Stein Variational Gradient Descent as Gradient Flow
Stein variational gradient descent (SVGD) is a deterministic sampling algorithm that iteratively transports a set of particles to approximate given distributions, based on a gradient-based update that guarantees to optimally decrease the KL divergence within a function space. This paper develops the first theoretical analysis on SVGD. We establish that the empirical measures of the SVGD samples weakly converge to the target distribution, and show that the asymptotic behavior of SVGD is characterized by a nonlinear Fokker-Planck equation known as Vlasov equation in physics. We develop a geometric perspective that views SVGD as a gradient flow of the KL divergence functional under a new metric structure on the space of distributions induced by Stein operator.
–––RESPONSE TO AUTHOR REBUTTAL––––
The author's response is very thoughtful, but unfortunately it doesn't fully address my concerns. 
The fact that the proof of Lemma 3.1 works in the compact case is fine as far as it goes, but substantially detracts from how interesting the result is. The proposed condition on E_{y\sim \mu}[max_x(g(x,y))] is not sufficient since the same condition also has to hold for mu', which is a much more challenging condition to guarantee uniformly. It seems such a condition is also necessary in a Wasserstein metric version of Lemma 3.1. So the fix is non-trivial. 
The changes to Theorem 3.3 are more straightforward. 
Another issue which I forgot to include in my review  is that the empirical measure converges in BL at a n^{-1/d} + n^{-1/2} rate, not the n^{-1/2} rate claimed at line 133. See [1] and [2]. 
Overall the paper requires substantial revision as well as some new technical arguments. I still recommend against acceptance. 
[1] Sriperumbudur et al. On the empirical estimation of integral probability metrics. Electron. J. Statist. Volume 6 (2012), 1550-1599. 
[2] Fournier and Guillin. On the rate of convergence in Wasserstein distance of the empirical measure. Probability Theory and Related Fields. Volume 162, Issue 3–4 (2015), pp 707–738. 
–––ORIGINAL REVIEW–––
Summary: The paper investigates theoretical properties of Stein variational gradient descent (SVGD). The paper provides asymptotic convergence results, both in the large-particle and large-time limits. The paper also investigates the continuous-time limit of SVGD, which results in a PDE that has the flavor of a deterministic Fokker-Planck equation. Finally, the paper offers a geometric perspective, interpreting the continuous-time process as a gradient flow and introducing a novel optimal transport metric along the way.
Overall, this is a very nice paper with some insightful results. However, there are a few important technical issues that prevent me from recommending publication. 
1. Theorem 3.2: Via Lemma 3.1, Theorem 3.2 assumes g(x,y) is bounded and Lipschitz. However, if p is distantly dissipative, then g(x,y) is not bounded. Thus, the result as stated is essentially vacuous. That said, examining the proof, it should be straightforward to replace all of the bounded Lipschitz norms with Lipschitz norms and then get a result in terms of Wasserstein distance while only assuming g(x, y) is Lipschitz. 
2. On a related note, the fact that the paper assumes eq. (7) holds “without further examination” (lines 78-79) seems ill-advised. As can be seen from the case of Theorem 3.2, it is important to validate that the hypotheses of a result actually hold and don’t interact badly with other (implicit) assumptions. The author(s) should more thoroughly discuss and show that the hypotheses in the two main theorems actually hold in some non-trivial cases. 
3. Theorem 3.3: The final statement of the theorem, that eq. (12) implies the stein discrepancy goes to zero as \ell -> infinity, is not true. In particular, if \epsilon_\ell^* goes to zero too quickly, there’s no need for the Stein discrepancy to go to zero. Again, I think this statement can also be shown rigorously using the upper bound on the spectral radius discussed in the first remark after Thm 3.3. However, this should be worked out by the author(s). 
Minor comments:
- \rho is never defined in Thm 3.3 
- In many results and remarks: instead of writing “Assume …, then …”, one should writing either “Assuming …, then …” or “If …, then …”.
- There were numerous other grammatical errors
- \phi_{\mu,p}^* was used inconsistently: sometimes it was normalized to have norm 1, sometimes it wasn’t. 
- The references need to be updated: many of the papers that are only cited as being on the arXiv have now been published