Paper: Casting Light on Large Generative Networks: Taming Epistemic Uncertainty in Diffusion Models (ICLR-2024)

Reviewer comment (Evaluation):
The motivation of estimating the epistemic uncertainty for image generation is unclear. The experiment result, *s.t.* different diffusion models would generate images with higher levels of variation on classes with fewer training data, is very much in line with our expectations as well as with empirical results even before computing the epistemic uncertainty. In other words, it’s unclear how the computation of epistemic uncertainty could be helpful in practice. (On the contrary, computing epistemic uncertainty under an active learning setting is very well-motivated.)

Author response:
We agree that the motivation behind epistemic uncertainty estimation for image generation should be better addressed in the text. To that end, we have the following text to the paper:  

Collecting data for image generation models can be a costly endeavor. Therefore, when seeking to enhance a model, leveraging epistemic uncertainty becomes a crucial factor in selecting new data points. This concept is frequently employed in Active Learning methodologies such as BALD Houlsby et al. 2011 and BatchBALD Kirsch et al. 2019. This underscores the relevance of applying our framework. It is essential to acknowledge that, at present, these models entail significant training costs. As a result, we do not offer active learning experiments. However, with the anticipation of future advancements in computational resources, there may be increased feasibility to explore these ideas.

Reviewer comment (Theory):
The estimation of epistemic uncertainty from the background and the methodology section focuses on the quantity $I_{\rho}(y_{t-1},\theta\vert y_t,x)$, where the time interval is 1 timestep. Meanwhile, the experiment section reported $I_{\rho}(z_0,\theta\vert z_5,x)$, which has a time interval of 5 timesteps. It’s unclear how the mutual information with an interval of multiple timesteps can be derived from the quantity with 1 timestep.

Author response:
To clarify, when running the experiments we leverage DDIM (Song et al. 2021) to generate images more efficiently. In doing so, we take larger steps in the reverse diffusion process than DDPM (Ho et al. 2020). Hence the timestep interval changes from 1 in the reverse diffusion process to 5. Please note this is highlighted in the text, “It’s important to note that we can only take steps of 5 through the diffusion process due to the 200 DDIM steps.”

Reviewer comment (Experiments):
Different classes are being used to compare epistemic uncertainty; a more apples-to-apples comparison would be to train multiple ensembles on the same class with different number of training instances, and computing PaiDEs on the generated images for that class.

Author response:
We acknowledge the interest in your experimental setup; however, due to the timeframe involved (more than 7 days for obtaining results), we lacked sufficient time and access to compute to thoroughly evaluate the idea. Currently, we are in the midst of expanding our work for a journal submission that will encompass your suggestion. The experiments detailed in the paper focused on isolating epistemic uncertainty within an active learning framework. In this context, the emphasis was on between-class epistemic uncertainty to inform the selection of new data points.

Reviewer comment (Theory):
Could the authors elaborate on the usage of 8 samples of random noise for the computation of epistemic uncertainty in Section 4.1? Given a particular component, 8 images generated from 8 different noise samples would represent aleatoric uncertainty instead of epistemic uncertainty.

Author response:
For every random noise sample, our approach generates a distinct estimate of epistemic uncertainty for each class $x_i$. To ensure the robustness of our epistemic uncertainty measure, we take the average across the 8 samples. This process is explicitly described in the paper, "We then average the ensemble’s epistemic uncertainty over these 8 random noise samples.". It's important to note that we do not analyze aleatoric uncertainty. We do not compare the images generated from different samples of random noise.

Reviewer comment (Theory):
Could authors explain the sentence in Section 4.1, “For bin 1300, we observe that epistemic uncertainty highlights different birds that could have been generated from our ensemble”? For a class with sufficient amount of training data, the variation among generated samples shall represent aleatoric uncertainty instead.

Author response:
In Figure 4, the plotted epistemic uncertainty on a per-pixel basis reflects the ensemble's component-wise variability derived from a single random noise sample. Aleatoric uncertainty is not considered in this figure as we consider only one sample of random noise and we are considering the variability within the ensemble.

Reviewer comment (Theory):
Could the authors provide an explanation (or a high-level intuition) for how the mutual information between *data* and *model parameter* could represent epistemic uncertainty — the uncertainty that captures the lack of knowledge? From Eq. (4), it’s makes sense to view the epistemic uncertainty as the difference between the total uncertainty and the aleatoric uncertainty; but the quantity of mutual information alone doesn't seem to say a whole lot about the level of ignorance.

Author response:
The mutual information criterion considered in the paper, is difference between total and aleatoric uncertainty (i.e. $I(X,Y) = H(Y)-E[H(Y|X)]$, EU=TU-AU) (Decomposition of Uncertainty in Bayesian Deep Learning for Efficient and Risk-sensitive Learning, Depeweg et al. 2018). Mutual information measures the amount of information gained about one variable from observing the other. In the context of Eq. 4, given an observation of $\theta_i$ what do we know about the distribution of $y$. One would expect that given many instances of a certain class in the training dataset that the ensemble components would produce relatively similar distributions of the output. This is conveyed in the paper, “Mutual information measures the information
gained about one variable by observing the other. When all of $\theta$’s produce the same $p_{\theta} (y_0|y_T, x)$, $I(y_{t−1}, \theta|y_t, x)$ is zero, indicating no epistemic uncertainty. Conversely, when said distributions have non-overlapping supports, epistemic uncertainty is high.”

Reviewer comment (Reproducibility):
Some of the details in the experimental setup are lacking. I was unable to find the number of ensemble particles used in the experiments.

Author response:
Number of Components:
 
While the number of components can be seen from Figures 2 and 6, we recognize that this information should have been more explicitly stated in the text. Therefore, we have included the following clarification:
 
We utilize an ensemble of 5 components, a number we found to be sufficient for estimating epistemic uncertainty in our context.

Reviewer comment (Evaluation):
The uncertainty distribution across different bins seems to be very similar without a huge difference. For example: see Fig 3. In Fig 3, even though labels in bin 1 are trained with a single datapoint, uncertainty is pretty small. It could be because ensemble particles only differ through random initialization. Authors might find the following work on alternate ensemble methods (ex: https://arxiv.org/pdf/2206.03633.pdf) and epistemic uncertainty methods (ex: https://arxiv.org/pdf/2107.08924.pdf, https://arxiv.org/pdf/2006.07464.pdf) useful.

Author response:
Uncertainty Distributions:
 
Epistemic uncertainty is a relative measure, and the community lacks consensus on the definition of groundtruth uncertainty. Consequently, a small value for epistemic uncertainty is not inherently incorrect. To our knowledge, we are pioneers in considering epistemic uncertainty in the realm of large-scale generative models, we interpret the observed differences between bin distributions in Figure 3 as indicative of the expected trend (i.e. bins with lower values are more skewed towards lower epistemic uncertainty). We hope to investigate further the inter-bin distributions in future research and our current findings provide valuable insights.

Reviewer comment (Evaluation):
In Table 1, possibly for the same reason as above, bins 1, 10, and 100 have very similar performance. Can authors offer some intuition on why this could be the case.

Author response:
Image Similarity in Table 1:
 
The model encountered challenges in generating accurate images for bins 1, 10, and 100, as illustrated in Figure 4. Consequently, it is understandable that the SSIM between components for these bins would exhibit similar values, irrespective of the branching point.

Reviewer comment (Presentation):
It might be useful to describe the approach in Section 3.1 via a diagram indicating which parts of the network are ensembled and which are frozen with pre-trained weights.

Author response:
Additional diagram for Frozen Weights:
 
To address this we have included the following text in the caption of Figure 1:
 
In our ensembles, networks taking class labels as input are randomly initialized and trained, with pre-trained encoders, decoders, and UNets for each component.