Thank you for your thoughtful questions and your positive evaluation.
Thank you for your thoughtful questions and your positive evaluation. We have added a notation section for easier reference to commonly used symbols.
**Why do you integrate it from $0$ to $N$?**
The first two bounds in Thm 1 apply to any $N$ and $T$. From those bounds, one can obtain rates for general $N$ and $T$. However, for the KSD to converge to zero, one needs both $N$ and $T$ to jointly go to infinity. If you assume $C^*< \infty$ and $\limsup_{N \rightarrow \infty} KL (p^N(0) ||\pi^{\otimes N})/N < \infty$, then one obtains a KSD bound of $O(\frac{1}{N} + \frac{1}{T})$. Thus, taking $T=N$ gives the optimal decay rate in this upper bound.
**What is $\mathcal{L}$ in Assumption 4?**
$\mathcal{L}(X)$ denotes the law of the random variable $X$. We have stated this explicitly in the `Notation' section.
**Growth condition in Assumption 2**
As discussed in Remark 3, the growth condition essentially interpolates between the exponential and Gaussian target distribution tail behaviors as $\alpha$ varies from $0$ to $1/2$, and *does not impose any convexity* on the potential $V$.
For analyzing discrete-time MCMC sampling algorithms, $V$ is typically assumed to be gradient-Lipschitz. Our growth condition required for Theorem 3 (which is focussed on the discrete-time case) is comparable to such smoothness assumptions, whereas Theorem 1 (for the continuous time case) does not require such a growth condition.
Functional inequalities (like log Sobolev and Poincare inequalities) are curvature or convexity-type conditions that are used in particular to obtain guarantees in the stronger Renyi and KL divergences for MCMC algorithms. One of the significant contributions of this work is that for the KSD results, we do not assume any convexity or functional inequality and thus our results are applicable to a wide range of potentials and kernels.
However, as also pointed out by Reviewer xwur, even for the population limit SVGD (infinite particle limit), obtaining KL guarantees is an interesting and open problem. Note that existing results require the Stein log Sobolev inequality which is not as well-understood as their classical counterparts.
**Table for comparison**
Since there is only one prior work (Shi and Mackey, 2024) on analyzing discrete-time finite-particle SVGD, such a table will not be too meaningful in our opinion. If you still insist on adding a table, we will be happy to oblige in the camera-ready version.
**Smoothness constant in Theorem 3**
Good point. Actually all the constants appearing in Assumption 2 are important in quantifying the convergence rates. However, the bounds would become quite clunky and hard to parse if we made all such dependence explicit. That is why we chose to only distill out the effect of the most important parameters $d$ and $N$ in the bounds.
**Wasserstein, KSD and CoD**
We have added a discussion in Remark 6 providing an intuitive understanding of this phenomenon. Convergence in KSD captures convergence of expectations for a class of test functions that is much smaller than that for Wasserstein convergence (see Gorham and Mackey, 2017 and Kanagawa et al., 2022). The latter class is large enough to be highly sensitive to the effect of growing dimension, thereby exhibiting the curse-of-dimensionality.
In principle, Theorem 3.2 from Kanagawa et al., 2024, could be used to get a Wasserstein bound based on KSD bound. However, Theorem 3.2 is extremely abstract and hence we work with the Matern kernel for obtaining tangible rates. A practical comparison of the different choices of kernels for SVGD algorithms, although interesting, is beyond the scope of this current work.
**Gaussian mixing time**
Definitely not, especially when sampling from Gaussian targets. However, we give the first poly-time complexity order for a deterministic sampling algorithm for non-parametric targets with minimal assumptions on the $V$ and $k$. For Gaussian targets and bi-linear kernels, with Gaussian initialization, Liu et al., 2024, provides more detailed results on convergence rates exploiting the parametric form (by tracking the flow of means and covariances) in comparison to the whole empirical distributions in the non-parametric case.
Improving the rates of SVGD or other deterministic particle methods or proving lower bounds for such methods provides a clearer understanding of deterministic and randomized sampling. The eventual goal is to mark the territories where deterministic or randomized sampling algorithms perform better.
**We sincerely hope that we have addressed many of your questions, in which case, we would greatly appreciate if you could increase your score as you see fit.**