deep generative model for noisy label classification and creating skepticism regarding the scalability and practical applicability of the proposed method for higher-resolution, real-world datasets.

Author response:
**Q3: The performance improvement over DivideMix is marginal, raising the question the necessity of employing a deep generative model for noisy label classification.**

**A3**: Performance improvement is dramatic when the noise rate is high. For example, when the noise rate is 50%, our method improves accuracy by 3.21% on FashionMNIST, 2.41% on CIFAR-10, and 4.58% on CIFAR-100. When the noise rate is low, the performance of baselines and the proposed method is relatively good. Thus, the improvement is marginal.

Reviewer comment (Experiments):
Incorporating a deep generative model in scenarios primarily focused on classification tasks can introduce a significant computational burden. The authors should compare the increased computational requirements of this additional modeling with existing baselines to provide a clearer perspective on its practical feasibility. Such a comparison is crucial for assessing the trade-offs between improved noise handling and increased computational demands, particularly for applications with limited resources or where efficiency is critical.

Author response:
**Q2: The deep generative model introduces a significant computational burden.**

**A2**: The proposed method does not increase the time complexity and memory because the encoder and the decoder introduced by our method are lightweight. The training time of the proposed method on CIFAR-10 is 13.68 hours, and the training time of DivideMix is 9.08 hours. The memory used for the proposed method is 7.06 GB, and the memory used for DivideMix is 6.99 GB.

Moreover, the inference time is the same as the time of baselines because the encoder and decoder can be discarded during inference.

Reviewer comment (Writing):
The use of "distillation" in the paper seems unclear, as it traditionally refers to transferring knowledge from a complex to a simpler model. If this process is not evident in the methodology, the term may be inaccurately applied. Using a simply trained network initially does not necessarily constitute distillation, which typically involves transferring knowledge from a more complex model. The authors should clarify or reconsider its use.

Author response:
**Q5: The word "distillation" traditionally refers to transferring knowledge from a complex to a simpler model.**

**A5**: Thank you for pointing out. To avoid confusion, we have changed the word to "selection" in the paper.

Reviewer comment (Writing):
In the context of noisy label classification, it is not typical to categorize noise types as "Worst," "Aggregate," "Random 1," "Random 2," and "Random 3" without clear definitions. These terms are not standard in the literature and require clarification for proper understanding.

Author response:
**Q6: It's not typical to specifically categorize noise types as noise types as "Worst," "Aggregate," "Random 1," "Random 2," and "Random 3".**

**A6**: These noise types are not categorized by us but by the original paper [1]. Their definition can be found in the original paper [3]. Specifically, "Random $i$" ($i\in\{1,2,3\}$) represents the $i$-th submitted label, "Aggregate" represents the aggregation of three noisy labels by majority voting, and "Worst" represents the dataset with the highest noise rate.

### References

[1] Wei, Jiaheng, et al. "Learning with noisy labels revisited: A study using real-world human annotations." *arXiv preprint arXiv:2110.12088* (2021).

Reviewer comment (Experiments):
Training a Deep Generative Model (DGM) for this task may impose a significant computational burden. The authors should consider whether the methodology could benefit from incorporating pre-trained DGMs to alleviate these computational demands.

Author response:
**Q4: Could the methodology benefit from incorporating pre-trained DGMs to alleviate these computational demands?**

**A4**: Thank you for the interesting question. No, our goal cannot be achieved if we use pre-trained DGMs. Our goal is to enable deep networks to understand the connections of noise transitions among different instances automatically (please refer to the answer in A1). To capture the connections, we need to obtain the joint distribution of all variables $p(\mathbf{X},\tilde{Y},Y,\mathbf{Z},\mathbf{N})$. 

To achieve this, the generation of the variable $\mathbf{N}$ is modeled by a Gaussian distribution. The generation of the latent variable $\mathbf{Z}$ is modeled by a Structure Causal Model (SCM): $Z_i:=\mathcal{W}^T_i \mathcal{Z}+N_i$, where $\mathcal{W}$ is a matrix used to represent the association among the latent variables. Two generators are used to model the generation of the instance $\mathbf{X}$ and the noisy label $\tilde{Y}$, respectively.

Moreover, our paper has analyzed the identifiability of the proposed method theoretically in Appendix C. We discuss the conditions required for identifiability. Specifically, when the number of distinct classes is large (more than the sufficient statistics of the latent variable $\mathbf{Z}$) and we have a confident example in each class, the latent variable $\mathbf{Z}$ can be identified up to permutation. In contrast, existing DGMs can not provide a theoretical guarantee for inferred latent variables. The accurate joint distribution $p(\mathbf{X},\tilde{Y},Y,\mathbf{Z},\mathbf{N})$ may not be obtained. Without the accurate joint distribution $p(\mathbf{X},\tilde{Y},Y,\mathbf{Z},\mathbf{N})$, the model can not capture the connections of noise transitions among different instances. Therefore, we can not use pre-trained DGMs.

Reviewer comment (Novelty):
There is a lack of discussion regarding related work on label-noise learning. While they use the clean example distillation method via the small-loss trick, it is essential to also survey this related work, such as sample selection methods, for a more comprehensive view of the field.

Author response:
**Q2: Need to discuss more related work for learning with noisy labels.**

**A2**: Thanks for your advice. We have added a paragraph in related work after introducing the noise-transition-based method to introduce other approaches, including sample-selection-based methods, other methods using generative models to facilitate learning with noisy labels, and the method leveraging the property of the label noise. Specifically, the paragraph is as follows.

> Some noise-robust algorithms select examples deemed likely to be accurate for training purposes. These selections are based on the memorization effect, which suggests deep neural networks initially learn dominant patterns before progressively learning less common ones. In noisy label environments, accurate labels often constitute the majority, leading networks to prioritize learning from examples with accurate labels, typically indicated by lower loss values. Co-Teaching employs this principle to identify low-loss examples as likely accurate. DivideMix uses a Gaussian Mixture Model to separate training examples into labeled and unlabeled sets based on their training loss, with the labeled set presumed to contain accurate labels. Additionally, some methods use generative models to facilitate learning with noisy labels. CausalNL and InstanceGM utilize instance-specific information to enhance classifier learning. Conversely, NPC focuses on the generative process of estimated clean labels, not the noisy data generation, using generative models for label calibration. Finally, SOP applies the sparse property of the label noise, i.e., incorrect labels are the minority, to prevent models from overfitting to label noise.

Reviewer comment (Experiments):
Some results, such as those for CIFAR-10 and CIFAR-10N, do not appear to be statistically significant, and the performance gain observed for Clothing1M is marginal. To further support the effectiveness of the proposed model, it would be beneficial to include experiments on various real-world datasets such as WebVision, ANIMAL-10N, and Mini-Imagenet, or to conduct repeated experiments on Clothing-1M.

Author response:
**Q3: Include experiments on various real-world datasets such as WebVision, ANIMAL-10N, and Mini-Imagenet.**

**A3:** Thanks for your advice. We are still running experiments on WebVision. We will post the results as soon as possible.

Reviewer comment (Experiments):
The paper primarily presents the classification performance without conducting an in-depth analysis of the contributing factors behind the performance improvements. A more detailed examination of specific elements that have positively influenced the overall performance is needed. This could include: 1) an ablation study comparing model training with only $L_{semi}$ and with the entire loss, 2) a sensitivity analysis regarding hyperparameters (e.g., $\lambda_{ELBO}$ and $\lambda_M$), 3) a comparison between end-to-end learning and alternative learning approaches, and 4) an ablation study based on the number of latent variables, among other potential analyses.

Author response:
**Q4: An ablation study and sensitivity analysis.**

**A4:** Thanks for your advice. For the ablation study, the method with only $\mathcal{L}\_{semi}$ is the same as DivideMix. Thus, the experiment results of the model only trained with $L_{semi}$ are the experiment results of DivideMix. For the sensitivity analysis, we have conducted the sensitivity analysis for $\lambda_{ELBO}$ and $\lambda_M$ on the CIFAR-10N dataset with noise type ''worst''. The experiment results are plotted as figures in Appendix E. We also list the experiment results as tables here:

| $\lambda_{ELBO}$ | 0.1              | 0.05             | 0.01                 | 0.001            | 0.0001           |
| ---------------- | ---------------- | ---------------- | -------------------- | ---------------- | ---------------- |
| Accuracy         | 93.00 $\pm$ 0.20 | 93.86 $\pm$ 0.10 | **93.87 $\pm$ 0.11** | 93.56 $\pm$ 0.15 | 93.59 $\pm$ 0.11 |

| $\lambda_{M}$ | 0.2              | 0.1              | 0.01                 | 0.001            | 0.0001           |
| ------------- | ---------------- | ---------------- | -------------------- | ---------------- | ---------------- |
| Accuracy      | 93.11 $\pm$ 0.21 | 93.44 $\pm$ 0.22 | **93.80 $\pm$ 0.13** | 93.53 $\pm$ 0.14 | 93.23 $\pm$ 0.17 |