Summary
This paper proposed a posterior approximation using a new Stein discrepancy, which does not require derivatives of the statistical model. For that purpose, the authors derived the new discrepancy, called gradient-free KSD, and studied its statistical and convergence behaviors theoretically.
Then the authors developed algorithms for differential equations, for which stable computation of derivatives is difficult, and a new sampling algorithm that bypass the Hessian calculations.
Strengths
- A new KSD that does not require gradients, similar to the idea of importance sampling, is proposed. This leads to new algorithms for differential equations where the gradient is difficult to compute and for problems requiring Hessian calculation.
- The authors studied the theoretical property of the proposed GF-KSD by extending the existing KSD theory.
- Not only theoretical analysis but also detailed numerical investigations of the choice of parameters and $q$ are carried out with the actual use in mind.
Weaknesses
- The writing style is such that the main paper alone is not complete, and it is assumed that the reader will read the Appendix. For example, Eq. 6 of Line 115 does not appear in the main text, and the tilted Wasserstein distance defined in Theorem 1 is introduced without any explanation of its properties in the main text.
- The writing style could be improved since the discussion about existing research and the explanation of the proposed method are mixed, making the paper difficult to read.
-Some parts are mathematically undefined or under-discussed
- In Definition 2, sup is undefined
- In Line 158, at last, $\not \to$ is undefined.
- I don't know how widely the tilted Wasserstein distance (TWD) in Theorem 1 is known to the general public, but there is no discussion of the properties of TWD. Therefore I could not understand how important Theorem 1 is, that is, how important it is when it is said that convergence of TWD leads to convergence of GF-KSD; even after reading the proof of Theorem 1, I could only understand that TWD is a convenient form of the usual Wasserstein distance, which is obtained after applying the triangle inequality.
- I could not understand the importance of the proposed method because I am not sure for what problems the proposed method is effective.
I agree that it may be useful for differential equation problems, but the authors only applied the method to very small models of Lotka-Volterra in the experiments. In such a setting, MCMC is the standard approach, and even if the model is high-dimensional, we can solve it efficiently by variational inference.
Also, although the combination with Stein variational inference seems interesting, I wondered if GF-KSD is really flexible enough to generate samples for complex real data under the two restrictions suggested in Section 3.
One restriction is that the tail of $q$ should not be far from the target distribution; the other is that it must not be high-dimensional.
I think the application to differential equations seems promising, so it would be better to find a problem setting where GF-KSD is more useful than MCMC and standard variational inference.
Questions
I would appreciate it if the authors would answer the concerns described in Weakness.
As for minor questions;
- What do the dotted and solid lines in Figure 1(b) correspond to?
- Is it required to adjust parameters of the Laplace distribution, KDE, and GMM for $q$ in some way ? If so, what is the recommended method ?
- Looking at Figure 3 (b), it seems that the number of samples ($n$) must be very large ($\log n=5$, i.e., $n=150$) even for low-dimensional problems such as d=8 in order for there to be any difference in energy distance. Is my understanding correct?
Rating
4: Borderline reject: Technically solid paper where reasons to reject, e.g., limited evaluation, outweigh reasons to accept, e.g., good evaluation. Please use sparingly.
Confidence
4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.
Limitations
The limitation of the proposed method is discussed in detail.