Metropolis Adjusted Microcanonical Hamiltonian Monte Carlo

We thank the reviewer for their time and thoughtful feedback. We understand that the main concerns relate to the lack of theoretical mixing guarantees and aspects of clarity in notation and presentation. We address both below.

Theoretical Contribution and Significance

We respectfully disagree with the assessment that this work lacks substantial theoretical contribution. A key theoretical novelty of our paper is the derivation of a valid Metropolis-Hastings acceptance rule for microcanonical dynamics, enabling the first asymptotically unbiased sampler based on MCLMC. This derivation is nontrivial, especially for MAMS with Langevin noise, as a standard procedure needs to be generalized to non-volume preserving dynamics. While we do not include formal mixing-time bounds (as in [1]), our primary goal is practical performance. As NeurIPS emphasizes empirical relevance and impact across ML subfields, we focus on showing that MAMS consistently outperforms NUTS — the de facto standard black-box sampler — across benchmark problems. We also study its scaling with the dimensionality and condition number. This addresses a real need in practical sampling and represents an important advance for the applied ML and probabilistic programming communities. Mixing analysis, especially in the microcanonical setting, is an interesting and challenging direction, and we agree that this is valuable future work. However, we believe that the current results — both theoretical and empirical — already justify publication. We ask the reviewer to consider this framing and the practical significance of our contribution in the NeurIPS context.

Presentation and notation

We appreciate the detailed list of clarity issues and have addressed them point-by-point below. We also note that other reviewers found the presentation strong (e.g., Reviewer DLZV: "All the results are stated very clearly, without overloaded notation, and with clear contributions."). Nevertheless, we fully agree that papers must be accessible across subfields and have revised the manuscript accordingly:

• x -> x’, we have clarified this in the text.
• Although not explicitly stated, d can be seen to be the dimension in the first line of the introduction, where we state, $x \in \mathbb{R}^d$. We have explicitly stated that it is the dimension.
• Velocity flip is necessary to make the proposal map $\varphi$ an involution. Indeed its effect is erased by the full refreshment, but we still keep it for consistency.
• MCLMC dynamics are a time-rescaled Hamiltonian dynamics, so the kinetic energy difference between two states z1 and z2 cannot be defined as K(z2) - K(z1), but is instead a nonlinear function K(z2, z1). This is why we have not directly defined K(z), but only its difference between the states. Note that it is only the difference that is relevant for the acceptance probability. We have now clarified this around Equations 6, 7 and 8.
• $\delta$ is a delta function so yes, it can be read as being evaluated at its argument (with all the usual subtleties of the delta “function”). We realize that the proof of Lemma 4.1 with delta functions might not appeal to the mathematically inclined audience, so we have now also included the reference to the other proofs from the literature, for example [1].
• Thank you for catching this, it was a typo.
• $\varphi$ in Equation 4 is defined by a leapfrog update, in Lemma 4.2 it is slightly more general, allowing for other split-type integrators. Thank you for pointing this out, we have clarified it in the text.
• Indeed this is a very sloppy notation, because $\Delta$ does not actually depend on z and z’, but rather only on z’’ = O(z). We instead should have defined $\Delta$(z’’) = K(BAB(z’’)) - K(AB(z’’)) + V(AB(z’’)) - V(B(z’’)) + K(B(z’’)) - K(z’’), we have now fixed this in the text.
• Thank you for catching this, we have indeed not defined it. We define Effective sample size (ESS) as: number of samples / $\tau_{int}$, from Equation 13. This is very standard, see for example [2].
• We agree this would be a nice illustration, we have added these curves to the appendix.

[1] Neklyudov, Kirill, et al. "Involutive MCMC: a unifying framework." International Conference on Machine Learning. PMLR, 2020. [2] Stan reference manual, Section 15.4: Effective sample size.

Limitations section

From a practitioner’s standpoint, a limitation of MAMS compared to NUTS is that it has not been so widely tested. This is understandable, given that we have just proposed MAMS, while NUTS has been state-of-the-art for years. This limitation is highlighted in the discussion.

We thank the reviewer for their thoughtful comments and for recognizing the technical soundness, clarity, and empirical strength of our work. We’d like to respond to the concern regarding novelty and the scope of comparisons.

Novelty and Significance

Our main contribution is the development of a new black-box gradient-based sampler that is asymptotically unbiased and consistently outperforms NUTS — currently the most widely used black-box MCMC method (e.g., in Stan [1]). We believe this alone is a very significant result, with a direct impact for practitioners. From a theoretical standpoint, the derivation of the MH acceptance rule for our microcanonical sampler (with and without Langevin noise) is non-trivial and novel. These derivations are necessary to ensure unbiasedness. Additionally, we propose a new automatic hyperparameter tuning scheme — including for trajectory length — which is key to making the sampler practical. One reviewer remarked that "best practices from the literature were incorporated at every level, including little tricks I wasn't familiar with..." and highlighted the tuning scheme as a non-obvious, effective contribution. We believe this underscores the care and originality in the algorithm’s design.

Comparison to Other Methods

Regarding the scope of baselines, we emphasize that our goal is to present a general-purpose black-box sampling scheme. While many gradient-based samplers exist in the literature, very few include both a sampler and a tuning mechanism that can be used out of the box — NUTS is the main such baseline. We therefore chose to focus on this meaningful and widely adopted comparator. Adding comparisons to methods that require significant hand-tuning would not be fair or informative in this context. That said, if the reviewer has specific methods in mind that meet the same black-box usability standard, we would be happy to discuss them.

Sensitivity to Initialization

We appreciate the question regarding sensitivity to step size and trajectory length initialization. We have now added a robustness experiment to the appendix: we repeated the tuning procedure using initial step sizes and $L$ values that were 10× larger and 10× smaller. The resulting tuned values and performance metrics (Table 1) were nearly unchanged, demonstrating that our scheme is not sensitive to initialization.

Conclusion

We hope this clarifies the novelty and significance of our contributions. In summary, the proposed method offers a new black-box unbiased algorithm that outperforms NUTS. Developing it required non-trivial derivation of the Metropolis-Hastings acceptance probability and a combination of tricks to make it black-box. The resulting algorithm is efficient, robust and applicable out of the box, and is intended to serve as a successor to NUTS. We believe these advances justify the paper's relevance and significance, and we respectfully ask the reviewer to consider revising their score in light of this clarification.

[1] Stan: A Probabilistic Programming Language, Bob Carpenter, Andrew Gelman, Matthew D. Hoffman, Daniel Lee, Ben Goodrich, Michael Betancourt, Marcus Brubaker, Jiqiang Guo, Peter Li, Allen Riddell

We sincerely thank the reviewer for the thoughtful and encouraging feedback. We address the points below.

Microcanonical vs. Canonical — Physical Background

We agree that it would be helpful to provide a clearer explanation of the physical intuition behind the equations 2 and 3 in the main text. While we discuss this in Appendix B, we now expand on it in the main text, as suggested.

MH weight

Thank you for pointing this out, we understand a clear statement of the final MH weight is valuable, especially if one wants to implement the algorithm. In simple terms, for every B or A update that the algorithm performs in the proposal, the energy difference must be calculated and added to the cumulative energy difference (B update energy difference is calculated by Equation 8, A update by Equation 6). Final MH acceptance probability is min(1, e^-total cumulative energy difference). We have clarified this in the paper and also moved the algorithm’s pseudocode to the main text.

Use of NUTS for Preconditioning

We agree that relying on NUTS for preconditioning may seem contradictory to our goal. In our experiments, we opted for NUTS-based preconditioning solely to isolate the sampler's ability to sample from its ability to precondition (which, granted, are strongly related). Preconditioning can be viewed as reparametrizing the parameter space, so we wanted to compare NUTS with MAMS on exactly the same problem, i.e., same target, same parametrization. In practice, however, we recommend using MAMS for preconditioning as well which is in fact the default behavior in our released implementation. We have clarified this in the updated manuscript.

Trajectory Length Adaptation

The trajectory length adaptation used in MAMS is not related to NUTS, but instead follows the approach proposed in [1] for unadjusted MCLMC. We have adapted it here in a natural way to work with the Metropolized variant. We now state this more clearly in the text.

Equation 3 and Hamiltonian Dynamics

Technically, Equation 3 does not describe Hamiltonian dynamics in the usual sense, as the transformation is not volume-preserving (the Jacobian is not unity). However, it does represent reparametrized Hamiltonian dynamics, where the time has been rescaled such that the velocity has unit norm. This connection allows us to define the change in energy along these trajectories, as discussed in Appendix B. We have revised the footnote to make this distinction more precise.

Involutive MCMC Reference

We appreciate the pointer to Neklyudov et al. (2020). We now cite this reference in the relevant section and briefly discuss its connection. Specifically, MAMS without Langevin noise is compatible with the involutive MCMC framework, while the stochasticity introduced by Langevin noise moves it outside of the class of deterministic involutive proposals.

Once again, we thank the reviewer for the detailed and constructive suggestions. We believe these clarifications will improve the paper’s readability and accessibility.

[1] Robnik, J., G. B. De Luca, E. Silverstein, and U. Seljak (2024). Microcanonical Hamiltonian Monte Carlo. The Journal of Machine Learning Research 24(1), 311:14696–311:14729

We thank the reviewer for their thoughtful and encouraging comments. We especially appreciate the recognition of the clarity, practical relevance, and completeness of our contributions.

We agree that including more applied or domain-specific benchmarks, particularly those relevant to the natural sciences would further strengthen the impact of our work. We see this as a valuable direction for future research and plan to explore it in subsequent work.

Regarding the connection to [4], indeed, MAMS without Langevin noise fits within the involutive MCMC framework presented in [4], as it involves a deterministic, involutive map. However, MAMS with Langevin noise falls outside this framework, as the Langevin proposal introduces stochasticity, which violates the determinism requirement in [4]'s construction. We have added an appropriate reference to [4].