Revealing Distribution Discrepancy by Sampling Transfer in Unlabeled Data

We appreciate your valuable feedback and suggestions on the limitations and potential improvements of our method. Here are our responses.

W1. General Feasibility

The estimation of the density ratio and likelihood ratio does not limit the applicability of our method. We transform the issue of unknown test sample labels into a problem of estimating these ratios, which are both straightforward to estimate. The density ratio can be estimated using a lightweight adaptor and Eq. (13) within any given hypothesis. The likelihood ratio can be derived analytically by minimizing the upper bound of the generalization error, as shown in Eq. (17), or it can be simplified to a constant value, along with the following experiments.

Specifically, in practical applications, we can simplify the algorithm using only the density ratio and setting the likelihood ratio to a constant value. As discussed in Line 188 of the paper, setting the likelihood ratio to a constant (e.g., 1) is equivalent to assuming a covariate shift in the data. In contrast, using the adaptive likelihood ratio estimation in Eq. (17) assumes a stronger semantic shift.

To clarify, we add corresponding experiments on comparing the performance before and after simplifying the likelihood ratio, and test them using ResNet18 trained on CIFAR10, with test datasets including SVHN, SEMEION, CIFAR10.1, CIFAR10 corrupted by Gaussian noise and adversarial samples (perturbation factor 0.01). The results show little difference in performance. For instance, both setups clearly distinguish between SVHN and CIFAR10. However, with the more accurate likelihood ratio estimation from Eq. (17), I-Div aligns more closely with human prior knowledge, such as recognizing smaller differences between CIFAR10, CIFAR10.1, and adversarial samples. These results demonstrate that even with a simplified likelihood ratio, our method remains effective and practical for various applications. The density ratio and likelihood ratio estimations are both robust and easy to implement, ensuring the general feasibility of our method in real-world scenarios.

Q1. Optimization Process

Your suggestion aligns well with our current direction, and we indeed have been considering similar approaches.

The current algorithm addresses the challenge of unknown class labels in the test samples using density and likelihood ratios for sampling transfer, as detailed in Eq. (4). This approach makes computational effort on estimating these ratios using an adaptor. Our preliminary idea is to treat both training and test samples as originating from an unknown labeling hypothesis. While we cannot know this labeling hypothesis, we do know the ground truth labels for the training data. We can explore the hypothesis space where this labeling hypothesis resides to derive an upper bound for Eq. (4). By applying the Legendre-Fenchel duality, we can simplify this upper bound and then minimize it to estimate the discrepancy between the two distributions. We would be happy to discuss any suggestions or thoughts you might have on this approach.

We appreciate the constructive suggestions and are committed to incorporating these insights into our future research to improve the applicability and efficiency. Thank you for your thoughtful review and guidance.

We appreciate your valuable feedback and constructive criticisms. Here are our responses to your identified questions.

Q1&2. Baselines and Backbones

The paper has already considered post-2021 algorithms, such as NNBD (ICML’21) [R1] and R-Div (NeurIPS’23) [R2] shown in Tables 1, 2, 4 and 5. To further address your concerns, we add H-Div (ICLR’22) [R3] and utilized updated network architectures (ResNet50, ViT, and CLIP) on diverse datasets (ImageNet, OID, Places365, StanfordCars, Caltech101, and DTD) to revise the paper. The additional experimental results are shown below.

It is important to note that the training data used for the pre-trained CLIP model is not accessible. According to Figure 4 in the CLIP paper [R4], CLIP achieves strong zero-shot performance on ImageNet. We, therefore, consider this dataset as part of the training data to estimate the differences between this dataset and the other test datasets.

The experimental results show that the AUROC values of I-Div better reflect the semantic similarity between datasets and ImageNet, aligning with human intuition. Specifically, ImageNet is more similar to OID and Places365 due to their shared everyday objects and scenes, leading to lower AUROC values of I-Div. Conversely, ImageNet is quite different from Caltech101 and DTD, which focus on specific object categories and textures, resulting in higher AUROC values of I-Div. However, the other algorithms show higher AUROC values across most datasets, indicating they struggle to capture the nuanced semantic and visual similarities between ImageNet and more diverse datasets. This results in less accurate distinctions compared to the I-Div algorithm, which aligns better with human intuition.

Q3. Performance of I-DIV

It appears there may have been a misunderstanding regarding the objective of our algorithm. Our goal is for the distribution discrepancy to align with human prior knowledge. As demonstrated in Tables 1 and 2, the effectiveness of the algorithm should be evaluated within the context of specific data. Achieving 100% distinction is not always feasible and does not necessarily indicate superior performance.

In cases where I-Div achieves 100% distinction, it suggests that, according to a given pre-trained network, the training and test data are certainly not from the same distribution. While we agree that achieving 100% classification accuracy in general is unrealistic, if the goal is to simply determine whether data samples come from the same distribution and have clear semantic differences, then achieving 100% distinction can be straightforward.

Specifically, in Table 1, the datasets show clear semantic differences, making it normal and straightforward for algorithms to achieve 100% distinction. This result is not due to model overfitting but rather reflects the inherent separability of the data categories. When datasets have distinct semantic boundaries, algorithms can easily detect these differences, especially with large dataset sizes that provide ample data for learning and differentiation. This phenomenon is consistent with the findings in the literature [R1-R3], where similar distinctions were observed, demonstrating that such results are typical in cases of clearly defined semantic classes. The clarity and volume of the data naturally lead to high accuracy in distinguishing between categories, confirming that the observed distinctions are a standard outcome under these conditions.

Conversely, in Table 2, which involves semantically similar datasets, I-Div achieves around 50% instead of 100% accuracy. This indicates that I-Div struggles to distinguish these samples, which is more intuitive. For example, the PACS dataset includes four domains with different styles, but consistent class semantics. Other algorithms like R-Div and MSP can distinctly separate all datasets, possibly because they do not rely on a pre-trained network and continually seek out differences during training. However, I-Div measures the discrepancy between two datasets based on a pre-trained network, considering the network predictive ability on the test data. If the network performs well on the test data, I-Div finds it challenging to distinguish between training and test data, aligning with our intended outcome.

[R1] Kato et al., Non-negative bregman divergence minimization for deep direct density ratio estimation, ICML, 2021.

[R2] Zhao et al, R-divergence for estimating model-oriented distribution discrepancy, NeurIPS, 2023.

[R3] Zhao et al., Comparing distributions by measuring differences that affect decision making, ICLR, 2022.

[R4] Radford et al., Learning transferable visual models from natural language supervision, ICML, 2021.

We appreciate your detailed feedback and thoughtful questions. Here are our responses to your raised concerns.

W1. Likelihood Ratio

It is impossible to completely solve this problem because the class labels of the test samples are unknown. However, we theoretically decompose this problem to identify solvable aspects and necessary assumptions. The density ratio can be estimated using Eq. (13) with a lightweight adapter by freezing the pre-trained network, while the likelihood ratio can be derived analytically under the weak assumption of minimizing the generalization error, as shown in Eq. (17). We conduct additional experiments on more complex data (ImageNet, OID, DTD, etc.) and networks (ResNet50, ViT, and CLIP) to demonstrate the effectiveness. Please refer to Table R1 in the PDF within the global response, or the table in our response to Reviewer fpGs.

We don't even need to estimate very accurately. Under a stronger covariate shift assumption, where all sample likelihood ratios are set to 1, reasonable results can still be achieved in practice, as shown in the experiments on CIFAR10 below.

W2. Computational Complexity

Estimating the density and likelihood ratios is actually very fast. This is because the density ratio can be estimated using a lightweight adapter, and the likelihood ratio can be quickly derived analytically. We add the following throughput experiments on large data and complex models to validate this.

For the density ratio in Eq. (9), the primary training time is spent on the adaptor for a given pre-trained network. The main bottleneck is the feature dimensions, which is the input size of the adaptor and the output size of the pre-trained network. The results below show the throughput for different dimensions across various pre-trained network architectures. ResNet18 has an input dimension of 28x28, while ViT-B/16 has 224x224. This demonstrates the efficiency for large datasets and complex models, as only the downstream adaptor is trained while the pre-trained network remains frozen. For the likelihood ratio, we can directly obtain its analytical solution using Eq. (17), making its time complexity minimal.

W3. Limited Scope

To address your concern, we add ResNet50 and ViT trained on ImageNet and CLIP to estimate the differences between this dataset and OID, Places365, StandfordCars, Caltech101, and DTD. For the complete results, please refer to Table R1 in the PDF within the global response. A portion of the experiments with ResNet50 is shown below. The outcomes align with human intuition.

Q1. Sample Quality

I-Div is highly stable and fast, and the experimental results in the paper are averaged over 100 runs. The experimental variance on CIFAR10 is shown in the table below. The lightweight adapter used to estimate the density ratio requires only a small number of samples, and the subsequent likelihood ratio can be directly derived analytically according to Eq. (17). Additionally, as shown in Figure 3, processing $M=1000$ samples on a single GPU takes only 7.09 seconds.

According to the experimental results in the table, if there is a significant difference between the training and test data distributions, the algorithm can easily identify the differences and converge quickly. When the test data is semantically similar to the training data, the stability is lower, but it improves rapidly as the number of samples increases. For more discussion on efficiency, please refer to our response to your W2.

Q2. Extreme Cases

In fact, our implementation already considers the issue of extreme values. Our core strategy is to ensure that the density ratio remains within a reasonable range by using appropriate activation functions and subsequently applying a proximal algorithm to keep the likelihood ratio within our defined interval. Additionally, we explain that the sample size does not affect the range of the likelihood. We also add experimental explanations in the table below regarding the impact of the number of samples $M$ on data with many class labels.

Specifically, in constructing the adapter (Eq. (9)), we introduce Softplus and GELU to ensure that the density ratio stays within a reasonable range (experimentally within (0.5, 5)). According to Eq. (17), the likelihood ratio depends on the density ratio and can be analytically solved using this formula. In practice, we project the final values to ensure they fall within our predefined interval, which is set to (0.5, 5) in our experiments.

Regarding the large data issue, the $M$ in Eq. (17) cancels out in the denominator of the second term, so the number of samples does not affect the likelihood value. Also, the number of class labels does not impact the likelihood value since the formula does not include them. Semantically different datasets have minimal bias and can be distinguished. However, semantically similar datasets may be misclassified as different distributions, presenting a lower AUROC, indicating that I-Div cannot distinguish between two subsets from CIFAR100.

Additionally, I suggest the authors make the code used in all experiments publicly available (if accepted).

Thank you for constructive suggestions. We appreciate the opportunity to address your concerns.

W1. Impact of Pre-trained Classifier

As discussed in Line 24, our goal is to measure the discrepancy between training and test datasets for a given pre-trained classifier, thereby assessing its applicability on the test data. Therefore, our work does not amplify errors. Rather, as shown in Eqs. (5) and (18), we use density and likelihood ratios to indirectly estimate and magnify the discrepancy.

However, different classifiers have varying generalization abilities, which affect their judgment on this matter. For instance, a classifier with strong generalization might consider samples from different domains as coming from the same distribution, while a classifier with weak generalization would not. Therefore, different pre-trained classifiers will result in different density and likelihood ratios, but this is expected. This can help us bypass the issue of unknown test sample class labels while amplifying the sensitivity to the distribution discrepancy.

W2. Consistent Setting

We made every effort to ensure consistency in the experimental setup. When conducting comparative experiments, we used the same network structure for all algorithms. Additionally, we provide the performance of the comparative algorithms with and without using class labels, as shown in the table below.

Specifically, in Tables 1, 4, and 5, the different algorithms all use ResNet18 as the backbone, while in Table 2, they use AlexNet. The following experiments on CIFAR10 show that I-Div achieves better performance in both settings. This is mainly because it is based on a pre-trained classifier and combined with the sampling transfer, indirectly estimating the network applicability on test samples. In contrast, the comparative algorithms directly measure divergence without considering the impact of the pre-trained classifier, leading to results that do not align with intuitive expectations.

Q1. Derivation of Eq. (16)

Theorem 3.2 derives an upper bound on the generalization error for distribution discrepancy w.r.t. the likelihood ratio. To minimize the generalization error bound, we extract the two terms related to the likelihood ratio and use $\gamma$ to balance them. With the constraints revealed by Eq. (15), we then derive Eq. (16).

Q2. Sample Size

In Figure 3, the first column represents 100 samples, but the overall performance is worse than that with 500 samples. The values of CIFAR10.1 and STL10 (where lower values are better), which are semantically related to CIFAR10, are similar for 100 and 500 samples and decrease with increasing the sample size. However, for other datasets unrelated to CIFAR10 (where higher values are better), the AUROC and AUPR values are higher with 500 samples.

Q3. Data Swap and Challenging Data

This is indeed a very interesting idea. In the revised version, we add experiments with data swapping and challenging datasets. By swapping the data, the network trains on noisy data and tests on clean data. Results show that higher noise levels reduce the ability to distinguish between datasets, aligning with our intuition. For experiments with more challenging data (ImageNet, OID, DTD, etc.) on complex models (ResNet50, ViT, and CLIP), please refer to Table R1 in the PDF within the global response, or the table in our response to Reviewer fpGs.

Q4. Intriguing Experiments

I-Div aligns more closely with human intuition, as one also struggles to differentiate adversarial samples. I-Div, based on a pre-trained network without adversarial defense, perceives adversarial samples as ``normal" and assigns them with high confidence scores, making it hard to distinguish between original and adversarial samples. Other algorithms detect subtle differences, concluding these samples are from different distributions.

Q5. Hyperparameter Analysis

Per your suggestion, in the revised version, we add parameter analysis for $\lambda$ and $\gamma$ using ResNet18 and CIFAR10. The results are shown below. The Lagrange coefficient $\lambda$ stabilizes beyond a certain value, as explained in Appendix B. The algorithm is not very sensitive to $\gamma$, with slightly better results at larger values, consistent with Theorem 3.2. Based on these findings, both parameters are set to 1 by default, as noted in Appendix D1.

L1. Measurement Differences

As discussed in Appendix A1, I-Div and R-Div are fundamentally different. R-Div measures the difference between training and test datasets using hypotheses from mixed data, requiring known test labels. I-Div measures this difference for a given hypothesis using density and likelihood ratios, even without test labels.

While it is impossible to address unknown test labels without any assumptions, our algorithm easily mitigates this by estimating density and likelihood ratios. In Eq. (13), the density ratio is straightforward to estimate using a lightweight adaptor. The likelihood ratio, derived analytically by minimizing the generalization error bound as shown in Eq. (17), can be estimated with a weak assumption. Even if the likelihood ratio is set to 1, as shown in the CIFAR10 experiments below, our algorithm remains effective.

Thank you for your insightful questions. We have carefully considered your comments and would like to provide the following responses:

Noisy Data:

There is indeed a turning point as you suggested. As shown in the table below, this turning point appears at a higher noise rate, such as 0.8.

We previously only presented results for the 0.3 to 0.7 noise range to highlight this interesting finding. In this range, it seems that our algorithm is not very effective at distinguishing between the training and test samples. However, this aligns with our intuition because even with these noise levels, the model still maintains a certain level of generalization to the test data. Specifically, since we added noise to the original data, this can be considered a form of data augmentation, where a certain amount of noise might even enhance generalization. The proposed I-Div relies on a pre-trained classification, which is closely tied to the model generalization ability.

When the noise level is significantly higher, such as at 0.8, the situation changes. At this point, the noise overwhelms the original data, making it difficult for the model to generalize. The noise introduces large variations that no longer match the true data distribution, leading to a clear separation between the training and test sets.

We are truly grateful for you pointing out this interesting aspect. We will continue to delve into the relationship between model generalization and data discrepancy in future research, especially in the context of noisy data.

Error Propagation:

$\widehat{r}$ and $\widehat{v}$ do indeed rely on the pre-trained classifier. I-Div is designed to measure the difference between training and test data for any given pre-trained network, which in turn helps assess its applicability on test data. This does not involve the type of error propagation typically seen in non-end-to-end training. If I-Div shows a significant discrepancy between two semantically similar datasets when applied to a given pre-trained classifier, this is not due to error propagation. Rather, it indicates the weak generalization ability of the classifier itself, meaning the classifier struggles to generalize well between those two datasets.

To further clarify, you can think of $\widehat{r}$ and $\widehat{v}$ as tools for analyzing the sensitivity of a given network to different data, similar to using a detector to identify out-of-distribution samples for a specific network. Therefore, the nature of the pre-trained classifier, whether it is highly accurate, somewhat flawed, or even based on randomly generated data, does not affect the validity of our approach. I-Div effectively evaluates the model generalization ability regardless of the specific characteristics of the pre-trained classifier. Even with a classifier trained on random data, our method can still measure how well the model generalizes across different datasets.

That said, in our experiments, we used a network trained on the training dataset. While this network generalization ability may not be exceptionally strong, it does possess some degree of generalization, allowing us to measure the differences between its training and test datasets. Naturally, different training methods and network architectures will result in different pre-trained classifiers due to varying levels of generalization ability. Consequently, this will also lead to different data discrepancy measurement results, as demonstrated in our Table 3.

Adversarial Samples:

The robustness of I-Div compared to R-Div is not simply due to differences in label settings. The following table highlights that the additional experiments provide further clarification on your point

Regardless of whether labels are present, many current methods, including R-Div, tend to maximize the differences between data (based on our experimental observations). These methods assume that any difference in data implies different distributions. This behavior persists even in labeled settings, where the methods detect even the slightest variations in input data and conclude they belong to different distributions.

However, I-Div does not fall into the trap of detecting these minor differences. Instead, it evaluates data discrepancies based on the model generalization ability. When given adversarial samples, I-Div focuses on the overall information that the model considers for generalization. If the model deems it should generalize to these samples, the perceived difference between the datasets becomes smaller.

Thank you for your feedback. If you have any more questions or concerns, feel free to reach out.