AVOID: Alleviating VAE's Overestimation in Unsupervised OOD Detection

Motivation: Thanks for giving us this chance to explain our motivation for focusing on unsupervised OOD detection.

1. A real-world use case could be autonomous driving and we want to identify whether the current driving state is similar or strange to the training set for ensuring the safety of decision-making. Although we could collect a lot of images or videos of driving in different scenarios, it would cost huge manual efforts to annotate each frame with a proper label. In that case, unsupervised OOD detection could avoid the labeling process and directly model the training set's distribution. Actually, under the setting of offline RL, a core direction of research focus is to identify the out-of-distribution state-action pairs and assign them a lower reward [1]. Moreover, as the RL setting always has a lot of states, e.g., millions of time steps, it could be impossible to label every state.

2. Motivation for VAE: VAE is estimating a lower bound (ELBO) of the marginal data log-likelihood unlike exact likelihood models like flow models and autoregressive models, thus the analyses on the relationship between ELBO and OOD detection performance are challenging and lacking. We tried to fill this research gap by identifying the two factors that contribute to the overestimation issue in ELBO-based OOD detection. Besides, we need to highlight that the training objective of diffusion models is also based on ELBO [2, 3], thus our analysis and proposed methods could be directly applied to diffusion models, where we have included the experiments on diffusion models in Appendix K.4.

   [1] Fujimoto, Scott, David Meger, and Doina Precup. “Off-policy deep reinforcement learning without exploration.” ICML, 2019.

   [2] Ho, Jonathan, Ajay Jain, and Pieter Abbeel. "Denoising diffusion probabilistic models." NeurIPS 2020.

   [3] Luo, Calvin. "Understanding diffusion models: A unified perspective." arXiv preprint arXiv:2208.11970 (2022).

Notation: Thanks for your suggestion and we have revised the notation with blue color.

Definition 1: Thanks for your thoughtful suggestions. We highly agree with your statement that even if $\mathcal{G}=\mathbb{E}\_{x\sim p_{id}}[\text{ELBO(x)}] - \mathbb{E}\_{x\sim p_{ood}}[\text{ELBO(x)}]>0$, there is still a certain chance that the overestimation issue could still happen on some OOD data samples. However, if $\mathcal{G}\leq 0$, most of the OOD data samples will be assigned unexpectedly higher ELBO values, which could be seen as a worst case of the failures for ELBO-based OOD detection. Thus, we propose to address this issue $\mathcal{G}\leq 0$ at first to develop a promising detection method for ELBO-based generative models, which is our focus in this paper.

Thanks for your constructive suggestions. We agree that a definition of over-confidence would be a stricter definition and we will add it in our revision. We also appreciate your understanding that the original definition is to make our theoretical analysis easier to understand, but we need to highlight that the conclusion of theoretical analysis in our paper still holds even if the definition is changed.

More notations: Yes, $p_\theta(x|z)$ denotes the decoder distribution paramterized with $\theta$ and $p(x)$ is the true distribution of $x$. We will highlight their definitions in our revision, and we need to highlight that the estimated data distribution $p_\theta(x)=\int_z p_\theta(x|z)p(z)$ may not be equal to $p(x)$ if the VAE is not a perfect generative model.

We are deeply grateful for the time and effort you've dedicated to reviewing our work. Your thoughtful insights and constructive comments are immensely valuable to us!

W1 & Q1: We much appreciate your deep thinking of the DEC method. Yes, we acknowledge that there could be many definitions that could satisfy property 1. Eq. 15 is only one of these definitions and is based on the SVD compressor. "$n_i \geq n_{id}$" means when a data sample is more complex than the in-distribution data measured by SVD, it would need more singular values to achieve the reconstruction error $|x_{recon}-x|<\epsilon$ with the SVD method. The "$(n_{id} - (n_i -n_{id}))/n_{id}$" guarantees that if and only if when $n_i=n_{id}$, $\mathcal{C}\_{\text{non}}(x)=1$, and for either $n_i>n_{id}$ or $n_i < n_{id}$, the $\mathcal{C}\_{\text{non}}(x)<1$. In that case, we could make sure the in-distribution data samples own a large $\mathcal{C}_{\text{non}}(x)$. We have also included the experiments of replacing the SVD method with other image compressors, JPEG and PNG compressors, to implement the DEC method in Appendix K.6.

We acknowledge that DEC based on the data compressors could have limitations. For instance, it could have no effect in alleviating the overestimation when detecting a vertically flipped in-distribution data as OOD, in which case we could only rely on the PHP method as shown in Table 7.

For concerns of "why scale by Ent-Mut of the ID distribution plus a KL term", we may guess it could be a misunderstanding since you cannot scale to Ent-Mut term directly, which is intractable. Recall $\text{Ent-Mut}(\theta, \phi, p) = \mathcal{H}\_p(x)+\mathcal{I}\_{q}(x,z)-\mathcal{I}\_{q,p}(x,z)$, to our knowledge, we cannot accurately estimate $\mathcal{H}\_p(x)$ without knowing the underlying ground truth distribution of a dataset. Besides, $\mathcal{I}\_{q}(x,z)-\mathcal{I}\_{q,p}(x,z)$ is an error term that related to the non-perfect VAE and this term is also unknown as we do not know the expression of the distribution of $p_\theta(z|x)$ in Eq. (5).

However, we can estimate the Ent-Mut term with the PHP method. As explained in Eq. (20), when $q_{id}(x)$ could be estimated by $\hat{q}\_{id}(z)$, the expectation on the ELBO becomes the Ent-Mut term. We admit with your potential underlying concern about the KL term $D_{\text{KL}}[q_{id}(x)||\hat{q}_{id}(z)]$ may still not be equal to 0. Our analysis for the overestimation issue appeals for further advanced methods in estimating it, and there are already existing works focusing on it, [1, 2] claims that a high-dimension input data is actually supported on a low-dimension manifold of $z$ and we could estimate it well if we find a proper dimension for $z$. Thus, our analysis could pave the way for further improvement in VAE-based OOD detection methods.

[1] Dai, Bin, and David Wipf. “Diagnosing and Enhancing VAE Models.” International Conference on Learning Representations. 2018

[2] Loaiza-Ganem, Gabriel, et al. "Diagnosing and fixing manifold overfitting in deep generative models." arXiv preprint arXiv:2204.07172 (2022).

W2 & Q3: Thanks for your suggestion and we have added the "reverse experiments" to Appendix K.2. Results show that our method could still work well in these cases.

Smaller comments: Thanks for your careful reading and we have revised them in the paper noted with blue color.

Q2: Yes, we could provide an explanation for it. For the PHP method, we add an experiment on Appendix K.1, where we can find the $q_{fashion}(z)$ is very distinguishable to $q_{mnist}(z)$ but the $q_{cifar10}(z)$ is not that distinguishable to $q_{svhn}(z)$, this could explain why PHP works better in FashionMNIST(ID)/MNIST(OOD) dataset pair but not that well in CIFAR10(ID)/SVHN(OOD) dataset pair. This provides an insight that learning a dataset-specific semantic latent space could achieve better OOD detection performance with the PHP method.

For the DEC method, we have previously conducted a "VFlip" experiment in Table 7 of Appendix F. In this experiment, the OOD data is set as the vertically flipped in-distribution data. Thus, the DEC method based on compressors cannot work at all, but our method could still rely on the PHP method to achieve promising performance on OOD detection.

We sincerely appreciate the time and effort you've put into reviewing our work. Your comments are incredibly valuable to us! We have taken your concerns into consideration and hope our responses effectively address your concerns.

W1: We absolutely agree with your suggestion that an ideal OOD detection model should be bidirectionally effective. Actually, the property you mentioned is one of the advantages of our method and we highlight the results of bidirectional comparison has been included in Figure 5(c-d) of first-time submission, indicating that our method can still be effective in detecting FashionMNIST as OOD when trained on MNIST as you suggest. On the contrary, the previous SOTA method $\mathcal{LLR}^{ada}$ tends to perform even worse than the basic OOD method with ELBO. To make a more comprehansive comparison, we have added more experimental results to demonstrate our method's bidirectional effectiveness in the Table 12 of Appendix K.2. The key insight is that, unlike baselines like HVK and $\mathcal{LLR}^{ada}$, the PHP and DEC are calibrations on the ELBO directly, which could improve or at least not hurt the performance of applying ELBO. Thus, when ELBO could already work well in some cases, our method could also perform well.

W2: Thanks for your recommendation of this interesting paper (Caterini et al. [1]) and we will cite this paper in our revision.

Firstly, we admit that the paper you recommend is relevant to ours, but we also need to highlight that the breakdown target of our method is different from the paper [1], which will be discussed below:

1. Caterini et al. breakdown the expectation on the log-likelihood $\log p_\theta(x)$ into two terms $\mathbb{E}\_{x\sim P}[\log p_\theta(x)]=-D_{\text{KL}}[P||P_\theta]-\mathcal{H}[P]$, which is not specified to VAEs.

2. In our work, we directly breakdown the expectation on the ELBO (lower bound of $\log p_\theta(x)$) as shown in Eq. (7): $\mathbb{E}\_{x\sim P}[\text{ELBO}(x)]=-D_{\text{KL}}[q(z)||p(z)] - [\mathcal{H}[P]+\mathcal{I}\_q(x,z)-\mathcal{I}\_{q,p}(x,z)]$ , which provides a lens to see what is happening in $D_{\text{KL}}[P||P_\theta]$ and thus operatable methods would be inspired.

Additionally, we want to show our sincereest gratitude for your insightful thinking of our paper as you claim the "special case" where ELBO equals the likelihood. Yes, it is correct that $\text{ELBO}(x)=\log p_\theta(x) =\log P_{ID}(x)$ when there is a perfect generative model to mimic the data distribution, and there could be

$\mathbb{E}\_{x\sim P_{ID}}[\log p_\theta(x)]-\mathbb{E}\_{y \sim P_{OOD}}[\log p_\theta(y)]$ $=\mathbb{E}\_{x\sim P_{ID}}[\text{ELBO}(x)]-\mathbb{E}\_{y \sim P_{OOD}}[\text{ELBO}(y)]$ $=[-D_{KL}(P_{ID}(x)||p_\theta(x))-\mathcal{H}(P_{ID})] - [-D_{KL}(P_{OOD}(y)||p_\theta(y)) - \mathcal{H}(P_{ood})]$ $=-\mathcal{H}(P_{ID}) + \mathcal{H}(P_{OOD}) + D_{KL}[P_{OOD}(y)||p_\theta(y)].$

In that case, we agree with you that calibration on the entropy term could further improve the OOD detection performance.

Paper [1] also provides insights of the likelihood-ratio methods for removing the influence of entropy term, but that relies on an assumption of a perfect VAE model satisfying $P_\theta=P_{ID}$ and find a reference model $R_\phi$ to satisfy $D_{KL}[P_{OOD}||R_\phi]\leq D_{KL}[P_{ID}||R_\phi]$.

However, we feel very sad to emphasize that acquiring a perfect model $P_\theta=P_{ID}$ could be difficult and even impossible for a long time in the future though it is theoretically possible [2,3]. Therefore, further breaking down the $D_{\text{KL}}[P||P_\theta]$ for ELBO-based methods is extreamly necessary.

In summary, the novelty of our analysis is to inspire an operatable way to alleviate the overestimation issue with a theoretical guarantee for a non-perfect ELBO-based generative model, e.g., VAE, which could be shorted as:

1. We could further investigate which factor will lead to a bad $D_{\text{KL}}[P||P_\theta]$, i.e., the mismatch between the aggerated posterior $q(z)$ and prior $p(z)$ or the mutual information gap $\mathcal{I}\_q(x,z)-\mathcal{I}\_{q,p}(x,z)$;
2. Our analysis provides a tracable way to fix the overestimation of VAEs even when it is not a perfect model to estimate $P_{ID}$, i.e., using the PHP method to replace the $p(z)$ with approximated $q_{id}(z)$ and DEC method to calibrate the Ent-Mut term.

[1] Caterini, Anthony L., and Gabriel Loaiza-Ganem. "Entropic issues in likelihood-based ood detection." I (Still) Can't Believe It's Not Better! Workshop at NeurIPS 2021. PMLR, 2022.

[2] Dai, Bin, and David Wipf. "Diagnosing and Enhancing VAE Models." International Conference on Learning Representations. 2018.

[3] Dai, Bin, Li Kevin Wenliang, and David Wipf. "On the Value of Infinite Gradients in Variational Autoencoder Models." Advances in Neural Information Processing Systems. 2021.

I appreciate the considerable effort the authors have put into revising the manuscript. My primary concerns, identified as (W1) and (W2), have been adequately addressed. Additionally, the revised compression algorithms for estimating DEC are more practical compared to the previous SVD-based method, which necessitated setting a $n_{\text{id}}$ for each model. Consequently, I will raise my review score.

The addition of experiments involving diffusions is a noteworthy enhancement. I suggest, for further improvement, to consider formulating a diffusion as a probability flow ODE, rather than a VAE. This approach enables the actual computation of data likelihood $p_\theta(\cdot)$ as mentioned in Song et al. [A]. For the comparisons in Table 17 of your final paper version, it would be beneficial to include this likelihood value as an OOD score and compare it against your method as a baseline. Please check [B] for reference, they perform the likelihood ratio on top of the ODE formulation of the diffusions. Although the model in Goodier and Campbell [B] differs from yours, they do report significantly higher AUC-ROC values.

References:

[A] Song, Yang, et al. Maximum likelihood training of score-based diffusion models. Advances in Neural Information Processing Systems 34 (2021): 1415-1428.

[B] Goodier, Joseph, and Neill DF Campbell Likelihood-based Out-of-Distribution Detection with Denoising Diffusion Probabilistic Models. arXiv preprint arXiv:2310.17432 (2023).

Thanks for your valuable suggestions! To better address your concerns about our method, please allow us to interchange the order of weaknesses in your comments.

W1: UMAP Visualization

Thanks for the suggestion of replacing t-SNE with UMAP visualization, and we have included the UMAP visualization of latent representations on FashionMNIST(ID) /MNIST(OOD) and CIFAR-10(ID)/SVHN(OOD) dataset pairs as shown in Fig.9 of Appendix K.1.

W2: When would the method not work well

Combining the results shown in Table 2 and also the visualization of latent representations in Fig.9, we can find that the PHP method works well in FashinMNIST(ID)/MNIST(OOD), but does not work well in CIFAR-10(ID)/SVHN(OOD). The underlying reason is that the aggregated posterior q(z) of FashionMNIST is quite distinguishable from that of MNIST, but there is a significant portion of the regions overlap between the aggregated posteriors of CIFAR-10(ID) and SVHN(OOD), which leads to the performance gains brought by PHP method is not very noticeable as shown in Table 2.

For the DEC method, it will not work well when the image complexity of ID/OOD datasets is similar, e.g. FashinMNIST(ID)/MNIST(OOD) shown in Table 2 and a special case in Table 7. For the special case of Table 7 that we detect the vertically flipped in-distribution data as OOD data, the DEC method based on SVD or other image compressors does not work, but we can still rely on the PHP method to work in this special case.

W3: Contribution of PHP and DEC

Continue analyzing the experimental results in Table 2, for CIFAR-10(ID)/SVHN(OOD), we can find that the DEC method can bring about an improvement in OOD detection performance, especially when PHP is not working well. Conversely, for FashinMNIST(ID)/MNIST(OOD), when the DEC method is not working well because of the relatively similar image complexity between ID/OOD datasets, the PHP method can bring about performance benefits.

Therefore, the complementarity between PHP and DEC methods can ensure that our combined approach works well in the majority of OOD detection scenarios.

W4: Computational efficiency of the proposed method

Thanks for your suggestion! We need to emphasize that both PHP and DEC methods are post-hoc methods with trained VAEs, which will not cause too many additional computation burdens. To make an intuitive efficiency comparison with performing OOD detection on VAEs with ELBO, we testify to the computation efficiency from two perspectives: training time and inference time, in Table 18 and Table 19 of Appendix K.5. Results show that the average computation time in scoring a data point on an A100 GPU is approximately 1.5x computation time than vanilla ELBO, which is still a very small number and could be much faster if we change the SVD to other image compressors or a binary search version. The additional time for training the LSTM of the PHP method is also very fast, taking about 30 min to converge.

W5: Comparison with ensemble methods

Thanks for your suggestions, we reproduce the ensemble method WAIC with an ensemble of 5 VAEs to test their performance (added to Table 2) and computation efficiency (in Table 18 and Table 19 of Appendix K.5).

Q1: Complex architecture

To demonstrate that our method can also work well on generative models with more complex architectures, we have included the experimental results of applying our method to the recently popular diffusion model in Table 17 of Appendix K.4. From the results, we can find that our method still works well on OOD detection with trained diffusion models, demonstrating the generalizability of our approach. More experimental details can be found in K.4.

Q2: Robustness on noisy data To evaluate the robustness of the proposed method on noisy datasets, we manually introduce varying scales of random noise into input data samples to create OOD datasets at different noise levels. Then, we apply our method to these noisy datasets and obtain the experimental results as shown in Table 21 and Table 22 of Appendix K.7. We can find that our method still works well on these noisy datasets.

Q3: Influence on real-world decisions or systems that rely on OOD detection

A real-world use case could be autonomous driving we hope to identify whether the current driving state is strange to the training set for ensuring the safety of decision-making. In this case, we should give lower confidence to the strange (OOD) states, otherwise, incorrect high confidence may hurt the safety. Actually, this could be the setting of offline RL, where a core direction of research focus is to identify the out-of-distribution state-action pairs and assign them a lower reward [1]. Moreover, as the RL setting always has a lot of states, e.g., millions of steps, it could be impossible to label every state, which appeals to unsupervised OOD detection.

[1] Fujimoto et al. "Off-policy deep reinforcement learning without exploration." ICML, 2019.