Rethinking the OoD Generalization for Deep Neural Network: A Frequency Domain Perspective

W1:

Thank you very much for your valuable suggestion about writing. We will make revisions and improve the paper further.

W2:

Via frequency-based analysis of two shifts, we empirically explain why the performance of some algorithms deteriorates rapidly in OoD environments and why some algorithms outperform others in one type of domain shift while fail in another one. Based on these analysis, we further propose our CFA.

By comparing the performance of different methods on two shifts, we can highlight the advantages of our CFA.

W3:

Shapley values of different frequency components is stored as two-dimensional matrix, as it can be seen from the heatmap of Figure 4. After the Fourier transform, the center point is the zero frequency component, and the frequency gradually increases from the center point to all directions. In order to process this two-dimensional information into one-dimensional information, we sum the two-dimensional shapley values ringly and each ring represents a frequency point.

The added and subtracted frequency components are derived from an average over training images and CFA is only used in the training process only. CFA helps the network learn more domain-generalizable features in the training stage. In testing stage, we use original testing images for fair comparison.

W4:

To make a strict ablation study, we run each experiment once with fixed random seed. In Table 2, we rerun each experiment with random seed for three times independently. We report the average of three runs, and its corresponding standard error.

W5:

The proof of nonlinearity is too complex to achieve, and the linear model can partially prove CFA's effectiveness.

W6:

Thank You for the reminder. $F$ means Discrete Fourier Transform. $f$ means the output of model. For classification task, it means the probabilities.

Q1 & Q2:

Sorry for the confusion, we will improve the writing further.

Q3:

The main purpose of listing the properties of the Shapley value here is to demonstrate that it is reasonable to use the Shapley value to analyze the importance of different frequency components. The properties of the Shapley value has been proved in work[1].

Missingness: The Shapley value of player i is zero, if ∀S ⊆ N, V (S ∪ {i}) = V (S). [1] L. S. Shapley. A value for n-person games. In Contributions to the Theory of Games, page 307–317, 1953.

In the third contribution bullet point, we will rephrase like "We conduct experiments to verify the effectiveness of our method on five baseline OoD algorithms on both diversity shift and correlation shift."

As for the acronyms, ERM first appears in the abstract and we have defined for ERM. For other acronyms, we will give the definition for clarity.

Q4: In equation 2, do you use the same random subset m in all the experiments, or is it chosen randomly each time?

It is chosen randomly each time.

Q5: Around equations 3 and 4, it would be good to explicitly define u and v as spatial coordinates and m and n as frequency coordinates.

Thanks for your advice. It's better to give more explaination of the equations.

Q6: It would be good to give some more description of the datasets that were used in the experiments, particularly how many train and test images, how large each image is, and how many classes there are.

Our implementation of experiments is based on the OoD-bench and DomainBed suits, which are open-source and details like image resolution, training setup for different datasets are fixed. Thus, we do not give more details in the paper. We will provide more experiment details in the supplementary.

[1] Nanyang Ye, Kaican Li, Haoyue Bai, Runpeng Yu, Lanqing Hong, Fengwei Zhou, Zhenguo Li, and Jun Zhu. Ood-bench: Quantifying and understanding two dimensions of out-of-distribution generalization, 2022.

[2] Ishaan Gulrajani and David Lopez-Paz. In search of lost domain generalization. In International Conference on Learning Representations, 2021.

https://github.com/facebookresearch/DomainBed

Q7: Is the “modified image” in Figure 6 an actual result of the inverse Fourier Transform, or just an illustration? It looks like just the edges of the input image, which I doubt would appear naturally as a result of the CFA method, but it would be good to specify if this is just an illustration (or even better to show an actual image resulting from the method).

The “modified image” is an illustration, which we use to emphasize our CFA pay more attention to the structure information. As in the visualization of modified image of its positive frequency bands, we can find more shape or structure information of the objects, which are more domain-invariant features.

W1: I have some doubts about the experimental results, it seems that results are reported (e.g. for Colored MNIST) only for a certain degree of correlation/ratio between color and digit. How does the proposed method behave in under different degrees of correlation (as in [1])

Colored MNIST is transforming from [1] as follows: [1] defines three environments (two training, one test) ; first, assign a preliminary binary label y˜ to the image based on the digit: y˜ = 0 for digits 0-4 and y˜ = 1 for 5-9. Second, obtain the final label y by flipping y˜ with probability 0.25. Third, sample the color id z by flipping y with probability p, where p is 0.2 in the first environment, 0.1 in the second, and 0.9 in the test one. Finally, color the image red if z = 1 or green if z = 0.

[1] Martin Arjovsky, Léon Bottou, Ishaan Gulrajani, and David Lopez-Paz. Invariant risk minimization, 2020.

W2: Related to the point above, the explanation about the nature of the shift in the different dataset is lacking. E.g. for MNIST, what ratio/correlation was used? For CelebA which attributes were considered? Etc.

According to [2], the OoD distribution shift can be divided into two patterns: diversity shift and correlation shift, referring to different reasons for domain shift. Diversity shift is caused by the appearance of new unseen environment features in the test set while correlation shift delineates the spurious correlation between environment features and labels.

For CelebA, [2 ]treats “hair color” as the classification target and “gender” as the spurious attribute.

For NICO, consisting of real-world photos of animals and vehicles captured in a wide range of contexts such as “in water”, “on snow” and “flying”. There are 9 or 10 different contexts for each class of animal and vehicle. This dataset simulates a scenario where animals and vehicles are spuriously correlated with different contexts.

[2] Nanyang Ye, Kaican Li, Haoyue Bai, Runpeng Yu, Lanqing Hong, Fengwei Zhou, Zhenguo Li, and Jun Zhu. Ood-bench: Quantifying and understanding two dimensions of out-of-distribution generalization, 2022.

W3: Comparison or references to relevant work in the debiasing/generalization fields are missing; e.g. [2,3,4]

We are reproducing LfF + BE proposed in [4] on CMNIST used in our paper for fair comparsion. But we find that the CMNIST used in OoD generalization fields is quiet different from that in debiasing fields. In OoD generalization, in our CFA, we only use object category as supervised information without extra arribute information as supervision. However, in [4], it seems that extra arribute information is needed in the training.

Q1: I think that your method could provide some sort of "explanability" in terms of visual interpretation of certain frequencies. Could you add some examples e.g. for ColoredMNIST or CelebA showing the reconstructed images for the most important frequency (both positive and negative)?

Thanks for your advice. In Figure 4, we reconstructed images of PACS. As it can be seen, the most important positive frequency is about shape or contour of the object, while the negative one is about the color of the object.

Q2: How the hyperparameters $\alpha$ and $\beta$ chosen? How robust is your method to changes in these values?

In the supplementary, we have evaluated the parameter sensitivity of CFA by changing one parameter and fixing the other. U-shaped parameter sensitivity curves are shown in Figure 9. From this figure, we can observe that suitable values of $\alpha$ and $\beta$ help to improve the generalization performances.

Q3: I suggest authors change the line "We introduce Shapley value" in the introduction as it seems to suggest that this paper proposes Shapley value rather then their novel application.

Thanks for your advice. To better and clear express, we rephrase this sentence as "We explore Shapley value".

W1:

[1] uses Shapley values to analyse the importance of frequency components to adversarial robustness, while we extend Shapley values to OoD generalization. [1] and our method focus on different research areas, but we both resort to Shapley values to analyse in frequency domain.

[5] finds that neural networks can exhibit different frequency biases, tending to adopt frequency shortcuts based on data characteristics because of simplicity-bias learning.They foresee that avoiding frequency shortcut learning may hold promise in improving generalization. But they do not give a concrete method to improve generalization yet. In our method, we precisely attribute which frequency components of training images are important for generalization via Shapley values and propose CFA to improve generalization.

W2:

Thanks for providing us more profound methods of the approximation of Shapley values. Intuitively, if we can attribute which frequency components of training images are important for generalization more precisely, CFA can have better performance in OoD generalization. We will try to use the improved approximation of Shapley values in [3] to refine our CFA.

W3:

In our method, the first step is to calculate Shapley values of existing Ood algorithms like ERM,IRM. Then, based on calculated Shapley values, we carry out CFA to improve corresponding OoD algorithms.

W4:

In Table 2, DFF proposed by Lin et al. (new SOTA in CVPR, 2023) is to modulate the frequency components to learn domain-generalizable features, which is the most similar one to our CFA. Thus, we mainly compare our CFA with DFF cross seven datasets.

W5:

Our implementation of experiments is based on the OoD-bench and DomainBed suits, which are open-source and details like image resolution, training setup for different datasets are fixed. Thus, we do not give more details in the paper. For readers nor familar with OoD-bench or DomainBed, we will provide more experiment details in the supplementary.

[1] Ishaan Gulrajani and David Lopez-Paz. In search of lost domain generalization. In International Conference on Learning Representations, 2021.

https://github.com/facebookresearch/DomainBed

As for the details of calculation of Shapley values, we should supplement further in the paper. For image resolution of 224, we divide the image to $16 \times 16$ patches in the frequency domain and sample 1000 times for each sample.

W6:

We implement more experiments with Vit on two datasets. As it can be seen, CFA stills have impressive performance. Experiments on other datasets are still running.

As for large-scale datasets, we alreary experiment on DomainNet, which is a dataset of common objects in six different domains and 345 categories (classes). Domainet has around 600,000 images, while ImageNet-R has 200 ImageNet classes resulting in 30,000 images. Thus, we believe that experiments on DomainNet is quite convincing.

W7:

We randomly sample m=1000 of all permutations for each data sample. N means all the permutations of image patches, while T means ramdomly remain some frequency components to calculate Shapley values.

Q1 is similar to W1, please see the response to W1.

Q2:

"Model-agnostic" is used to describe a class of algorithms that do not make any assumptions about the underlying model. The model-agnostic algorithms view the model as a black box and focus on understanding the relationship between the input and the output of model. Our CFA can be directly adopted to any model without modifying the structure of the network. In addition, the calculation of Shapley values in our paper also meets the definition of "model-agnostic", as it focuses on understanding the relationship between the input and the output of model without attention to the internal model.

Q3:

In OoD algorithms, 224 is a common-used resolution. For fair comparisons with other OoD lgorithms, we do not experiment on different resolutions.

Q4:

According to Castro et al [2], with the increasing of sampling times m, the average error for the estimation of Shapley value is decreasing. But considering the time complexity analysis below, we set m=1000 to better balance efficiency and stability.

Q5 is similar to W6, please see the response to W6.