Second order quantitative bounds for unadjusted generalized Hamiltonian Monte Carlo

TL;DR

The paper develops a non-asymptotic, entropy-based analysis for unadjusted generalized Hamiltonian Monte Carlo (gHMC), situating the method within a hypocoercivity framework to derive explicit KL contraction bounds. By combining a discrete modified entropy with a detailed study of the Verlet integrator and momentum-refresh steps, it shows second-order accuracy in the step size $\delta$ and provides dimension-dependent complexity bounds: $\mathcal{O}( d \varepsilon^{-1/4} \log(d\varepsilon^{-1}) )$ gradient evaluations in the general case and $\mathcal{O}( (d/\varepsilon)^{1/4} \log(d\varepsilon^{-1}) )$ in weakly interacting mean-field regimes. The analysis requires a log-Sobolev inequality for $\pi$, regularity on the potential $U$ (uniform bounds on up to fourth derivatives), and moment control via Lyapunov techniques; these conditions are shown to hold in mean-field-type potenitals under suitable bounds. Collectively, the results yield explicit, dimension-aware convergence rates and invariance-approximation errors for a broad class of unadjusted gHMC schemes, advancing understanding of entropy-based convergence for nonconvex, high-dimensional sampling.

Abstract

This paper provides a convergence analysis for generalized Hamiltonian Monte Carlo samplers, a family of Markov Chain Monte Carlo methods based on leapfrog integration of Hamiltonian dynamics and kinetic Langevin diffusion, that encompasses the unadjusted Hamiltonian Monte Carlo method. Assuming that the target distribution $π$ satisfies a log-Sobolev inequality and mild conditions on the corresponding potential function, we establish quantitative bounds on the relative entropy of the iterates defined by the algorithm, with respect to $π$. Our approach is based on a perturbative and discrete version of the modified entropy method developed to establish hypocoercivity for the continuous-time kinetic Langevin process. As a corollary of our main result, we are able to derive complexity bounds for the class of algorithms at hand. In particular, we show that the total number of iterations to achieve a target accuracy $\varepsilon >0$ is of order $d/\varepsilon^{1/4}$, where $d$ is the dimension of the problem. This result can be further improved in the case of weakly interacting mean field potentials, for which we find a total number of iterations of order $(d/\varepsilon)^{1/4}$.

Theorems & Definitions (41)

• Theorem 1
• Proposition 1
• Theorem 2
• Remark 1
• Theorem 3
• Remark 2
• proof : Proof of Theorem \ref{['thm:main_KL']}
• proof : Proof of \ref{['Prop:dissipation']}
• Lemma 1: Dissipation from velocity randomization
• proof