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 proposed a new way of implementing the Hamiltonian Monte Carlo method. This new approach restricts the distance a single HMC step will travel, hence to avoid unnecessary oscillations in low probability region, as well as to reduce the correlations between samples. The proposed algorithm has been tested against several well-known examples for HMC, and compared with existing method. IT can be seen that the proposed algorithm outperform the pain HMC, as well as NUTS(no-U-turn-sampling).
strengths_and_weaknesses: Strengths: The idea is innovative and effective in dealing with the limitations the author discussed. The algorithm is carefully designed to ensure reversibility while achieving fixed-distance traveling. The numerical experiments are representative and convincing.

Weaknesses: Quantitative analysis of the algorithm is lacking. For example, it is curious to know how sensitive the performance of the FDHMC algorithm is with respect to the parameter of the fixed-distance,
questions: 1. Please provide the argument for the statement of "the expected (rather than exact) step of the initial and final positions is at $i+\epsilon/2$. (L151-L152).

2. How does the fixed-distance change for each example, since it is determined by the procedure discussed in the first paragraph of Sec. 5. Also how sensitive is the performance with respect to the distance parameter?


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 paper proposes a new Hamiltonian Monte Carlo algorithm for probabilistic inference in which each step of the proposal simulates the Hamiltonian dynamics for a fixed distance traversed in state space rather than a fixed number of steps. The paper motivates this alternate algorithm and provides the details of the algorithm and a proof of correctness. Empirical results comparing the new FDHMC algorithm with classical HMC and NUTS are also provided showing improved effective sample size per gradient estimation.

The motivation for the new FDHMC algorithm (paraphrasing the authors) is that the existing HMC algorithm traverses through high probability regions with very high momentum and thus doesn't spend enough time in these regions and when it does stop in the high probability regions it tends to get stuck there producing highly correlated samples.

strengths_and_weaknesses: The proposed FDHMC algorithm certainly appears novel. [This reviewer has read and reviewed a large number of variants of HMC but not quite this one!]

Most current work on improving HMC algorithms seem to be focused on improving the curvature of the state space which appears to be a critical challenge in probabilistic inference using any kind of gradient-based methods. This paper doesn't address the curvature issue. The proposed FDHMC algorithm assumes a unit mass for the momentum. All of the examples in the evaluation seem to be selected such that unit mass matrix would work well for them. Hence the the paper scores low on relevance.

In terms of quality and clarity the paper is very well written and the mathematical proofs and algorithms are very easy to follow. 

The two main motivations of the paper Limitation 1 and 2 are presented somewhat informally. I am able to follow the main arguments that the authors are making, but still I would have preferred to see a more mathematical justification. At the very least the empirical evaluation should somehow directly tie in the higher ESS numbers to these limitations somehow. Otherwise it could well be that the higher ESS numbers are explained by something else entirely. For example, the tuning methodology of distance traversed could have a huge influence. (As we can see in the paper that the tuning of HMC seems to produce even better results than NUTS!)

Also, while the experimental results demonstrate higher ESS numbers they don't show that the samples are exploring the high probability region of the posterior space. It might be worth showing the unnormalized posterior probabilities of the samples. In the case of logistic regression it is also customary to show the log likelihood of held out test data using the posterior samples of $\alpha$ and $\beta$.

questions: How would one incorporate learning of the mass matrix into FDHMC? If the mass matrix is not unit does this pose any complications in terms of calculations of the remaining distance?

Doesn't NUTS address Limitation 2?

I would suggest to the authors that they pick some model with a complex multimodal posterior where each mode has roughly equal probability mass. If they can show that FDHMC explores all the modes better than HMC/NUTS then one could argue that it is addressing the limitations that are claimed to be addressed.

Another minor suggestion would be to include some results on a hierarchical model. These are customary to show in papers on probabilistic inference :) 

Minor Points:

- I would suggest to use a different letter other than `p` for the superscript of `q` on lines 87, for example, and in related equations. `p` is used for momentum elsewhere and this is a bit confusing.

- In equation 2 the Jacobian should refer to $\cal{F}$ and I feel that the variables should be q,r and in the partial derivative rather than q and p.

limitations: Not aware of any negative societal impact of this work.
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 paper proposes fixed distance Hamiltonian Monte Carlo (FDHMC), a variation of HMC that simulates the equation of motion for a fixed distance rather than a fixed amount of time. The paper establishes the theoretical foundations of FDHMC, and demonstrates its advantages over HMC and NUTS with numerical experiments.
strengths_and_weaknesses: ## Strengths

- The idea is novel, and the theoretical foundations are clearly presented with excellent organization of the materials.
- The proposed FDHMC algorithm makes intuitive sense, and can be easily implemented efficiently. This means FDHMC has the potential to make a big impact due to the widespread applications of HMC.

## Weaknesses

 While the theoretical foundations are excellently presented, I have some concerns regarding the numerical evaluation. While the motivation for FDHMC makes sense intuitively, I would like to see more evidence from numerical experiments to confirm the advantages of FDHMC over HMC/NUTS:

- In the experimental results in Section 5, vanilla HMC outperforms NUTS in many cases, sometimes by a very large margin. This seems contradictory to many of the existing results. The authors seem to indicate this is due to a simple tuning heuristic for HMC (L242-243) but I find this claim to be strange and surprising. Given the experiments are done using customized implementations and NUTS is quite a complicated algorithm, the results do not seem convincing to me. I would like to ask the authors to reimplement some of the experiments using readily available and more thoroughly tested probabilistic programming language (PPL) packages (e.g. NumPyro which has a NUTS implementation where all parameter tuning are automated and readily available tutorials on models like the Neal's funnel) and re-verify that HMC still outperforms NUTS.
- The paragraph on parameter tuning is quite confusing. Why is the total simulation duration in HMC chosen to be 2 (L239-240)? What is 2 here, and why is this number independent of the model? For FDHMC, there is some discussion on how to pick the fixed distance, but what about the step size? Is it picked the same way as HMC? What would happen if we apply some of the automated parameter tuning utilities available in commonly used PPL packages?
- In the supplementary materials the authors present an algorithm to automatically tune the hyperparameters involved in FDHMC. Is there any reason why this algorithm is not used for the experiments in the main paper?

I would be happy to bump up my score if the above concerns regarding evaluation can be properly addressed, but in its current form I still have concerns over accepting the paper.

questions: See above
limitations: The main limitation I see is the potential challenge in tuning the hyperparameters involved in FDHMC. The authors have made some discussions on how to address this limitation. But the discussions are not entirely convincing since the baseline results involving HMC and NUTS do not seem convincing. It would be ideal if the authors can:

- Establish baseline results involving HMC and NUTS using more thoroughly tested PPL packages, using the provided parameter tuning scheme (e.g. for NUTS which is fully automated in most cases).
- Study how sensitive the performance of FDHMC is to the choice of hyperparameters (e.g. if with some simple heuristics FDHMC can already outperform HMC and NUTS then the FDHMC performance is probably robust).
- Demonstrate how the automated parameter tuning presented in the supplementary can further improve performance.
review_title: null

Meta-review: metareview: This paper proposes a new variant of Hamiltonian Monte Carlo. Rather than using a fixed number of iterations (as in the original HMC) or choosing the step-size adaptively (as in NUTS) the paper simulated the dynamics until a fixed *distance* has been traversed. The paper gives some arguments why this might be a good idea and a careful proof of detailed balance for thew new proposed algorithm. Reviewers agreed the algorithm seemed correct and the numerical results were compelling, but there were some existing concerns about the implementation of baseline algorithms and some questions about the technical details of the algorithm. Given the importance of HMC and the novelty and agreed correctness of this work, I recommend acceptance and urge the authors to consider the clarifying questions asked by the reviewers as opportunities for improving the paper. Also, additional evidence for experiments (e.g. perhaps comparing to another implementation of NUTS/HMC) would also be helpful.
confidence: Certain
Decision: Accept