Venue: NeurIPS 2022 Conference

Reviews:

final_score: rating: 6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
review_content: summary: The authors show how to use (continuous) simulated tempering in conjunction with a PDMP sampler (in particular, the zig-zag sampler) to sample from multi-modal distributions. In the process that they propose, once the process hits $\beta=1$, it spends an amount of time there that is an exponential rv; this ensures that the marginal distribution of the inverse temperature is a mixture of a continuous distribution on [0,1) and a point mass at 1, which allows obtaining samples directly rather than from importance/rejection sampling. Experiments show significantly improved performance compared with the non-tempered algorithm, as well as how the root-mean-square error and thinning efficiency depends on the value chosen for alpha (parameter controlling how much time the process spends at $\beta=1$).
strengths_and_weaknesses: The authors propose a simple, easy-to-use extension of PDMP (specifically the zig-zag sampler) that significantly improves its performance on multi-modal objectives in simulations. The simulations show how the parameters can be tuned to minimize the error. The paper is generally clear.

However, only the zig-zag sampler was considered in this work. Furthermore, one limitation of the simulated tempering-based approach is the necessity of estimating the normalizing constant (discussed in section 3.4). 
questions: * Line 61-62: "this means we can obtain samples from $\pi(x)$ rather than having to resort to importance sampling to correct samples drawn at different temperatures" - in what sense is the proposed approach better than using correction using importance sampling?
* Line 66-67: Can you explain or give a reference for the issue with the HMC sampler?
* Line 156-157: Can't you use importance sampling no matter what \beta is, by using the ratio of the unnormalized distribution at the temperature of the sample and at temperature 1?

Minor notes:

* line 171: "important" sampling -> "importance"

limitations: Yes.
review_title: null

final_score: rating: 6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
review_content: summary: The manuscript proposes to use a modification of the ZigZag sampler (or any almost any other PDMP sampler) to sample from complex (eg. multimodal) target distributions. The approach proceeds by introducing a "temperature" parameter $\beta \in [0,1]$ and an extended distribution with a Dirac mass at $\beta=1$ (corresponding to the target distribution of interest) and run a slightly-modified ZigZag sampler on the extended target. Samples obtained at temperatures $\beta < 1$ can still be used by using an importance sampling scheme, if necessary.
strengths_and_weaknesses: **Strength:**

The text is well-written and very easy to follow. Furthermore, the approach is quite natural and, to the best of my knowledge, novel. 

**Weaknesses:** 
The method is simple to implement (which is an advantage) and its working principles are intuitive. Unfortunately, I have not found the numerical experiments extremely convincing. As a matter of fact, after reading the paper, I am not concince that I would like to give the method a try (even if it is simple to implement). See next section for more detailed remarks. As of now, I am leaning towards a "weak reject" since I am finding the empirical evaluation section to not be up to the standard of NeuRIPS.
questions: 1. Choosing the distribution $p(\beta)$ is indeed crucial in not-entirely-trivial situations. I think that this section should be expanding greatly. Choosing $\kappa(\beta) \propto 1/Z(\beta)$ is indeed a good strategy, I think, but quite hard to implement in practice in difficult situations. How do the authors think this should be done in practice? There are about 3 lines describing this and one would expect a much more thorough discussion since it is likely to have a very important impact on the method.

2. Implementing the Zig-Zag, and many other PDMP sampler, is not trivial (eg. simulating the event time, etc..) A metric of the type (accuracy)/(compute time) should be reported. For example, (accuracy) / (number of target or gradient-of-target evaluations) seems like an appropriate metric.

3. It is very standard and straightforward to use SMC methods, or parallel-tempering schemes, to tackle multi-modal situations. Furthermore, it is not that difficult to (almost) automatically tune these methods. I think that it is crucial to compare the proposed method to these standard alternatives (and report ESS or MSE per compute time!). Since the method is relatively simple (which is good), it seems natural to expect very thorough numerical experiments.

**minor remarks:**
1. Are we assuming $\kappa(\beta)=1$ so that $\int_{[0,1]} p(\beta) d\beta=1$? 

limitations: NaN
review_title: null

final_score: rating: 8: Strong Accept: Technically strong paper, with novel ideas, excellent impact on at least one area, or high-to-excellent impact on multiple areas, with excellent evaluation, resources, and reproducibility, and no unaddressed ethical considerations.
review_content: summary: This article presents a principled way to do continuous tempering in PDMP samplers, with positive probability of sampling from the actual target distribution. It discusses its ergodicity, implementation and optimisation with respect to the tempering distribution. The method is illustrated on a series of challenging examples.
strengths_and_weaknesses: Strengths:
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1. This article is well written, and beside its contributions, offers a good overview of the zig-zag sampler. 
2. The heuristics of the method are clear, and its symbiosis with PDMP samplers made clear.
3. The method is theoretically grounded, and the article clearly describes the sampling mechanism (thinning) for each example.
4. The applicability of tempering for transdimensional sampling is particularly convincing.


Weaknesses:
------------------
1. Because the paper insists that the method applies to any PDMP sampler, a quick review of PDMP samplers in general (as in, a short summary of, e.g., Fernhead et al. [2018]), allowing the reader to confirm that it indeed generalises, would have been welcome in the supplementary material.

2. While the experimental validation is thorough, the reported metrics hardly allow to understand which sampler is best due to the fact we are lacking information on the run time of each method. This is probably the main issue with the article. Given that the RMSE of the MCMC estimates is roughly speaking $O(\frac{1}{\sqrt{ESS}})$, and that $ESS$ should increase in average linearly with the run time of the experiment (supposing that the average time to simulate an event is constant), I believe a better metric to report would be $\frac{RMSE}{\sqrt{T}}$, where $T$ is the time (e.g., in seconds) to get 50,000 samples from the chains, similarly, the absolute error should then be divided by the time taken (please adapt the $\sqrt{T}$ terms the way you deem best to match the super efficiency of PDMPs). Would this be feasible? Alternatively (and perhaps better), instead of giving yourself a number of iterations in the chain, you could give yourself a time budget for each chain (for example the average time taken by your experiments) and report the error achieved in this amount of time. I tend to believe this could marginally change the conclusions as it would take into account the thinning efficiency directly into the result.

3. The estimator of Section 3.3 --- which I believe is essentially a Rao-Blackwellisation of the proposed method for a specific choice of tempering prior --- does not seem to be empirically evaluated at all. Instead, a less performing version is used which almost surely does not sample from the target. It would have been interesting to also check if the "sacrifice" of using a poorly adapted tempering prior was compensated by being able to use the intermediate samples in the case when there is a point mass at $\beta = 1$. However, this point is less important than the point 2 above.

Some minor comments and typos:
---------------------------------------------
1. The reference style changes between the main paper and the supplementary, making it hard to jump from one to the other. 
2. The citations in the main paper are breaking the text, and it seems like the authors perhaps used "\citet{}" from natbib everywhere. This probably should be corrected for readability. For example "piecewise deterministic processes (PDMPs) Davis [1993]" should probably read "piecewise deterministic processes (PDMPs) [Davis, 1993]" or similar depending on the chosen style.
3. The text alternates between of "piece-wise" and "piecewise"
4. "called a piecewise deterministic processes" -> "called piecewise deterministic processes"
5. "The idea is that allowing mixing across this set will improve mixing" -> probably meant "moving across this set"
6. "transitions from the continuous" -> "transitions between ..."
7. "for an exponential period of time": this sounds like it stays there for a long time, where as you approximately meant "exponentially distributed"
8. "This will involve additional post-sampling computation of the importance weights at the samples taken from the Zig-Zag trajectory." I believe this cost will probably be negligible compared to the cost of simulating from the Poisson process, so this sentence feels overly pessimistic to me.
9. In the supplementary, the box plots are not linked to the text.
10. "Zig-Zag $\alpha = 1$ and importance" -> "Zig-Zag ($\alpha = 1$) and importance"
11. "and we allow a pointmass at  $\beta = 1$" -> "and we allow a pointmass at $\beta = 1$"?
questions: 1. I am rather uncertain how Section 3.4 on the calibration of $\kappa$ differs from Section 2.1. of Yao et al. [2020]. It seems to me that Section 3.4 is a direct special case of the latter. Can you explain how your method is different from this? Note that I understand this is not the main contribution of the article, but believe this kinship between the two methods should be highlighted in case I am not mistaken.

2. The use of $t$ and $s$ in Section 3 of the supplementary material is somewhat fuzzy: $t$ sometimes is a free parameter (or the current time), and sometimes denote the next exent. Clarifying the notation (for instance by calling the event time $\tau$ as per the main text) would be useful. Referring to the supplementary material at the time of introducing the Zig-zag sampler would be useful too.

3. I believe that the proof that $\omega$ is an invariant distribution for the given rate (Theorem 1) is correct. However, I believe this to follow more from Proposition 7 of [6] than directly from Theorem 2 and 3; a comment on the other PDMP sampler would also be useful. Moreover, for completeness, it may be useful in the supplementary material to insist that $\omega$ is the **unique** stationary distribution of the chain.


limitations: One of the main limitations of PDMP is the need to sample from a non-homogeneous Poisson process. This is however thoroughly discussed in the supplementary material.
review_title: null

Meta-review: metareview: The authors show how to combine a clever temperature schedule for simulated tempering with a Zig-Zag sampler (though the authors suggest this could be expanded to arbitrary PDMP samplers, it was mentioned that this may not be completely obvious), so that the marginal distribution of the inverse temperature is a mixture of a continuous distr. on [0,1) and a point mass at 1--- allowing taking samples directly and obviating the need for importance/rejection sampling. The theory proves stationarity of the resultant chain, and the original simulations demonstrated improvement over the vanilla non-tempered chain. At the request of the reviewers, the authors also compared to other tempering chains and showed encouraging results (even though as was noted, comparison is a bit hard for some methods, since e.g., some are more suited for parallel computation). 

confidence: Certain
Decision: Accept