Detecting Out-of-Distribution Samples via Conditional Distribution Entropy with Optimal Transport

Thanks for your valuable comments, which helps us improve the manuscript.

Answers to Weaknesses

Weakness 1. "The organization and writing of the paper need to be improved. Notations are..."

A1.1 & A1.2 We aim to ensure the paper is self-contained and presented clearly. While there is room for improvement, particularly in the points you highlighted, we plan to enhance these aspects in the revised version.

A1.3 Please see the Def 2.1(P2) for the definition of OOD detection.

Our definition aligns with your viewpoint that OOD detection is indeed a binary classification problem. In fact, the entropic regularized OT in our paper is used to model the mass transport between test inputs and ID training data, which is a crucial aspect of our method. Therefore, we agree that it is better to rephrase the sentence that you mentioned to as "we leverage optimal transport with entropic regularization [1] for measuring the discrepancy of empirical distributions between test inputs and ID data." in P4 (Sec 3.1). We will modify it in the revised version.

[1] Cuturi, Marco. "Sinkhorn distances: Lightspeed computation of optimal transport." Advances in neural information processing systems 26 (2013).

A1.4 We clarify that our approach is model-agnostic, meaning that the testing procedure is applicable to different model architectures and training losses, including cross entropy loss and contrastive training loss (as shown in Table 1). This has been confirmed in the line of research on distance-based OOD detections. Also, the advantage of contrastive training over cross entropy was discussed in previous works (e.g., KNN and SSD). In our implementation, we employ supervised training, for being in line with existing distance-based OOD detection approaches (e.g., KNN and SSD) and for empirically fair comparison.

We will incorporate the above discussion to make a smooth transition between Sections 3.2 and 3.3.

Weakness 2. "The methodology is somewhat vague. It remains..."

A2. Please see Q1 for the calculation of scoring function. The $\nu$ means the empirical distribution of a set of test inputs. It is worth noting that our method only uses ID data to train model and obtain training data features without requiring any OOD validation set. Following with distance-based methods (KNN and SSD), we use the trained model to extract the feature of a test input. Then, the distances between a test input and training samples can be calculated by some distance metrics, such as Euclidean distance and cosine distance.

Answers to Questions

A1. We apologize if the calculation of our proposed scoring function has caused confusion.

As detailed in Section 3.2, we introduce a novel scoring function called conditional distribution entropy (Definition 3.5), derived from the optimal transport plan. In this context, we interpret the mass transported from one test input to training data as a conditional distribution, a concept further explained in our discussion on the part of Uncertainty Modeling. Additionally, to quantify the uncertainty of a test input being an OOD sample, we suggest using the entropy of the conditional distribution as the scoring function. For an intuitive understanding of the uncertainty in the transport plan, we refer to Figure 1. Furthermore, we theoretically explore how entropic regularized optimal transport affects the conditional distribution entropy (Proposition 3.6).

A2. Current challenges faced by existing distance-based methods in handling OOD samples that closely resemble ID in difficult OOD tasks arise from their simplistic utilization of distance information, as seen in methods like SSD or KNN. These approaches rely solely on the distance from test samples to either the centroid of training data or some ID samples. In contrast, our approach leverages the empirical distribution of training data and test inputs. By integrating pair-wise distance information with distributional insights through optimal transport, our method develops a scoring function that effectively captures the distinctions between OOD and ID samples. This enables more accurate discrimination of OOD samples that are closely located to ID data.

W4. Results on large-scale dataset, e.g., ImageNet.

Answer to W4

Evaluation on large-scale ImageNet. We take ViT-B/16 as the backbone and follow the parameter settings of KNN. We use ImageNet-1k as the ID dataset and five datasets (in Table 1) as the OOD datasets. The average performance is reported in the table below.

The results show the superiority of our proposal over large datasets and models. We will incorporate these results in the revised manuscript.

W5. Reasons of using CIFAR10 vs CIFAR100 (ID vs OOD).

Answer to W5

In our study, CIFAR10 vs. CIFAR100 are employed in three contexts: hard task, unsupervised detection, and ablation studies. Since classes share multiple similar semantics in these two datasets, they were employed as a challenging task to fully evaluate the OOD detection method in previous works, such as KNN [3], SSD [4], and CIDER [5].

In line with those studies, we chose it for evaluating our method and for comparison with baselines.

References

[1] Du, Xuefeng, et al. "VOS: Learning What You Don't Know by Virtual Outlier Synthesis." International Conference on Learning Representations. 2021.

[2] Ming, Yifei, Ying Fan, and Yixuan Li. "Poem: Out-of-distribution detection with posterior sampling." International Conference on Machine Learning. PMLR, 2022.

[3] Sun, Yiyou, et al. "Out-of-distribution detection with deep nearest neighbors." International Conference on Machine Learning. PMLR, 2022.

[4] Sehwag, Vikash, Mung Chiang, and Prateek Mittal. "SSD: A Unified Framework for Self-Supervised Outlier Detection." International Conference on Learning Representations. 2020.

[5] Ming, Yifei, et al. "How to Exploit Hyperspherical Embeddings for Out-of-Distribution Detection?." The Eleventh International Conference on Learning Representations. 2022.

Thanks for your insightful comments and approval of the general research direction.

Let's try to summarize your comments into five weak points.

W1. Our method is similar to the method that you have mentioned (Garg et al.).

W2. Our techniques look similar to the techniques of the method that you have mentioned (Garg et al.).

W3. Some evaluation setup details are not clear.

W4. Results on large-scale dataset, e.g., ImageNet.

W5. Reasons of using CIFAR10 vs CIFAR100 as (ID vs OOD).

Answer to W1

In fact, our work differs significantly from the study that you have mentioned (Garg et al., 2023) because they operate under different problem settings. The work by Garg et al. (2023) assumes the availability of an OOD validation set. However, it is not practical to assume the explicit knowledge of unknowns due to the vulnerability of OOD inputs [1, 2].

In contrast, our method is distance-based and does not require any OOD validation set. That is why we compared our method with the latest distance-based approaches, such as KNN [3] and SSD [4], as they share the same problem setting but differ from Garg et al. (2023).

We all know it is not fair to compare a method with prior knowledge to one without. But we can compel the comparison to happen, by using Place365 as OOD validation set for Garg's method. The result shows that our method still outperforms Garg's method in all cases but Place365. Notice that our method is without the aid of OOD validation set.

Answer to W2

We would like to argue the techniques of our method are significantly different from those in Garg's paper.

1. The Usage of Statistical Distance. The work (Garg et al. 2023) directly uses the dual form of KL divergence, a statistical distance, to estimate the chance of a test input being an OOD sample. In contrast, we don't directly use the Wasserstein distance (another statistical distance) to detect an OOD sample. Instead, we formulate the problem as an entropic regularized OT problem and define a novel scoring function over the optimal transport plan to measure the uncertainty of a test input being an OOD sample.

2. The Role of Dual Form. Despite the involvement of the dual form concept in both papers, it serves a distinct role in each. In Garg's work, the dual form of the KL divergence is employed to avoid the estimation of the density function, and the final dual function can be viewed as a function approximator. Differently, our method utilizes the dual problem of entropic regularized OT to efficiently obtain the optimal transport plan. So, the dual form of our work cannot take care of the representation learning like they did.

Answer to W3

In Same Setup. Our work put an emphasis on utilizing empirical distributions with geometric information for OOD detection, which is overlooked by existing distance-based approaches, such as KNN [3] and SSD [4].

In our experiments, we have tried to evaluate the effect of the cardinality of a test input batch to the performance of OOD detection, as shown in Figure 2. In this setting, an extreme case is to handle a test input as a batch (i.e., batch cardinality is one). Here, we provide more results in the table below.

The above results show: 1) The empirical distribution information of test inputs is beneficial for OOD detection; 2) As the batch cardinality increases, our method outperforms the distance-based method SSD.

We will incorporate these results in the revised manuscript.

More Evaluations

Given our method's distance-based nature, our comparisons primarily focused on other distance-based methods, including SOTA baselines like KNN and SSD. Notably, the pioneering work by Hendrycks et al. (2019) is not typically regarded as a baseline in current literature discussing distance-based methods, such as KNN and SSD. Consequently, we did not include it in our baseline comparisons.

However, we are open to incorporate more comparisons into our paper if deemed necessary. To reproduce OE (Hendery et al. 2019), we use CIFAR100+Tiny ImageNet as ID data, following its settings (See Sec. 4.2.2 & 4.3 in OE). We report the performance comparison with OE (Hendery et al. 2019) as follows.

Clarity on comments

We clarify that the term “OOD validation set” in our comments refer to auxiliary OOD data, as presented in the seminal work (Hendery et al. 2019). We feel sorry about the confusion and misunderstanding caused by this terminology.

It is known that the auxiliary OOD data mainly includes three categories:

1. Generating OOD data from ID sample
2. Subset of OOD test data
3. Another non-ID and non-OOD-test data

Here, we reclaim the method in that paper (Garg et al. 2023) is significantly different from our method, due to the usage of OOD data during training.

In the paper (Garg et al. 2023 ), KL dual divergence is computed by training data and the subset of OOD test data, which has been described in P3 as follows: “The key idea is that estimating the divergence measure in its dual form is naturally informative about the subset of samples in the test set that are OOD.”, also affirmed by their official implementation. In contrast, our method does not require any auxiliary OOD data during the training stage. We guess that this confusion may have led to your misunderstanding.

In particular, the paper by Garg et al. (2023) also utilizes the first type of auxiliary OOD data, i.e., generating OOD data from ID samples, to analyze the generalization of the estimator.

More, your viewpoint: “there is no reason to not utilize outlier exposure in any method.”, is open to discussion in OpenReview.

Thanks for your insightful comments and suggestions, which help us to improve the manuscript.

Answer to Weakness 1

Please see answer to Question 1.

Answer to Weakness 2

Please see answer to Question 3.

Answer to Weakness 3

We agree that the readability of paper is very important. We will incorporate more visual examples as follows in the revised manuscript (we feel sorry about that we can't provide figure in comments.), to enhance the readability of our paper.

Showcase. When $\mu$ and $\nu$ are identical empirical distributions, the transport of a test input is exactly, suggesting the smallest entropy in the respective conditional distribution.

Answer to Question 1

Dimension of Encoding. We provide the ablation study on the dimension of encoding in the table below. Here, we adopt the ResNet18 as the backbone and the same dataset settings as Table 1.

We can find the effect of feature dimension on performance is not monotonic on a single model. For example, on the backbone ResNet18, the performance decreases when the feature dimension exceeds 1024. However, a higher feature dimension yields better performance for larger models, such as ResNet50.

Answer to Question 2

We clarify all test inputs are with the same input dimension of training samples. Thus, we describe the dimension of ID data used in our empirical study.

Answer to Question 3

Thanks for your comments. As you have pointed out, there are several works (Serrà, Joan, et al. 2019; Schirrmeister, Robin, et al. 2020; Ren, Jie, et al. 2019) that employ the likelihood ratio for OOD detection. Notably, (Serrà, Joan, et al. 2019; Schirrmeister, Robin, et al. 2020;) necessitate training models beyond ID data, requiring additional knowledge, in addition to using multiple samples for making decision on a test point. The work by (Ren, Jie, et al. 2019) doesn't require extra data, but it computes the likelihood ratio by training a background model on perturbed ID data. In contrast, our method doesn't have access to additional data, such as generated data or OOD data, for model training. Despite there being some differences, we agree that it would be an interesting line to consider the likelihood ratio in-depth for OOD detection.

In the sequel, we have tried to discuss the strategy of taking the whole test set in making decisions, covering comparisons with other distance-based competitors, addressing limitations, and contemplating potential future directions.

Differentiating with methods not using empirical distribution. We discussed the relationship between our method and existing distance-based methods. For example, SSD, it uses Mahalanobis distance as a detection metric. As mentioned in page 13, we remark that the Mahalanobis distance can be viewed as a special case of OT. Another baseline KNN, which is a degenerated version of our method (as described in page 9) when OT is eliminated from our method, using the $k$-th distance to ID sample as a detection metric.

Limitations. Nevertheless, relying on the entire test input may pose challenges in terms of efficiency, which is the cost paid for performance enhancement. In the current manuscript, we have investigated how the scale of test inputs impacts performance. We split the whole test set into multiple batches with varied batch sizes. As presented in Figure 2, our method benefits from an increase in the size of the test batch and shows a gradual rise in performance until convergence.

Thus, it is of interest to further study the balance between the batch size and the performance variance. It would also give the promise for adapting the techniques to different applications with various performance requirements and affordable computational resources. Also, when test data arrives in batches, another interesting question arises: Can we leverage the information from previously detected batches to enhance OOD detection in subsequent batches?

Future works. Shifting our attention to training samples, a mirroring problem arises: how to effectively utilize training samples, which remains underexplored in existing distance-based methods that use all training samples.

One possible solution is to find effective representations in the feature space. Following this line of thought, in our study, one way that we can try to explore these effective representations in the probability space, such as the barycenters of empirical probability distribution.

Another possible solution is to generate virtual samples as effective representations to substitute for all training samples. Certainly, the application of generation is not limited to this. It is known that a kind of research on OOD detection involves accessing known OOD samples during the training stage (OOD exposure). However, assuming the explicit knowledge of unknowns is not practical due to the vulnerability of OOD inputs. This leads to a potential problem: Can we randomly generate virtual OOD samples for OOD exposure?

These discussions can be integrated into a revised manuscript.

References

Serrà, Joan, et al. "Input Complexity and Out-of-distribution Detection with Likelihood-based Generative Models." International Conference on Learning Representations. 2019.

Schirrmeister, Robin, et al. "Understanding anomaly detection with deep invertible networks through hierarchies of distributions and features." Advances in Neural Information Processing Systems 33 (2020): 21038-21049.

Ren, Jie, et al. "Likelihood ratios for out-of-distribution detection." Advances in neural information processing systems 32 (2019).

Answer to Weakness 3

Thanks. We acknowledge the slight notation discrepancy you pointed out, and we will address it in the revised version as follows:

Modified Uncertainty Modeling. In Definition 3.1, it shows that the transport plan $\mathbf{P}$ can be viewed as a joint probability distribution $\pi(\mu,\nu)$, with marginals $\mu$ and $\nu$. Given a transport plan in the form of a matrix, we use two random variables, $U$ and $V$, to describe the randomness of the mass distributed in rows and columns, respectively. For a test input $v \in dom(V)$, there is a corresponding column in the transport plan, which is essentially a conditional probability distribution $\pi_{U|V}(u|v)$ computed by joint probability $\pi(u,v)$ and marginal probability $\pi(v)$, as defined in Definition 3.4. Accordingly, the entropy of the conditional probability distribution indicates the level of uncertainty regarding a test input belonging to the OOD. To this end, formally, we define the transport that happens on the test input $v$ as a random event, represented by a random variable $T$ that follows the conditional distribution, i.e., $T\sim \pi_{U|V}(u|v)$. Then, the score function is denoted as the entropy of conditional distribution $\mathbb{H}(U|V=v)$, as defined in Definition 3.5.

Answer to Questions

Incorporating optimal transport (OT) into our approach allows us to go beyond traditional distance-based methods that rely solely on the distance information of training samples and test inputs. Our method harnesses OT to capture empirical distribution information in the feature space, enhancing out-of-distribution (OOD) detection.

To assess the impact of optimal transport (OT) on OOD detection performance, we conducted an ablation study. The results are presented in Table 4, where the row labeled (w/o OT) represents calculations based solely on the distance between a test input and training samples. The distance metric employed in this case is the mean Euclidean distance.

We are encouraged that reviewer finds our work novel, effective, and with extensive experiments. Thanks for your valuable comments and suggestions, which we address below:

Answer to weakness 1

Thanks for your suggestions. We have added more experimental results for larger models and datasets as suggested.

• Evaluation on ResNet50. Following the experimental settings of SSD, we alter the backbone to ResNet50 and use the CIFAR100 as ID dataset. The average performance over five datasets (in Table 1) is presented in the table below.

• Evaluation on ImageNet with ViT. Following the parameter settings of KNN, we take ViT-B/16 as the backbone. We use ImageNet-1k as the ID dataset and five datasets (in Table 1) as the OOD datasets. The average performance is presented in the table below.

The above results show the superiority of our method under the settings of larger models and datasets, which is consistent with the result shown in the manuscript. We will incorporate these results in the revised version.

Answer to weakness 2

We agree that doing more ablation studies is good for analyzing and improving the performance, interpretability, and robustness of the work. In the sequel, we provide the results of ablation studies as suggested.

• Ablation on Distance. We enhance our ablation studies by examining the impact of different distance metrics on out-of-distribution (OOD) detection performance. As shown in the table below, it is evident that Euclidean distance and Cosine distance produce comparable results, given their transformability into each other. In general, Euclidean distance served as the cost metric for optimal transport (OT) due, in part, to its well-analyzed theoretical properties. In this paper, we adopt Cosine distance to align with normalized feature training.

• Ablation on Detection Metric. We improve our ablation studies by examining the effect of detection metrics, specifically the scoring function, on out-of-distribution (OOD) detection performance. The 'Max' metric represents the maximum value of the transport plan for each test input. The 'Probability Distance' metric refers to the inner product over the transport plan and distance. The 'Mean' metric indicates the average value of distances between a test input and training samples. The empirical results highlight the efficacy of our uncertainty modeling and underscore the superiority of our proposed conditional distribution entropy scoring function.

These ablation studies are conducted on the same settings as in Table 1. The reported result is the average performance over five OOD datasets. We will incorporate the above results in the revised manuscript.