Towards Unifying Hamiltonian Monte Carlo and Slice Sampling
We unify slice sampling and Hamiltonian Monte Carlo (HMC) sampling, demonstrating their connection via the Hamiltonian-Jacobi equation from Hamiltonian mechanics. This insight enables extension of HMC and slice sampling to a broader family of samplers, called Monomial Gamma Samplers (MGS). We provide a theoretical analysis of the mixing performance of such samplers, proving that in the limit of a single parameter, the MGS draws decorrelated samples from the desired target distribution. We further show that as this parameter tends toward this limit, performance gains are achieved at a cost of increasing numerical difficulty and some practical convergence issues. Our theoretical results are validated with synthetic data and real-world applications.
sampling, with a non-uniform conditional slicing distribution. Based on this relationship, we develop theory to analyze the newly proposed broad family of auxiliary-variable-based samplers. We prove that under this special family of distributions for the momentum in HMC, as the distribution becomes more heavy-tailed, the one-step autocorrelation of samples from the target distribution converges asymptotically to zero, leading to potentially decorrelated samples. While of limited practical impact, this theoretical result provides insights into the properties of the proposed family of samplers. We also elaborate on the practical tradeoff between the increased computational complexity associated with improved theoretical sampling efficiency. In the experiments, we validate our theory on both synthetic data and with real-world problems, including Bayesian Logistic Regression (BLR) and Independent Component Analysis (ICA), for which we compare the mixing performance of our approach with that of standard HMC and slice sampling.
This paper derives a framework in which a type of HMC and slice sampling form special cases (in one dimension), motivating a family of HMC samplers with generalized Gaussian momentum distributions, which are shown similarly to be associated to particular generalizations of slice samplers.
The connection between these two separate auxiliary variable Monte Carlo methods is interesting; some previous work has been done in this general area (e.g. Slice Sampling on Hamiltonian Trajectories, ICML 2016), although the specific connection presented in this paper is novel as far as I’m aware.  Overall the quality of the presentation is fairly good. One point I found confusing was that the general set-up for MG-HMC and MG-SS (including explicit statements of the algorithms) is given in Section 3 entirely for the one-dimensional case. In Section 5 and beyond, the methods are used in multidimensional cases, without much explanation given as to how the algorithms/derivations as stated in the previous sections adapt for more than one dimension (e.g. when Hamiltonian trajectories are not necessarily closed) - perhaps it could be made explicit in the main paper exactly how the samplers generalize to more than one dimension? Some description of how this generalizes to more than one dimension is given in the appendix, although there is relatively little detail and/or justification of the points made here as far as I can see - i.e. what happens if the potential isn’t decomposable across dimensions, and what does “hyper-diamond” mean in the footnote of p10? It’s also not clear to me what is meant by “almost certainly” in line 457 - there are cases where the trajectories are not space-filling (e.g. Gaussian kinetic and potential energies).  The theoretical results in the main paper are clearly presented, and demonstrate a nice property of the one-dimensional analytic MG slice sampler for a large class of potential functions. However, there appears to be some discrepancy between the results as stated in the main paper, and the assumptions used in the proofs in the appendix. For example, it seems to be the case that the chain is assumed to be in its stationary distribution in order to prove Theorem 1, but this is not mentioned in the main paper - it would be good to get clarification from the authors on this point, and whether there are any other conditions needed to obtain the results as stated in the main text.  The analytic toy example of the exponential distribution is a nice addition that demonstrates some of the theoretical results explicitly in a particular case. Again, it is not clear to me from the paper whether these results would generalize to more than one dimension. It is mentioned in the paper that when the slice region is disconnected, there may be some discrepancy between the performance of analytic MG-HMC and analytic MG-SS, although it is not made clear whether the ACF results earlier in the section would be expected to generalize. It’s also not clear to me whether these results generalize to multidimensional versions of the samplers.  My two main concerns about the paper are: i) whether the set-up of Sections 2 and 3 is particular to the one-dimensional case, and whether the multi-dimensional implementation of MG-HMC described in Section 5 bears much relation to the analytic MG-HMC samplers, and generalized slice samplers, described earlier in the paper; ii) what is the full list of conditions needed to ensure the validity of some of the results in the appendix, and whether the statements of results in Section 4 of the main paper reflects these conditions.  The appendix appears to contain quite a few typos that should be corrected e.g. line 455 “If uniformly sample”, line 464 “in each of the case”, line 465 “in cases more than one dimension” etc.  With regards to experiments, the simulation studies provide a nice illustration of the one-dimensional theoretical results discussed earlier, and indicate where numerical issues cause practical performance to deviate from the theoretical expectation. The Bayesian logistic regression experiment also demonstrates that improved mixing can be achieved when taking a value of the parameter "a" not equal to 0.5, which nicely demonstrates that the general MG-HMC framework introduced yields samplers that can out-perform HMC with traditional Gaussian momentum.  Minor points: Line 166: MG-SS mentioned before being introduced - possible typo? Line 259: In the second equation, I think there is a missing sign(p)_d on the right-hand side.