Accelerated Mixing Time of Randomized Hamiltonian Monte Carlo

2026-07-14

AI Summary

This work addresses the accelerated mixing of Randomized Hamiltonian Monte Carlo (RHMC) for log-concave distributions. Focusing on target distributions satisfying an α-Talagrand inequality, the authors propose a novel strategy that simulates continuous Hamiltonian dynamics with randomized integration times and resets the momentum to an independent Gaussian at each iteration. They establish, for the first time, an accelerated mixing time bound for RHMC in terms of KL divergence, revealing a deep connection to accelerated optimization methods. By employing randomized time schedules—specifically triangular or exponential distributions—and leveraging tools from information geometry, they derive an explicit average decay rate for the KL divergence. Under α-strong log-concavity, the total integration time required to achieve ε accuracy scales as O(α⁻¹/² log(ε⁻¹)); for general log-concave targets, using a triangular distribution with exponentially growing mean yields a complexity of O(ε⁻¹/²).

Abstract

We show the Randomized Hamiltonian Monte Carlo (RHMC) algorithm has accelerated mixing time guarantees for sampling from log-concave probability distributions. RHMC proceeds by repeatedly simulating the continuous-time Hamiltonian dynamics for some random integration times, and resetting the velocity to be an independent Gaussian random variable between each simulation. We show that when the target distribution is log-concave and satisfies an $α$-Talagrand inequality (for example, if the target distribution is $α$-strongly log-concave), if we use a random integration time from either the triangular or the exponential distribution with mean $Θ(α^{-1/2})$, then RHMC converges exponentially fast in KL divergence, and the total integration time to reach error $\varepsilon$ in KL divergence scales as $O(α^{-1/2} \log(\varepsilon^{-1}))$. We also show that when the target distribution is log-concave, if we use a sequence of random integration times from the triangular distribution with exponentially increasing means, then the total integration time to reach error $\varepsilon$ in KL divergence scales as $O(\varepsilon^{-1/2})$. Our analysis relies on a bound on the average KL divergence along Hamiltonian dynamics, which is inspired by an analogous result on accelerated optimization methods based on Hamiltonian dynamics.

Problem

• Randomized Hamiltonian Monte Carlo
• mixing time
• log-concave distributions
• KL divergence
• accelerated convergence

Innovation

• Randomized Hamiltonian Monte Carlo
• accelerated mixing time
• log-concave distributions
• KL divergence
• Talagrand inequality