Rare Event Probability Learning by Normalizing Flows

1. I am not convinced of the novelty of this approach. This paper mostly cites pre 2021 papers. Please clarify the relation to more modern approaches such as [1,2].

2. The flows are trained with the reverse KL. This comes with some caveats. First one assumes differentiability of the function $g$. Please comment on whether this is realistic. Furthermore, the reverse KL is known to be mode seeking. I think for most applications in the field of rare event sampling it is crucial to cover all the modes of a density. There has been some recent line of work for normalizing flows such as [3] to overcome this but this seems like a major limitation.

3. Similarly, the evaluation should also include some measure of the distance to the true measure and not only the estimated probability. As far as I understand the paper, this should be possible.

4. Please also cite relevant papers such as [4], who introduced a kind of log det schedule for covering multimodal distributions, which I think is related to way the different $\Omega_{a_i}$ are constructed.

5. This paper does not come with any code. Do the authors intend to make their code public? Appendix C does not suffice for reproducibility in my opinion.

6. The heuristic why MCMC wont cut it for this problem makes sense for vanilla MH. But if one takes gradient informed steps such as HMC or MALA, I am not sure why this rationale outlined in section 3.3 should hold true. What is the proposal for MCMC taken in the experiments?

[1] A Flow-Based Generative Model for Rare-Event Simulation, Gibson et al

[2] Conditioning Normalizing Flows for Rare Event Sampling, Falkner et al

[3] Flow Annealed Importance Sampling Bootstrap, Midgley et al.

[4] Deep Probabilistic Imaging: Uncertainty Quantification and Multi-modal Solution Characterization for Computational Imaging , Sun et al.

• Novelty: I don’t find much novelty in the proposed paper. The technique presented by the authors has already been explored in many prior works in different fields, particularly in physics, where rare event sampling is often a challenging problem (see below).

• Related Works: Despite many prior works combining normalizing flows with importance sampling, and beyond, exist, this paper lacks a dedicated Related Work section. Several seminal works have been completely overlooked despite their significant contributions to the field of normalizing flow-aided importance sampling in statistical physics [1], chemistry[2], and quantum field theory[3,4,5].

• Annealed Importance Sampling: There is no reference to annealed importance sampling [6], which I believe is highly tight to the idea of the paper. Besides [6], several relevant works [7,8,9] perform annealed importance sampling within the context of normalizing flows, falling within the same category as the CRAFT method referenced in the paper, though only marginally. What these methods do closely aligns with what the authors propose in the paper: instead of learning the target distribution in one step, they 'anneal' towards that distribution by learning and sampling from intermediate distributions, ensuring that the final learned probability density has as much support as possible, including regions where the target density is small enough to fall within the rare event regime. I believe it is crucial for this paper to be published in this or any other venue to highlight the connection to these (and the previously referenced works).

• Rare Event Sampling: A recent paper [10] discusses similar behaviors in training normalizing flows and combining them with importance sampling to ensure full support over the target density, including rare event regions. I would find it interesting if the authors commented on this work within the context of their findings. Some of the metrics and tools proposed in [10], such as the mode-dropping estimator, could also be used to assess the performance of a sampler in approximating regions of low probability where a shallow sampler is likely to lose some of the probability mass.

• Idea of Anchor Points: The notion of anchor points has implicitly been explored in some of the prior works mentioned above, albeit with a slightly different connotation that may have escaped the authors' attention. For instance, in the paper by Kanwar et al. [4] (Fig. 4), the authors use a technique very similar to what is suggested in this paper, although with slightly different connotations (e.g., they use previously trained flow-based models as starting (anchor) points to sequentially train more challenging distributions).

• Additional Related Works: Other closely related works, such as [11], are not mentioned in the manuscript despite having similar titles. This may cause confusion for potential readers.

• Experiments: I find the results presented in the paper not entirely convincing. Although the authors compared their approach to a large set of baselines, this alone does not seem sufficient to claim the superiority of the proposed method. I am surprised that the proposed approach is not compared against prior works, such as Annealed Importance Sampling with Normalizing Flows [7], and naive RealNVP training with a sufficiently large number of couplings and no anchor points.

As a side note, I strongly recommend that the authors conduct an extensive literature search to include and acknowledge existing prior works, and eventually, compare and discuss potential differences and similarities

Freezing seems to provide only a marginal advantage over non-freezing. The main advantage as the authors proposed is in the speed, but that's not particularly central to the paper as speed is measured by function calls and not wall clock time. If we remove step 5 from NOFIS then most of the method is not particularly distinguishable from standard normalizing flows.

In addition, if we're looking for just samples from the proposal distribution, what's the advantage of using NFs over other generative models? If there is a lack of distinguishing feature then the middle portion on NFs specifically might not be needed in lieu for a general generative model construction.

1. The experiments only contain up to dimension 62, and the paper does not explain why sampling rare events at this dimension is difficult. How the comparison may look like if we compare the method with traditional sampling methods, like metropolis sampling.
2. The method's speedup and precision improvement are not clear from the languages used in the text.
3. The experiments in Figure 2 and Figure 3 look unrelated to rare event sampling but show the effectiveness of the method approximating a given distribution. It will be beneficial to get more ideas on what these figures tell us.