Rare event modeling with self-regularized normalizing flows: what can we learn from a single failure?

Thank you for your feedback. You raise an important point r.e. justifying CalNF’s increased complexity with corresponding increased performance.

First, you are correct that the ensemble approach has fewer hyperparameters (one fewer than our method) and is easier to parallelize. We have added discussion of these points to the limitations section of our revised paper.

Second, we would like to clarify point (ii) in your review: in our implementation, the ensemble model is trained on both nominal and target data (using a label to distinguish between the two, implemented similarly to $c$ in our method). We include a table below (and in the appendix of our revised paper) that shows the results of training the ensemble model on the target data alone. We find that excluding the nominal data decreases the performance of the ensemble model on three out of four problems. In addition, all of the regularization-based methods also require training on nominal as well as target data.

Third, while the results in Table 1 shows that our method achieves modest performance gains over the ensemble method on the posterior learning task (as you point out in your review), Table 2 provides important context for how posterior learning affects downstream tasks like anomaly classification. The results in Table 2 show that our method yields better, more consistent performance than all other methods on this downstream anomaly classification task.

Finally, regarding online or real-time applications: our method is designed primarily with post-event failure analysis (i.e. “post-mortem” analysis), but if we were to deploy CalNF in real-time (e.g. for failure detection), we would pre-train on any available data offline, then use CalNF for online inference to classify new examples as nominal or failure. While CalNF is more complex to train than an ensemble model, it is faster at inference time (since we only need to evaluate one model with the optimized label rather than evaluating all mixture members). In this application, we would expect the improved performance of CalNF on downstream tasks like failure detection (in Table 2) to be even more important.

Does this help clarify your questions from your review? We are happy to follow up if you have further questions or if there are any points that you are not satisfied with.

Thank you for your review and feedback! We have included responses to your questions below along with a revised version of our paper (with changes highlighted in blue).

Weakness 1 (lit review): Thank you for this feedback. We had discussed prior work in the introduction and background sections of our paper, but we agree that there should be a focused subsection on related work. We have added this subsection in our revised paper.

Weakness 2 (baselines): We selected baselines that we believe are representative of prior work on this problem. Most of the work in the last few years on few-shot generative modeling has focused on image generation (see Abdollahzadeh et al., 2023 in our bibliography for a survey). Unfortunately, these methods for few-shot image generation are typically not applicable to non-image domains, since they rely on pre-training on massive open image datasets and it is not clear how they would be applied when large domain-specific datasets are not available. We have included further discussion of this in the revised related work section. If you have suggestions for specific methods for us to consider, we would be happy to discuss those.

Weakness 3 (increasing ensemble size): While increasing the ensemble size generally improves performance on problems with plentiful data, this effect is much less strong in data-constrained settings. This is because each ensemble member is trained on a random subsample of the target dataset, and when this dataset is small there are only a limited number of possible distinct subsamples (e.g. only 35 possible combinations for $N_t=4$, as in the ATC problem). Even before $K$ reaches this limit, each new subsample becomes more and more similar to previously-drawn subsamples. This is why our results in Fig. 11 show that the performance of our method increases slightly, but not dramatically, as $K$ increases. We expect the same pattern to hold for the ensemble method, since it relies on the same subsampling method.

Q1: Thank you for catching this typo. The self-regularization term is included to encourage minimizing the divergence between candidate posteriors, and so Equation (5) should have a plus sign. We have fixed this in the revised paper.

Q2: While increasing diversity between ensemble members may be a good strategy when lots of training examples are available, it does not work well in the data-constrained setting. As we show in Lemma 2 in our paper, in the absence of any regularization the optimal (maximum likelihood) ensemble members are those that approximate a delta function at each training example (i.e. overfit to their respective subsamples). In this case, even two ensemble members trained on subsamples that differ by a single data point would have KL divergence approaching $\infty$, since one would assign near-zero probability mass to the example that differs. In short, encouraging diversity in the sparse-data regime encourages overfitting of each ensemble member.

Fig. 12 illustrates this counterintuitive result empirically by varying both the regularization strength $\beta$ and the size of the failure dataset. We find that when the failure dataset is very small, a larger regularization strength ($\beta=10$) yields better performance, but smaller regularization strengths perform better when the failure dataset is large. We have added some discussion of this point in the appendix of our revised paper (page 20, immediately prior to Fig. 12).

Q3: You are correct on the typo in Equation 3; we have fixed this in our revised paper. For $L_{cal}$, we have revised this equation to be more consistent in referring to $c$ as the variable and $c^*$ as its optimized value.

(continued in following comment due to space limitations)

Thank you for your review and feedback! We have included responses to your questions below along with a revised version of our paper (with changes highlighted in blue).

Complexity & added training cost (Weaknesses 1 and 2, Question 3): You ask about suitability to robotics applications. We would like to emphasize that while our method is more complex to train, there is no penalty at inference time compared to any other normalizing flow-based method. Because of this, we believe that our method would be suitable for real-time applications like robotics, where the normalizing flow trained using CalNF could be used for downstream tasks like anomaly detection or control. We have added some discussion of this point to the revised paper.

We also acknowledge that our method is slightly more complex to implement than a vanilla normalizing flow (we provide an example implementation in the supplementary materials). As such, CalNF would not be the best choice for a problem with plentiful failure data, but our experiments (both comparisons and ablations) show that this additional complexity is needed to achieve good performance on data-constrained problems.

Sensitivity to hyperparameters $\beta$ and $K$ (Question 2): Our results in Fig. 11 suggest that our method is not very sensitive to the choice of $K$, likely because there are so few data points (e.g. the SWI problem only includes 4 failure examples, so increasing $K$ just leads to sampling additional subsets that are statistically similar to those already sampled). Our results in Fig. 12 suggest that we may have been able to achieve better performance by tuning $\beta$ (e.g. higher $\beta$ seems perform better on very small failure datasets), but we wanted to avoid accidentally overfitting to our limited datasets through excessive hyperparameter tuning (the UAV and ATC datasets in particular only have a small amount of available real-world data).

Practically, we fixed $K=5$ and $\beta=1$ for all of our experiments and found that we achieved good performance without further fine-tuning. We would recommend that practitioners start with these values, then potentially tune $\beta$ depending on how much failure data is available (with larger $\beta$ helping to avoid overfitting when very few failure examples are available).

Generalization to new data (Weakness 4 and Question 1): You are correct that the ability to generalize beyond the training dataset is important. In fact, the inability of existing methods to generalize well based on limited failure datasets (due to overfitting) is the primary motivation for our method. In our experiments, we quantify how well our method generalizes to new data by reporting our model’s performance on a test set of previously-unseen failure examples. These empirical results (Tables 1, 2, and 3) show that our method generalizes well to previously-unseen failures.

There is still an open question on how well our method would adapt to new failure examples from an entirely different distribution than what is seen during training, but this limitation applies to any learned generative model.

Risk estimates or probability bounds (Weakness 3): We included this issue in the limitations section of our original paper. We agree with you that obtaining these risk estimates would be useful in many practical scenarios, and look forward to addressing the theoretical and practical challenges of obtaining accurate probability bounds from a very small number of examples in future work. If you have any thoughts or suggestions on this future direction, we would be happy to hear them!

Does that help answer your questions? We are happy to follow up if you have further questions or if there are any points that you are not satisfied with.

Thank you for your review and feedback! We have included responses to your questions below along with a revised version of our paper (with changes highlighted in blue).

1. Calibrating at the same time as training: Since the loss function used for calibration ($L_{cal}$) is the same as the third term in the loss function used for training the normalizing flow ($L$), it would be equivalent to optimize $c^*$ and the parameters of the flow jointly. In Algorithm 1, we put the model and calibration updates on two separate lines for clarity, but they could be implemented as one joint update. We have added some discussion to this effect.
2. Referencing Fig. 4: thank you for catching this mistake. We now discuss Fig. 4 on page 6 of the revised paper.
3. Real-valued calibration label: Our method fully supports continuous values for the calibration label $c^*$, the observations $x$, the latent variables $z$, and the context variables $y$, and all of our experiments use continuous values for these variables. We have added a note on this to the problem statement in our paper.
4. Additional details on anomaly rejection and few-shot image learning experiments: Thank you for this feedback. We have added additional information on these experiments to our revised paper.

Does that help answer your questions? We are happy to follow up if you have further questions or if there are any points that you are not satisfied with.

Thank you for your review and feedback! We have included responses to your questions below along with a revised version of our paper (with changes highlighted in blue).

1. Cases when failure events are drastically different from nominal events: This is an interesting question! The implicit assumption behind our work is that there is a benefit from learning a shared representation for both the nominal and failure events. In Theorem 1, we show that our method provides an implicit regularization between the nominal and failure events. This could be either a benefit or a drawback (depending on whether there is actually some structural similarity between the nominal and failure distributions). We have added a remark in the limitations section of our paper discussing this point.
2. How did we select $K$ and $\beta$? We did not spend much time tuning either parameter: we used $K=5$ and $\beta=1$ on all problems. Our results in Fig. 11 suggest that our method is not very sensitive to the choice of $K$, likely because there are so few data points (e.g. the SWI problem only includes 4 failure examples, so increasing $K$ just leads to sampling additional subsets that are statistically similar to those already sampled). Our results in Fig. 12 suggest that we may have been able to achieve better performance by tuning $\beta$ (e.g. higher $\beta$ seems perform better on very small failure datasets), but we wanted to avoid accidentally overfitting to our limited datasets through excessive hyperparameter tuning (the UAV and ATC datasets in particular only have a small amount of available real-world data). As a result, we kept $K$ and $\beta$ fixed in our experiments and found that to be sufficient.
3. Extending the proposed method to high-dimensional data? The main bottleneck in expanding to high-dimensional data (e.g. images) comes from evaluating the underlying normalizing flow. To address this issue, we can leverage existing work on adapting normalizing flows for image data, such as the Glow architecture (Kingma & Dhariwal 2018). In Section 4.5 of our paper, we demonstrate applying our method to both the MNIST and CIFAR-10 image datasets using the Glow architecture for the underlying flow. More work is certainly needed to improve on these results on image modeling, but we hope that these experiments demonstrate how our method might be adapted to high-dimensional image data.
4. Are normalizing flows the optimal solution? This is a good question. The general idea of our paper (self-regularization of an ensemble model) could likely be applied to other generative models, but our use of KL divergence to regularize between candidate posteriors means that we need to be able to evaluate the learned likelihood $q_\phi$ for different labels $c$ in addition to drawing samples from the learned distribution. Since evaluating the likelihood for GANs and diffusion models is not straightforward, normalizing flows are a natural choice for our method. There are also other benefits to normalizing flows for downstream tasks; for example, the ability to evaluate the likelihood allows them to be adapted easily to anomaly classification applications.

Does that help answer your questions? We are happy to follow up if you have further questions or if there are any points that you are not satisfied with.