ConjNorm: Tractable Density Estimation for Out-of-Distribution Detection

We thank reviewer MqR8 for your valuable comments. We updated the submission accordingly. Please kindly find the detailed responses below.

Q1: The main search problem of optimal coefficient for OOD scoring is a hyperparameter search, which may constrain the practicality of the proposed score, and How to choose $p$.

A1: Similar to DICE [d], we adopt a validation set of Tiny ImageNet as the auxiliary OOD data for tuning $p$. we empirically define the search space of $p$ as (1,3]. We also observe that SOTA post-hoc OOD detection methods [a,b,c,d,e,f] also come with (one or more) hyper-parameters, where their searched values vary across datasets as well.

We would like to argue that the tuning $p$ is reasonable and necessary for density-based OOD detection since the (latent feature) distribution of different datasets could not be necessarily the same as each other. By simply adjusting the value of the norm coefficient, we can succeed in finding the (relatively) most suitable distribution from the exponential family for each ID dataset in a computationally efficient manner, which is exactly one of the strengths of our method.

[a] Extremely simple activation shaping for out-of-distribution detection. ICLR 2023

[b] LINe: Out-of-Distribution Detection by Leveraging Important Neurons. CVPR 2023

[c] Out-of-distribution detection with deep nearest neighbours. ICML 2022.

[d] Dice: Leveraging sparsification for out-of-distribution detection. ECCV 2022

[f] Out-of-Distribution Detection via Conditional Kernel Independence Model. NeurIPS 2022

Q2: For long-tailed OOD detection when the ID training data exhibits an imbalanced class distribution, I guess the accuracy of the importance sampling-based estimation may decrease given a limited number of tail-class predictions. Elaborate why their method still outperforms in the long-tailed scenarios.

A2: We agree with your insightful intuition. The mentioned class imbalance is widely recognized as a challenging setting. In response to your valuable comments, we have conducted the following experiments.

As shown in the Table below, all methods that involve the use of ID training data suffer from a decrease in their averaged OOD detection performance when the ID training data is with class imbalance. Note that we keep using the network pre-trained on the long-tailed version of CIFAR-100 for fair comparison. Even so, our method consistently outperforms in both scenarios, which implies the robustness of our method. We suspect the reason is that the flexibility of the norm coefficient provides us with the chance to find a compromised distribution from the exponential family.

Thanks again for your valuable comments. We have revised the submission accordingly by adding the discussion above. Please refer to Section A.12 in the revision for details.

We're grateful for your quick feedback during this busy period. We would like to emphasize that

1. it is important and necessary to tune $p$ indeed since it is reasonable for the ID distribution $p(x)$ to vary from dataset to dataset. We kindly note that previous methods impose a fixed distributional assumption for all datasets (Gaussian in GEM and Maha; Gibbs-Boltzmann in Energy). However, different datasets should posse a different $p(x)$, based on the optimal results achieved: p=2.2 for CIFAR-10, 2.5 for CIFAR-100, 1.5 and 1.8 for ImageNet-1k on ResNet50 and MobileNetv2.

2. while the OOD detection performance depends on the value of $p$, it can be seen from Figures 4.c and 4.c that the performance of our method remains comparable to SOTA in CIFAR-10 when $p \in [1.6,2.4]$ and consistently outperforms SOTA in CIFAR-100 when $p \in [1.8,3.0]$.

We agree with the reviewer that currently searching for the optimal p is suboptimal as the auxiliary data is required for cross-validation. It is definitely more practical if an OOD detector can adjust its hyper-parameter(s) with only access to its ID data though this is still an open problem in the OOD detection domain.

Inspired by your insightful comments, we consider one of our future works to synthesize OOD samples with powerful generative models such as stable-diffusion models as alternative sources for auxiliary OOD validation data. The corresponding exploration will be conducted very soon.

Q2: Maybe it's worth adding a section summarizing the paper's technical novelty.

A2: We thank you for your advice. The contributions of our method are summarised as follows:

• It is always non-trivial to generalize from a specific distribution/distance to a broader distribution/distance family since this will trigger an important question to the optimal design of the underlying distribution ($\clubsuit$).To answer this question, we explore the conjugate relationship as a guideline for the design. Compared with other hand-crafted choices, our proposed $l_p$ norm is general and well-defined, offering simplicity in determining its conjugate pair. By searching the optimal value of p for each dataset, we can flexibly model ID data in a data-driven manner instead of blindly adopting a narrow Gaussian distributional assumption in prior work, i.e., GEM and Maha.

• Our proposed framework reveals the core components in density estimation for OOD detection, which was overlooked by most heuristic-based OOD papers. In this way, The framework not only inherits prior work including GEM and Maha but also motivates further work to explore more effective designing principles of density functions for OOD detection.

• We demonstrate the superior performance of our method on several OOD detection benchmarks (CIFAR10/100 and ImageNet-1K), different model architectures (DenseNet, ResNet, and MobileNet), and different pre-training protocols (standard classification, long-tailed classification and Contrastive learning).

We included the summary above in Section A.11 of the revision.

Q3: List out all the assumptions made

A3: Thank you for your advice. The assumptions made in our method are given as follows:

• Assumption 1. The ID class prior is uniform, i.e., $\hat{p}_{\boldsymbol{\theta}}\left(k\right)=\frac{1}{K}$.

Assumption 1 helps our method work without true knowledge of the ID class prior distribution. We note that the assumption is also made in many post-hoc OOD detection methods either explicitly or implicitly [b]. Experiments in Section 4.4.2 show that our method still outperforms in long-tailed scenarios.

• Assumption 2. $g_\varphi(\cdot)=const$ and $\psi(\cdot) = \frac{1}{2}\|\|\cdot\|\|_{p}^{2}$

Assumption 2 helps to reduce the complexity of the exponential family distribution. While it is possible to parameterize the exponential family distribution in a more complicated manner, our proposed simple version suffices to perform well.

We listed the assumptions above in Section A.12 of the revision.

[b] Detecting Out-of-distribution Data through In-distribution Class Prior. ICML 2023

Q4: it would be good if the authors could make the notations more distinguishable.

A4. Thanks for your advice. We have improved our used notations in the revised version.

We thank Reviewer LYsh for your thorough comments. As to the weaknesses and minor issues you pointed out, we took them very seriously, and have updated parts of the paper to improve it. Our response is as follows.

Q1.1: theoretical justification

A1.1:

Setup. Let $\mathcal{P}\_{\text{in}}$ and $\mathcal{P}\_{\text{out}}$ are the underlying distributions for ID and OOD data. We use $p\_{\text{in}} (\mathbf{z})$ and $p\_{\text{out}} (\mathbf{z})$ to denote the probability density function where the input $\mathbf{z}$ is sampled from the feature embeddings space $\mathcal{Z}$. We model ID data as a mixture of ID-class conditioned exponential family distributions, i.e.,

$

p\_{\text{in}}\left(\mathbf{z}\right) = \frac{1}{K} \sum\_{k=1}^{K} p\_{\text{in}}\left(\mathbf{z} | k\right) = \frac{1}{K} \sum\_{k=1}^{K} \frac{\exp(-d\_\varphi(\mathbf{z},\boldsymbol{\mu}(\boldsymbol{\eta}\_k)))}{\int \exp(-d\_\varphi(\mathbf{z}',\boldsymbol{\mu}(\boldsymbol{\eta}\_k))){\rm d}\mathbf{z}'}

$

Inspired by open-set recognition [a], we treat OOD data as a whole and model it as a single exponential family distribution parameterized by $\boldsymbol{\eta}\_{\text{out}}$, i.e.,

$

p\_{\text{out}}\left(\mathbf{z}\right)=\frac{\exp(-d\_\varphi(\mathbf{z},\boldsymbol{\mu}(\boldsymbol{\eta}\_\text{out})))}{\int \exp(-d\_\varphi(\mathbf{z}',\boldsymbol{\mu}(\boldsymbol{\eta}\_\text{out}))){\rm d}\mathbf{z}'}.

$

We consider the following measure to how well our method distinguishes ID data from OOD data:

$

D:=\mathbb{E}\_{\mathbf{z} \in \mathcal{P}\_{\text{in}}} [\hat{p}\_{\boldsymbol{\theta}}\left(\mathbf{z}\right)] - \mathbb{E}\_{\mathbf{z} \in \mathcal{P}\_{\text{out}}} [\hat{p}\_{\boldsymbol{\theta}}\left(\mathbf{z}\right)]

$

Main Result. We have the following bound:

$

\mathbb{E}\_{\mathbf{z} \in \mathcal{P}\_{\text{in}}} [\hat{p}\_{\boldsymbol{\theta}}\left(\mathbf{z}\right)] - \mathbb{E}\_{\mathbf{z} \in \mathcal{P}\_{\text{out}}} [\hat{p}\_{\boldsymbol{\theta}}\left(\mathbf{z}\right)] \le \alpha

$

where $\alpha: = \frac{1}{K} \sum_{k=1}^{K} \sqrt{\frac{1}{2}d\_\varphi(\boldsymbol{\mu}(\boldsymbol{\eta}_k),\boldsymbol{\mu}(\boldsymbol{\eta}\_{\text{out}}))}$. Please refer Section A.10 in the revised manuscript for detailed proof.

Main Result bounds the measure $D$ in terms of Bregman divergence between $\boldsymbol{\mu}(\boldsymbol{\eta}\_k)$ and $\boldsymbol{\mu}(\boldsymbol{\eta}\_{\text{out}})$. It can be observed that $D$ will converge to 0 as $\alpha \rightarrow 0$. This indicates that the performance of our method can be guaranteed by a sufficiently discriminative feature space where the averaged Bergman divergence between ID-class means and OOD data mean is sufficiently large. Note that this theory is empirically justified by our results in Section A.5 where CIDER are more beneficial to our method than SupCon with the former learning more powerful feature representations than the latter.

In the future, we will delve deeper into the theoretical understanding of our method as our future work.

[a] Adversarial reciprocal points learning for open set recognition. TPAMI, 2021.

Q1.2: Explain why the algorithm tends to perform well for small p values

A1.2: Since the searching process of the coefficient $p$ is data-driven, the optimal value of $p$ should vary from dataset to dataset. Therefore, while extensive experiments show that small P values tend to be beneficial to OOD detection on CIFAR and ImageNet, this observation does not necessarily hold for all datasets..

We thank Reviewer WvTp for your constructive comments. To address some of your concerns, we have updated our paper. Please see below for our point-to-point rebuttal.

Q1: Clarify “Without loss of generality, we employ latent features z extracted from deep models as a surrogate for the original high-dimensional raw data $\mathbf{x}$. This is because $\mathbf{z}$ is deterministic within the post-hoc framework”

A.1: Thanks for pointing out the potentially confusing description.

• Since this paper mainly focuses on the feature space for determining OOD data, we use different notations to explicitly discriminate raw input data $\mathbf{x}$ from the latent feature $\mathbf{z}$ to make things clearer. Note that some prior papers sometimes use $\mathbf{z}$ and $\mathbf{x}$ interchangeably.
• To clarify, the term "deterministic" here means that, for any given input image x, we can always have a deterministic representation z since the pre-trained encoder is fixed in the setting of post-hoc OOD detection. This motivates us to choose the lower-dimensional feature space as a suitable surrogate of the raw data space X for more computationally efficient density estimation.

Q2: Discuss the role of Deep generative models (DGMs) for flexible density estimation in OOD detection

A.2: Thanks for your kind suggestion.

• We agree that using DGMs for density estimation is, of course, a valid and intuitive option. However, aligning with our response in A.1, this practice requires training DGMs from scratch to reconstruct high-dimensional $\mathbf{x}$, therefore bringing more computational overheads.

• Please kindly note that our paper focuses on the task of Post-hoc OOD detection where only pre-trained models at hand are expected to be used to detect OOD data from streaming data at the interference stage.

• We also explore the possibility of integrating pre-trained Diffusion models [a,b] into zero-shot class-conditioned density estimation based on Eq.(1) in [c]. Unfortunately, the computation is intractable due to the integral. Although authors in [c] use a simplified ELBO for approximation, there is no theoretical guarantee that the ELBO can align with the data density not to mention the computational-inefficient inference of diffusion models. We will leave this challenge as our future work.

[a] Scalable Diffusion Models with Transformers. ICCV 2023.

[b] High-Resolution Image Synthesis with Latent Diffusion Models. CVPR 2022

[c] Your Diffusion Model is Secretly a Zero-Shot Classifier. ICCV 2023.

Thanks again for the valuable suggestion. Accordingly, we have added the discussion on DGMs for density estimation in Section A.8 of the revised version.

Q3: Tell more about why learning the natural parameter of an Exp. Family is intractable

A.3: Thank you for your advice. given the fact that $\int \hat{p}_{\boldsymbol{\theta}}\left(\mathbf{z}|k \right) {\rm d}\mathbf{z} =1$, we then have:

$\int \exp \left\lbrace \mathbf{z}^\top\boldsymbol{\eta}\_k-\psi(\boldsymbol{\eta}_k)-g\_{\psi}(\mathbf{z})\right\rbrace{\rm d}\mathbf{z}=1$

This means that, for any known $\psi(\cdot)$ and $g\_{\psi}(\cdot)$, one can learn the natural parameter $\boldsymbol{\eta}_k$ by solving the following equation:

$\exp \psi(\boldsymbol{\eta}_k)=\int \exp \left\lbrace \mathbf{z}^\top\boldsymbol{\eta}\_k-g\_{\psi}(\mathbf{z})\right\rbrace{\rm d}\mathbf{z}$

Since the right side of the equation includes the integral over latent feature space that is high-dimensional, learning the natural parameter of an Exp. Family is said to be intractable.

We add the elaboration above in Section A.9 of the revised version.

Dear Reviewer WvTp,

Thanks a lot for your time in reviewing and insightful comments, according to which we have carefully revised the paper to answer the questions. We sincerely understand you’re busy. But since the discussion due is approaching, would you mind checking the response and revision to confirm where you have any further questions?

We are looking forward to your reply and happy to answer your further questions.

Best regards

Authors

We appreciate the insightful comments provided by Reviewer EBSh. Please see our responses to your concerns below.

Q1: I think this is an explicit assumption on the prior distribution, which contradicts $\spadesuit$ ( how can we obtain a tractable estimate for $\Phi (k)$ without presuming any particular prior distribution of $\hat{p}_{\boldsymbol{\theta}}\left(\mathbf{z}|k \right)$ )

A.1: Thanks for your insightful comments. We would like to note that: 1) the mild assumption of the exponential family is introduced to guide the design of density function ($\clubsuit$), where we search against the given dataset for the best choice of $l_p$ to determine the optimal density function. 2) $\spadesuit$ targets at the computation of the normalizing constant $\Phi(k)$ in Eq.(11) that involves the integral over high-dimensional feature space. Different from prior work that specifies $\hat{p}_{\boldsymbol{\theta}}\left(\mathbf{z}|k \right)$ as a pre-defined distribution, e.g., Guassian, to simplify $\Phi(k)$ as a known value, we alternatively design an importance sampling-based estimator of $\Phi (k)$ as the solution to $\spadesuit$ without loss of generalization. Note that our proposed estimator does not rely on any prior knowledge of data distribution and therefore can be ideally applied to any forms of density functions.