Paper: Dataset Distillation via Adversarial Prediction Matching (ICLR-2024)

Reviewer comment (Novelty):
The adersarial prediction matching seems to be close to DD [1]. Could the authors summarize the differences and improvements compared to DD?

Author response:
Since the reviewer did not provide a detailed explanation of why our method is close to DD, we attempt to **summarize the differences between our method and DD, as well as our improvements**, from **three key aspects**.

First, **fundamentally, our problem formulation for solving the dataset distillation task differs from DD's proposed formulation**. Specifically, DD formulates the dataset distillation task as a bi-level optimization problem, aiming to minimize the generalization error on the original data caused by models trained on the distilled data. In contrast, we introduce an innovative imitation-based surrogate objective. This objective optimizes distilled data to enable proxy student models trained on it to directly emulate the predictions of proxy teacher models, trained on the original dataset, on the authentic data distributions.

Second, **the adversarial strategy we introduced for effectively optimizing the formulated prediction-matching objective provides a novel intuition for synthetic samples, distinct from existing methods**. Specifically, the objective defined by Equation (3) encourages synthetic samples to capture informative features recognized by the teacher but absent in the current synthetic samples. This reduces the gap in contained features between the original and the distilled datasets.

Third, **our optimization for the synthetic data relies on optimizing a single-level loss function defined in Equation (4) with remarkably low memory complexity**. In contrast, even if DD truncates the inner optimization of training models on the distilled data to only one-step gradient descent, it still requires a complex nested optimization when calculating the gradient for updating synthetic data.

Finally, it is important to note that we did not employ DD as a baseline method in our paper. This decision was based on **the significant performance gap between DD and our method**, as well as **the high computational and memory complexity of DD, which limits its applicability in various scenarios we evaluated**.

Reviewer comment (Theory):
If the loss function in line 6 of algorithm 1 is based on batch, then why is the memory complexity in section 3.3 only the graph size? The batch size should be counted as well.

Author response:
We appreciate your meticulous attention to detail and the opportunity to underscore one of our key contributions: a significant reduction in memory complexity compared to TESLA.

As outlined in the first paragraph of Section 3.3, we explicitly highlighted that **all computational operations performed on each synthetic sample** in our loss function, as defined by Equation (4), **are consolidated through addition**. This unique characteristic allows us to **independently compute the gradient of the loss with respect to each sample, $u_i$, without interference from other samples $u_{j \neq i}$**. In other words, **we can calculate the loss independently for each synthetic sample and update it using the derived gradient**. This explains why we do not need to factor in the batch size $B$ when considering our memory complexity. It also elucidates why our method offers a flexible trade-off between memory and runtime.

In contrast, TESLA and all existing variations of MTT are unable to avoid the batch size in their memory complexity. This is because the **training trajectories are determined by batches of synthetic samples**, leading to the loss and updating gradient being inseparable for a single synthetic sample.

Reviewer comment (Experiments):
The authors should have more experiments on NAS. Because NAS is a practical application of dataset distillation, the experiments on Table 3 are not sufficient.

Author response:
Thank you for your valuable suggestion. **In response, we conducted additional experiments for NAS (Neural Architecture Search) on CIFAR-100 and Tiny-ImageNet**. We randomly selected three groups of 100 networks from NAS-Bench-201 for these experiments. Specifically, we compared our methods with other baselines by training the 100 different architectural models on synthetic datasets with an IPC of 10 for 200 epochs. The table below presents the averaged Spearman's correlations between the rankings obtained from the synthetic datasets and the ground-truth rankings. The results clearly demonstrate that **our method continues to outperform other baselines on both evaluated NAS tasks**. We will include these results in our final print.

|        | Random    | DM        | FrePo     | MTT       | **Ours**      |
|--------|-----------|-----------|-----------|-----------|-----------|
| CIFAR-100 | 0.15±0.03 | 0.11±0.05 | -0.11±0.08 | 0.39±0.03 | **0.61±0.06** |
| Tiny   | 0.19±0.05 | 0.18±0.06 | -0.12±0.11 | 0.24±0.04 | **0.42±0.06** |

Reviewer comment (Experiments):
The study of memory cost presented in Figure 3 could be more detailed. For instance, a comparison of complexity versus batch size or number of steps relative to the baseline TESLA would be informative.

Author response:
We appreciate your valuable suggestion. **In fact, we offer a detailed comparison of our memory complexity concerning batch size and the number of steps relative to the baseline TESLA in the initial paragraph (Low Memory Complexity) of Section 3.3**. Due to space constraints, we chose not to redundantly include this information in the caption of Figure 3. Following your recommendation, we will consider adding a brief reference in the caption to assist readers in quickly locating this section.

Reviewer comment (Presentation):
Visualization of the distilled ImageNet dataset.

Author response:
Due to the limited elaboration, we found that we did not fully grasp the specific weaknesses and questions raised by the reviewer. If you consider these points to be significant weaknesses or questions, could you please provide additional details to assist us in providing a more thorough response?

Reviewer comment (Experiments):
There seems to be a performance gap between the CNNs and the ViTs. Could authors have the cross-archi experiments on the distilled images trained with ViT (train on ViT and test on CNNs).

Author response:
We appreciate your attention to the detail. This performance gap are generally exist in all the evaluated baseline method according to our reported results in Table 2. This is because all of the compared method (including ours) employ the ConvNet model as the proxy for generating synthetic samples. Due to the dissimilarity of the structures between ViT and ConvNet, all the evaluated results demonstrate a better generalization ability among models which also involving convolutional layers (e.g., ResNet18 and VGG) but exhibit a relatively poor performance on ViT. 

For your reference, **we conduct additional experiments of using ViT as the proxy model for DM, MTT and our method for distilling CIFAR-10 with an IPC of 50 and implement the cross-architecture experiments on randomly initialized ViT, ConvNet and ResNet18**. The results are presented in the following table. According to the results, although the results, obtained on ConvNet and ResNet 18, are remarkably lower than those obtained by using ConvNet as the proxy, **our method still exhibit the outstanding generalization performance among other baselines**. this observation underscores **the better ability of our method to capture essential cross-architecture features for persisting training effects**.

|          |   DM    |   MTT   |   **Ours**   |
|----------|---------|---------|----------|
|   ViT    | 32.5±0.3| 36.8±0.6| **44.8±0.4** |
| ConvNet  | 44.6±0.6| 51.2±0.4| **54.4±0.1** |
| ResNet18 | 24.6±3.1| 51.8±0.8| **56.7±0.5** |

Reviewer comment (Evaluation):
The SOTA of ImageNet-1k should be [2] instead of TESLA. The claim of SOTA in ImageNet-1k is not objective.

Author response:
Thank you for your attention to detail. To the best of our knowledge, SRe^2L is developed for distilling data exclusively for ResNet models on the Tiny ImageNet and ImageNet-1k datasets. This specific application is somewhat because SRe^2L is a direct adaptation of model inversion and batch normalization statistics alignment techniques initially proposed for data-free knowledge distillation tasks [B].

Notably, according to the experimental results reported by SRe^2L and its released code, we identify 3 reasons to not include SRe^2L in our formal evaluations: 1. the crafted data is shown to be effective primarily for training ResNet models with batch normalization layers, 2. SRe^2L is not effective for distilling other baseline datasets like CIFAR-10/100, and 3. **Most importantly**, when using synthetic data generated by SRe^2L for ImageNet-1k for model training, **it is necessary to involve CutMix data augmentation [C] to avoid a significant performance drop**. 
**This means that the amount of data actually used for model training is far greater than what is reported (e.g., IPC=50), and it also requires more epochs as well as training time for models to converge**.

Given these limitations and the differing application scenarios, we believe it would not be fair to include SRe^2L in comparisons with other evaluated baselines that follow the same distillation protocols.

[A] Zeyuan Yin, Eric Xing, and Zhiqiang Shen. Squeeze, recover and relabel: Dataset condensation at imagenet scale from a new perspective. In NeurIPS, 2023. 1, 2, 3, 4, 5,

[B] Hongxu Yin, Pavlo Molchanov, José M. Álvarez, Zhizhong Li, Arun Mallya, Derek Hoiem, Niraj K. Jha, Jan Kautz: Dreaming to Distill: Data-Free Knowledge Transfer via DeepInversion. CVPR 2020: 8712-8721

[C] Sangdoo Yun, Dongyoon Han, Sanghyuk Chun, Seong Joon Oh, Youngjoon Yoo, Junsuk Choe: CutMix: Regularization Strategy to Train Strong Classifiers With Localizable Features. ICCV 2019: 6022-6031

Reviewer comment (Experiments):
The paper highlights a significantly reduced storage requirement for the proposed method compared to MTT. It would be beneficial to include a comparative analysis of this result with other existing dataset distillation (DD) frameworks to provide a broader context and assess the method's efficiency against a wider range of approaches.

Author response:
Thank you for your suggestion. To clarify, we would like to first emphasize that in the caption of Table 1, we explicitly noted that, apart from our method and TESLA, all other evaluated baseline methods have missing entries in Table 1's results. This omission is due to the fact that **these methods, when faced with a given computational resource (a server equipped with a TESLA A100), were unable to perform data distillation on the evaluated datasets under these specific scenarios**. We have elaborated extensively on the reasons behind the infeasibility of these methods for data distillation in these scenarios in Appendix A.2. It is important to note that due to the space limit of the paper, we specifically chose to highlight our method's efficiency improvements by comparing it to the only effective competitor, TESLA, using the challenging ImageNet-1k dataset, with a focus on memory usage and runtime.

**Regarding storage requirements**, it is worth mentioning that **only our method and training trajectory matching-based methods (including MTT, FTD, and TESLA) have such demands**. This is because both our method and these MTT-based methods are grounded in the teacher-student paradigm, where imitation-based objectives are defined to transfer essential information from the original datasets to their distilled counterparts, preserving the training effect. However, **the effective utilization of information from the teacher model empowers our two frameworks to consistently outperform other methods in the majority of evaluated scenarios**. Furthermore, since we do not need to store snapshots of teacher models trained on the original data, **our method fundamentally reduces storage requirements to just $\frac{1}{epoch}$ compared to MTT-based methods**.

Reviewer comment (Experiments):
While the proposed method offers a more efficient dataset distillation (DD) framework, it raises questions about its scalability. It would be valuable to see DD results using the original resolution of ImageNet, as the resolution setting is not explicitly mentioned in the paper (and it is assumed that the authors present results with reduced resolution).

Author response:
Thank you for your attention to detail. In fact, **at the beginning of Section 4.1**, titled "Experimental Setups," **we explicitly state that we adhere to the protocol of previous arts (FRePo and TESLA) by resizing ImageNet-1K images to 64 × 64 resolution in our evaluations to ensure fair comparisons**.

Based on the analysis and experimental results we provided in response to the previous question, our method demonstrates **the capability to use the ResNet18 model as the proxy for distilling ImageNet-1k in its original resolution**. Additionally, we want to reiterate that, as per our memory complexity analysis in Section 3.3, our method, along with SRe^2L, can independently calculate and update the loss for each synthetic data point. Our memory complexity is primarily dominated by the partial derivative of logits difference between a teacher and a student for a single synthetic sample, which involves only a trivial backpropagation through the two models. Mathematically, this leads to our method having a memory complexity that is only twice that of SRe^2L. However, while TESLA is the only generally defined dataset distillation method that can be applied to distil 64x64-sized ImageNet-1k (when IPC is larger than 10), attempting to use it for the original resolution ImageNet-1k leads to out-of-memory (OOM) errors on our TESLA A100 server

Reviewer comment (Experiments):
It is advisable to include results on training models directly on ResNet rather than transferring to it, as there are existing DD methods that operate in this specific setting. Comparing against these methods would provide a more comprehensive assessment of the proposed method's scalability.

Author response:
Thank you for your attention to detail. To the best of our knowledge, we have **only identified one related work SRe^2L** [A], which was presented at NeurIPS 2023, **focusing on distilling data exclusively for ResNet models** on the Tiny ImageNet and ImageNet-1k datasets. This specific application is somewhat because SRe^2L is a direct adaptation of model inversion and batch normalization statistics alignment techniques initially proposed for data-free knowledge distillation tasks [B].

Notably, according to the experimental results reported by SRe^2L and its released code, **we identify 3 reasons to not include SRe^2L in our formal evaluations**: 1. the crafted data is shown to be effective primarily for training ResNet models with batch normalization layers, 2. SRe^2L is not effective for distilling other baseline datasets like CIFAR-10/100, and 3. **Most importantly**, when using synthetic data generated by SRe^2L for ImageNet-1k for model training, **it is necessary to involve CutMix data augmentation [B] to avoid a significant performance drop**. 
**This means that the amount of data actually used for model training is far greater than what is reported (e.g., IPC=50), and it also requires more epochs as well as training time for models to converge**.


Given these limitations and the differing application scenarios, we believe it would not be fair to include SRe^2L in comparisons with other evaluated baselines that follow similar distillation protocols.

Additionally, we want to emphasize that **in Appendix A.4, we conducted a study using ResNet18 as a proxy model to generate distilled data for CIFAR-10 and Tiny ImageNet and evaluate their generalization ability on various architectural models including ResNet18**. The experimental results indicate that **our method outperforms the results reported by SRe^2L when training ResNet18**. For your reference, we **conducted additional experiments using ResNet18 as the proxy to generate distilled data for CIFAR-100 and ImageNet-1k (224x224)**, with a distillation budget set to IPC=50 for all four benchmark datasets. The results are presented in the table below. We can see from the results that our method performs slightly worse than SRe^2L on ImageNet-1k but remarkably outperforms SRe^2L on CIFAR-10/100 and Tiny ImageNet.

|         | CIFAR-10 | CIFAR-100 | Tiny ImageNet | ImageNet-1k |
|---------|----------|-----------|---------------|-------------|
| SRe^2L  | 52.8     | 35.8      | 41.1          | 46.8        |
| Ours    | 63.2     | 59.4      | 54.8          | 43.5        |

[A] Zeyuan Yin, Eric Xing, Zhiqiang Shen: Squeeze, Recover and Relabel: Dataset Condensation at ImageNet Scale From A New Perspective. NeurIPS 2023

[B] Hongxu Yin, Pavlo Molchanov, José M. Álvarez, Zhizhong Li, Arun Mallya, Derek Hoiem, Niraj K. Jha, Jan Kautz: Dreaming to Distill: Data-Free Knowledge Transfer via DeepInversion. CVPR 2020: 8712-8721

[C] Sangdoo Yun, Dongyoon Han, Sanghyuk Chun, Seong Joon Oh, Youngjoon Yoo, Junsuk Choe: CutMix: Regularization Strategy to Train Strong Classifiers With Localizable Features. ICCV 2019: 6022-6031

Reviewer comment (Experiments):
Can you provide a detailed efficiency comparison of your proposed method with other existing dataset distillation (DD) frameworks, particularly in terms of storage requirements?

Author response:
Thank you for your suggestion. To clarify, we would like to first emphasize that in the caption of Table 1, we explicitly noted that, apart from our method and TESLA, all other evaluated baseline methods have missing entries in Table 1's results. This omission is due to the fact that **these methods, when faced with a given computational resource (a server equipped with a TESLA A100), were unable to perform data distillation on the evaluated datasets under these specific scenarios**. We have elaborated extensively on the reasons behind the infeasibility of these methods for data distillation in these scenarios in Appendix A.2. It is important to note that due to the space limit of the paper, we specifically chose to highlight our method's efficiency improvements by comparing it to the only effective competitor, TESLA, using the challenging ImageNet-1k dataset, with a focus on memory usage and runtime.

**Regarding storage requirements**, it is worth mentioning that **only our method and training trajectory matching-based methods (including MTT, FTD, and TESLA) have such demands**. This is because both our method and these MTT-based methods are grounded in the teacher-student paradigm, where imitation-based objectives are defined to transfer essential information from the original datasets to their distilled counterparts, preserving the training effect. However, **the effective utilization of information from the teacher model empowers our two frameworks to consistently outperform other methods in the majority of evaluated scenarios**. Furthermore, since we do not need to store snapshots of teacher models trained on the original data, **our method fundamentally reduces storage requirements to just $\frac{1}{epoch}$ compared to MTT-based methods**.

Reviewer comment (Experiments):
How does your method compare in terms of efficiency when using original ImageNet image resolutions? Is the reduced resolution setting explicitly mentioned in the paper?

Author response:
Thank you for your attention to detail. In fact, **at the beginning of Section 4.1**, titled "Experimental Setups," **we explicitly state that we adhere to the protocol of previous arts (FRePo and TESLA) by resizing ImageNet-1K images to 64 × 64 resolution in our evaluations to ensure fair comparisons**.

Based on the analysis and experimental results we provided in response to the previous question, our method demonstrates **the capability to use the ResNet18 model as the proxy for distilling ImageNet-1k in its original resolution**. Additionally, we want to reiterate that, as per our memory complexity analysis in Section 3.3, our method, along with SRe^2L, can independently calculate and update the loss for each synthetic data point. Our memory complexity is primarily dominated by the partial derivative of logits difference between a teacher and a student for a single synthetic sample, which involves only a trivial backpropagation through the two models. Mathematically, this leads to our method having a memory complexity that is only twice that of SRe^2L. However, while TESLA is the only generally defined dataset distillation method that can be applied to distil 64x64-sized ImageNet-1k (when IPC is larger than 10), attempting to use it for the original resolution ImageNet-1k leads to out-of-memory (OOM) errors on our TESLA A100 server

Reviewer comment (Experiments):
Could you consider providing results on training models directly on ResNet (without transfer learning), as some DD methods operate in this specific setting? What insights can you offer on the efficiency and performance of your method in this scenario?

Author response:
Thank you for your attention to detail. To the best of our knowledge, we have **only identified one related work SRe^2L** [A], which was presented at NeurIPS 2023, **focusing on distilling data exclusively for ResNet models** on the Tiny ImageNet and ImageNet-1k datasets. This specific application is somewhat because SRe^2L is a direct adaptation of model inversion and batch normalization statistics alignment techniques initially proposed for data-free knowledge distillation tasks [B].

Notably, according to the experimental results reported by SRe^2L and its released code, **we identify 3 reasons to not include SRe^2L in our formal evaluations**: 1. the crafted data is shown to be effective primarily for training ResNet models with batch normalization layers, 2. SRe^2L is not effective for distilling other baseline datasets like CIFAR-10/100, and 3. **Most importantly**, when using synthetic data generated by SRe^2L for ImageNet-1k for model training, **it is necessary to involve CutMix data augmentation [B] to avoid a significant performance drop**. 
**This means that the amount of data actually used for model training is far greater than what is reported (e.g., IPC=50), and it also requires more epochs as well as training time for models to converge**.


Given these limitations and the differing application scenarios, we believe it would not be fair to include SRe^2L in comparisons with other evaluated baselines that follow similar distillation protocols.

Additionally, we want to emphasize that **in Appendix A.4, we conducted a study using ResNet18 as a proxy model to generate distilled data for CIFAR-10 and Tiny ImageNet and evaluate their generalization ability on various architectural models including ResNet18**. The experimental results indicate that **our method outperforms the results reported by SRe^2L when training ResNet18**. For your reference, we **conducted additional experiments using ResNet18 as the proxy to generate distilled data for CIFAR-100 and ImageNet-1k (224x224)**, with a distillation budget set to IPC=50 for all four benchmark datasets. The results are presented in the table below. We can see from the results that our method performs slightly worse than SRe^2L on ImageNet-1k but remarkably outperforms SRe^2L on CIFAR-10/100 and Tiny ImageNet.

|         | CIFAR-10 | CIFAR-100 | Tiny ImageNet | ImageNet-1k |
|---------|----------|-----------|---------------|-------------|
| SRe^2L  | 52.8     | 35.8      | 41.1          | 46.8        |
| Ours    | 63.2     | 59.4      | 54.8          | 43.5        |

[A] Zeyuan Yin, Eric Xing, Zhiqiang Shen: Squeeze, Recover and Relabel: Dataset Condensation at ImageNet Scale From A New Perspective. NeurIPS 2023

[B] Hongxu Yin, Pavlo Molchanov, José M. Álvarez, Zhizhong Li, Arun Mallya, Derek Hoiem, Niraj K. Jha, Jan Kautz: Dreaming to Distill: Data-Free Knowledge Transfer via DeepInversion. CVPR 2020: 8712-8721

[C] Sangdoo Yun, Dongyoon Han, Sanghyuk Chun, Seong Joon Oh, Youngjoon Yoo, Junsuk Choe: CutMix: Regularization Strategy to Train Strong Classifiers With Localizable Features. ICCV 2019: 6022-6031

Reviewer comment (Experiments):
Given that your method aims to offer a more efficient DD framework, what scalability challenges or considerations have you encountered when dealing with the original resolution of ImageNet images?

Author response:
Thank you for your attention to detail. In fact, **at the beginning of Section 4.1**, titled "Experimental Setups," **we explicitly state that we adhere to the protocol of previous arts (FRePo and TESLA) by resizing ImageNet-1K images to 64 × 64 resolution in our evaluations to ensure fair comparisons**.

Based on the analysis and experimental results we provided in response to the previous question, our method demonstrates **the capability to use the ResNet18 model as the proxy for distilling ImageNet-1k in its original resolution**. Additionally, we want to reiterate that, as per our memory complexity analysis in Section 3.3, our method, along with SRe^2L, can independently calculate and update the loss for each synthetic data point. Our memory complexity is primarily dominated by the partial derivative of logits difference between a teacher and a student for a single synthetic sample, which involves only a trivial backpropagation through the two models. Mathematically, this leads to our method having a memory complexity that is only twice that of SRe^2L. However, while TESLA is the only generally defined dataset distillation method that can be applied to distil 64x64-sized ImageNet-1k (when IPC is larger than 10), attempting to use it for the original resolution ImageNet-1k leads to out-of-memory (OOM) errors on our TESLA A100 server

Reviewer comment (Experiments):
Can you elaborate on the choice of not providing results on models trained directly on ResNet, especially when other DD methods operate in this mode? What insights can you provide on the scalability and effectiveness of your approach under this setting?

Author response:
Thank you for your attention to detail. To the best of our knowledge, we have **only identified one related work SRe^2L** [A], which was presented at NeurIPS 2023, **focusing on distilling data exclusively for ResNet models** on the Tiny ImageNet and ImageNet-1k datasets. This specific application is somewhat because SRe^2L is a direct adaptation of model inversion and batch normalization statistics alignment techniques initially proposed for data-free knowledge distillation tasks [B].

Notably, according to the experimental results reported by SRe^2L and its released code, **we identify 3 reasons to not include SRe^2L in our formal evaluations**: 1. the crafted data is shown to be effective primarily for training ResNet models with batch normalization layers, 2. SRe^2L is not effective for distilling other baseline datasets like CIFAR-10/100, and 3. **Most importantly**, when using synthetic data generated by SRe^2L for ImageNet-1k for model training, **it is necessary to involve CutMix data augmentation [B] to avoid a significant performance drop**. 
**This means that the amount of data actually used for model training is far greater than what is reported (e.g., IPC=50), and it also requires more epochs as well as training time for models to converge**.


Given these limitations and the differing application scenarios, we believe it would not be fair to include SRe^2L in comparisons with other evaluated baselines that follow similar distillation protocols.

Additionally, we want to emphasize that **in Appendix A.4, we conducted a study using ResNet18 as a proxy model to generate distilled data for CIFAR-10 and Tiny ImageNet and evaluate their generalization ability on various architectural models including ResNet18**. The experimental results indicate that **our method outperforms the results reported by SRe^2L when training ResNet18**. For your reference, we **conducted additional experiments using ResNet18 as the proxy to generate distilled data for CIFAR-100 and ImageNet-1k (224x224)**, with a distillation budget set to IPC=50 for all four benchmark datasets. The results are presented in the table below. We can see from the results that our method performs slightly worse than SRe^2L on ImageNet-1k but remarkably outperforms SRe^2L on CIFAR-10/100 and Tiny ImageNet.

|         | CIFAR-10 | CIFAR-100 | Tiny ImageNet | ImageNet-1k |
|---------|----------|-----------|---------------|-------------|
| SRe^2L  | 52.8     | 35.8      | 41.1          | 46.8        |
| Ours    | 63.2     | 59.4      | 54.8          | 43.5        |

[A] Zeyuan Yin, Eric Xing, Zhiqiang Shen: Squeeze, Recover and Relabel: Dataset Condensation at ImageNet Scale From A New Perspective. NeurIPS 2023

[B] Hongxu Yin, Pavlo Molchanov, José M. Álvarez, Zhizhong Li, Arun Mallya, Derek Hoiem, Niraj K. Jha, Jan Kautz: Dreaming to Distill: Data-Free Knowledge Transfer via DeepInversion. CVPR 2020: 8712-8721

[C] Sangdoo Yun, Dongyoon Han, Sanghyuk Chun, Seong Joon Oh, Youngjoon Yoo, Junsuk Choe: CutMix: Regularization Strategy to Train Strong Classifiers With Localizable Features. ICCV 2019: 6022-6031

Reviewer comment (Theory):
Why having a form of Equation (3) to train the synthetic samples? I understand it might become easier to split the loss into two components if using Equation (3), and also it might have a similar form with Equation (1), but taking the logarithm over the loss values still look uncommon to me.

Author response:
**Weakness 1: Why have a form of Equation (3) to train the synthetic samples? I understand it might become easier to split the loss into two components if using Equation (3), and it might have a similar form with Equation (1), but taking the logarithm over the loss values still looks uncommon to me.**

**Answer:**

We greatly appreciate the thorough review by the reviewer and the opportunity to clarify the benefits of taking the logarithm of the probability-weighted logits difference in Equation (3). As mentioned in our response to Weakness 2 above, we formulate the objective for crafting our synthetic data as Equation (3) to simultaneously create "hard samples" that satisfy our defined conditions for all involved training snapshots (checkpoints). This formulation helps prevent inductive bias towards any specific training stage.

Taking the logarithm of the term $p(u|y_{u}) d\left( f_{\theta^{\mathcal{T}}}\left(u\right) , f_{\theta_e^{\mathcal{S}}}\left(u\right) \right)$ in this equation offers **two main advantages**. First, as you've noted, it **decouples the maximization of logits difference in Equation (4) from the maximization of $p(u|y_{u})$ (substitutable by $p(y_u|u,\theta^{\mathcal{T}})$)**, allowing us to flexibly tradeoff optimization between the two by introducing an additional hyperparameter $\alpha$. We also discuss its effectiveness in Appendix A.5.

Second, leveraging the properties of the logarithmic function and the chain rule, we can express the derivative of $\ln(A(u))$ w.r.t u as $\frac{\partial}{\partial u}[\ln(A(u))] = \frac{1}{A(u)} \cdot \frac{\partial A(u)}{\partial u}$. Compared to directly taking the derivative of $A(u)$ w.r.t u, taking the logarithm of $A(u)$ and then obtaining the derivative naturally weights it by the value of $A(u)$. **This trick is widely applied in scenarios where gradients need to be weighted by the value of the function itself**. For instance, in the design of loss functions, applying a logarithmic transformation to the output (e.g., taking the log-likelihood loss) can alter the gradient scale, contributing to the stability of the training process. In the loss function defined in Equation (4), **taking the logarithm of logits difference can also automatically enlarge the gradient for samples with smaller differences**.

Reviewer comment (Experiments):
I noticed that the authors use soft label instead of one-hot label. However, some baseline methods in Table 1 and Table 2 do not apply this. I think it would be beneficial to indicate this point when comparing with other baselines.

Author response:
**Weakness 3: I noticed that the authors use soft label instead of one-hot label. However, some baseline methods in Table 1 and Table 2 do not apply this. I think it would be beneficial to indicate this point when comparing with other baselines.**

**Answer:**

Thank you for your valuable suggestion. Regarding Tables 1 and 2, the results for all baseline methods are accurately derived by adhering to their specific default protocols. Among them, FRePo and RCIG utilize optimal soft labels, which they have learned, to create their distilled dataset. In contrast, other methods use standard one-hot labels. We will make this distinction clear in our final manuscript.

Moreover, to convincingly demonstrate that our synthetic images encapsulate the most informative features essential for achieving our prediction-matching goal and maintaining the training effect, we **reconstruct distilled datasets for DM, MTT, and FTD by substituting one-hot hard labels with soft labels (logits) output by our pretrained teacher models**. We then train models using our adopted $L1$ loss. The following table presents the test accuracies (%) of ConvNet models trained on these logits-labeled distilled datasets, along with a comparison of the variations relative to the results in Table 1. The results demonstrate that **our methods still outperform other baselines in all the evaluated scenarios**. This observation emphasizes that **our synthetic images are effective for incorporating the essential features that are critical for meeting our prediction-matching objective** and **can better utilize teachers' logits for preserving the training effect**.

| Dataset         | DM               | MTT              | FTD              | **Ours**             |
|-----------------|------------------|------------------|------------------|------------------|
| CIFAR10-ipc 50  | 60.7±0.2 ↓2.3    | 71.1±0.4 ↓0.5    | 73.4±0.3 ↓0.4    | **75.0±0.2**         |
| CIFAR10-ipc 500 | 73.8±0.2 ↓0.5    | 79.3±0.2 ↑0.7    | 79.8±0.4 ↑1.1    | **83.4±0.3**         |
| CIFAR100-ipc 10 | 23.2±0.5 ↓6.5    | 39.2±0.5 ↓0.9    | 43.2±0.2 ↓0.2    | **44.6±0.3**         |
| CIFAR100-ipc 50 | 44.5±0.1 ↑0.9    | 49.1±0.3 ↑1.4    | 51.1±0.3 ↑0.4    | **53.3±0.2**         |
| Tiny-ipc 10     | 15.8±0.3 ↑2.9    | 24.2±0.3 ↑1.0    | 25.4±0.3 ↑0.9    | **30.0±0.3**         |
| Tiny-ipc 50     | 23.5±0.3 ↑2.3    | 31.5±0.2 ↑3.5    | 35.5±0.3 ↑4.0    | **38.2±0.4**         |

Reviewer comment (Evaluation):
Figure 5(b): it seems that the performance will drop when using more checkpoints, which looks counterintuitive to me: the approximation outperforms the original loss objective in terms of the final performance. It would be beneficial to provide more analysis on this point.

Author response:
**Weakness 4: Figure 5(b): it seems that the performance will drop when using more checkpoints, which looks counterintuitive to me: the approximation outperforms the original loss objective in terms of the final performance.**

**Answer:**

We really appreciate your attention to detail and pointing out this counterintuition! We carefully studied this phenomenon and found out that **this slight performance drop is attributed to that we did not correspondingly tune the learning rate used for updating the synthetic when deriving the results of involving more than five checkpoints for calculating the loss**. That is, we maintained to use the learning rate initially set for involving five checkpoints. However, our defined loss function calculates the summation of the logits differences caused by synthetic samples between the teacher and multiple checkpoints, rather than the average. Therefore, introducing additional checkpoints alters the final loss, necessitating an adjustment in the learning rate for updating the synthetic data. Following this analysis, we carefully **recalibrated the learning rate across various scenarios involving different numbers of checkpoints, retested the performances, and reported the results in the table below**. The experimental outcomes remain consistent with our original paper's assertion: setting the number of involved checkpoints higher than five does not lead to significant performance changes. We greatly appreciate your attention to the detail of our work and will correct the Figure 5 (b) in our final print.

| checkpoint    | 1           | 3           | 7           | 10          | 12          | 15          | 20          |
|---------------|-------------|-------------|-------------|-------------|-------------|-------------|-------------|
| Acc (%)       | 67.8±0.5    | 74.2±0.3    | 74.9±0.4    | 75.1±0.2    | 75.0±0.1    | 75.3±0.2    | 75.1±0.3    |