[Paper Review] Second order quantitative bounds for unadjusted generalized Hamiltonian Monte Carlo

Evan Camrud, Alain Durmus+2 more|arXiv (Cornell University)|Jun 15, 2023

Markov Chains and Monte Carlo Methods4 citations

TL;DR

This paper establishes second-order quantitative convergence bounds for unadjusted generalized Hamiltonian Monte Carlo (gHMC) using a perturbative discrete version of the modified entropy method. Under a log-Sobolev condition and mild potential regularity, it proves relative entropy decays as $\mathcal{O}(\delta^2)$, yielding a total iteration complexity of $\mathcal{O}(d/\varepsilon^{1/4})$ to achieve accuracy $\varepsilon$, improving to $\mathcal{O}((d/\varepsilon)^{1/4})$ for weakly interacting mean field potentials.

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 $(d/\varepsilon)^{1/4}$.

Motivation & Objective

• To establish quantitative convergence rates for unadjusted generalized Hamiltonian Monte Carlo (gHMC) in terms of relative entropy.
• To extend existing hypocoercivity-based entropy methods to discrete, perturbative schemes for gHMC.
• To derive explicit iteration complexity bounds for gHMC under log-Sobolev and regularity conditions on the potential.
• To analyze the impact of potential structure (e.g., mean field) on convergence speed.
• To provide second-order error bounds in relative entropy, improving upon first-order schemes.

Proposed method

• Adapts the continuous-time modified entropy method to discrete, perturbative gHMC dynamics using a splitting integrator.
• Introduces a discrete Lyapunov function based on relative entropy with respect to the target measure $\pi$.
• Uses a perturbation analysis to control the discrepancy between the discrete gHMC kernel and the ideal Hamiltonian dynamics.
• Employs a weighted norm $\mathfrak{W}$ and a function $\mathbf{M}$ to bound higher-order error terms in the generator.
• Applies the log-Sobolev inequality to control the decay of relative entropy over iterations.
• Derives bounds on error terms via Taylor expansion and integral remainder estimates for the leapfrog integrator.

Experimental results

Research questions

• RQ1 Can second-order convergence bounds be established for unadjusted gHMC in relative entropy under a log-Sobolev condition?
• RQ2 How does the discrete, perturbative nature of gHMC affect the entropy decay rate compared to continuous dynamics?
• RQ3 What is the total iteration complexity of gHMC to achieve a target accuracy $\varepsilon$ in high dimensions?
• RQ4 How does the structure of the potential (e.g., weakly interacting mean field) influence the convergence rate?
• RQ5 Can the modified entropy method be adapted to discrete, non-reversible MCMC schemes like gHMC?

Key findings

• The relative entropy of the gHMC iterates decays at a rate of $\mathcal{O}(\delta^2)$, with a constant depending on the initial distribution and potential regularity.
• The total number of iterations required to achieve accuracy $\varepsilon > 0$ is $\mathcal{O}(d/\varepsilon^{1/4})$ under standard log-Sobolev and regularity conditions.
• For weakly interacting mean field potentials, the iteration complexity improves to $\mathcal{O}((d/\varepsilon)^{1/4})$.
• The analysis establishes second-order error bounds in the relative entropy, improving upon first-order schemes like Euler-based methods.
• The method is robust to weaker assumptions: replacing the fourth-order regularity condition with a third-order one yields $\mathcal{O}(\delta^2)$ bounds instead of $\mathcal{O}(\delta^4)$.
• The results are derived via a discrete, perturbative adaptation of the continuous-time modified entropy method, validated through integral error bounds and generator analysis.