Are the assumptions of Lemma 1 really satisfied by SGLD? I don't see why the Markov chain induced by SGLD would have the posterior of interest as its stationary distribution, even if the step sizes are appropriately annealed.
More generally, the paper makes several strong claims of "fast posterior simulation" and "guaranteed superiority of $\tilde{q}{\eta, \phi}^{(t)} o v e r
q\phi$". However, the results stated in Lemma 1 to 3 could be seen as motivation for the proposed approach but do not constitute rigorous theoretical guarantees for the actual algorithm.
Finally, it is not clearly justified why we can use samples from all T
iterations (instead of just the samples from the T
th iteration) to estimate the gradients. Since the marginal distribution of the Markov chain is different for each time 0 ≤ t ≤ T
, using samples from all T
time steps would seem to add an additional layer of approximation compared to the objective stated in Equations 4--6. It would help to move Algorithm 1 (and any mention of practical approximations needed to make the algorithm work in practice) into the main body of the paper.
Minor comments
MCMC SGLD and VI are not proper nouns (i.e. it should be "MCMC algorithms" instead of "MCMCs")
P. 1: what does "$$q(z)-mixed Markov chain" mean?
P. 2: two-way -> two
P. 2: by (richly) ... functions -> by a (richly) ... function
P. 2: "inspirition-driven designs"?
P. 2: because marginal -> because the marginal
P. 3: "are variables on a Markov chain" sound grammatically incorrect to me
P. 4: markov -> Markov
3: "are variables on a Markov chain" sound grammatically incorrect to me P.