Dimensionally Tight Bounds for Second-Order Hamiltonian Monte Carlo
Hamiltonian Monte Carlo (HMC) is a widely deployed method to sample from highdimensional distributions in Statistics and Machine learning. HMC is known to run very efficiently in practice and its popular second-order "leapfrog" implementation has long been conjectured to run in d 1 4 gradient evaluations. Here we show that this conjecture is true when sampling from strongly log-concave target distributions that satisfy a weak third-order regularity property associated with the input data. Our regularity condition is weaker than the Lipschitz Hessian property and allows us to show faster convergence bounds for a much larger class of distributions than would be possible with the usual Lipschitz Hessian constant alone. Important distributions that satisfy our regularity condition include posterior distributions used in Bayesian logistic regression for which the data satisfies an "incoherence" property. Our result compares favorably with the best available bounds for the class of strongly log-concave distributions, which grow like d 1 2 gradient evaluations with the dimension. Moreover, our simulations on synthetic data suggest that, when our regularity condition is satisfied, leapfrog HMC performs better than its competitors -both in terms of accuracy and in terms of the number of gradient evaluations it requires.
This paper looks at the "running time" of HMC.   It seems one of the prime contributions of this work is a relaxation of a condition given in prior work (which this reviewer has not read) to accommodate a more general regularity structure on the target. Namely, Assumption 1 in the present work is a kind of Lipschitz assumption on the Hessian but based on some apparently weaker norm. Under this condition, and importantly strongly concave targets, the authors look at the number of "gradient evaluations" required in HMC.   I think this is interesting from the theoretical point of view, and perhaps useful - albeit the nature of the problems addressed are not so interesting alone and thus the practical impact of this work is limited I would guess.  I have some concerns and I don't think the paper is really ready for publication in NIPS. These are minor in some sense.  Firstly, the idea of "run time" from the start is pretty vague. It seems it is only in Theorem 1 that running time is defined explicitly as "the number gradient evaluations". Even this is perhaps not completely transparent.  The Algorithm 1 has a fixed running time determined by i_{max}, T, \eta, and it is independent of d in terms of the number of gradient evaluations. So it seems the notation between this and, for example, Theorem 1, inconsistent.   Line 107 - why do I have only p_1 and p_2, I am confused?  What is the "^\prime" symbol, surely it is used for multiple purposes or something is wrong here?  What is the norm used in the Theorem 1 to measure X_{i_{max}} and Y? Perhaps this is mentioned somewhere but I missed it. Same question in the definition of a warm start and in the Section 5. This seems rather critical? And then what is a "high probability" in Theorem 1. I don't really follow this theorem in a rigorous sense nor its proof outline - I think there are some problems with a lack of defining an appropriate distance here.  I didn't read the supplementary material.  Personally, for this reviewer, the formatting with a lot of "in-line" notation made the paper a little hard and frustrating to read. But this is certainly not a criticism for rejection - rather just a comment that may aid the authors in presenting a revision.  In summary, I think there is a lot of potential with this result and it would be of theoretical interest to people working in this particular direction in this area. However, from my reading I didn't get the feeling it was polished enough and perhaps suitable enough for NIPS although I am happy to be corrected.   ----  Added after author rebuttal:  I am happy with the changes. I don't think you need the "_2" subscript on the norm if you define it - previously you had the Euclidean norm with "_2" in some places and someplace without. For notation, there are a lot of places I don't see the notation as necessary or even understandable. For example, the \dagger in Algorithm 1 seems completely unnecessary. As does the \tilde in the warm start definition. I don't follow the \tilde{O} notation - I guess this is to distinguish it from O^\star which only captures the dimension terms, but then just plain O( ) notation is also used so the  \tilde{O} is unknown to me (perhaps its separate but standard notation I should know?). Anyway, these are just minor issues really; but I saw a lot of probably unnecessary notation which could probably be worked on in later versions (e.g. the superscript notation later on in the leapfrog integrator section (7 in the arXiv) is a bit much.  My main concern is the length and the suitability for NIPS in terms of acceptance. I like the idea of the result. But unadjusted HMC is not really a practically chosen algorithm. And the length of the supplementary material - which I tried in the rebuttal period to follow but had to stop due to time constraints - is too long. I don't believe supplementary material's purpose is to contain the bulk of an article, which is the case when the main contribution of the article is a theorem and the supplementary contains the proof. So while I think the paper has merit, I think it needs to benefit from a proper review of the entirety of the technical details in a more suitable venue (e.g. some probability journal). I note, for example, similar studies on unadjusted Langevin appear in AAP for example.