[Paper Review] Uniform minorization condition and convergence bounds for discretizations of kinetic Langevin dynamics

Alain Durmus, Aurélien Enfroy+2 more|arXiv (Cornell University)|Jul 30, 2021

Markov Chains and Monte Carlo Methods53 references4 citations

TL;DR

This paper establishes uniform minorization and geometric drift conditions for a broad class of time-discretized kinetic Langevin dynamics, enabling convergence bounds in total variation and $V$-norm that scale linearly with the timestep. The key contribution is a stepsize-uniform geometric ergodicity result, crucial for analyzing Poisson equation solutions and ensuring reliable sampling in molecular dynamics and Bayesian inference.

ABSTRACT

We study the convergence in total variation and $V$-norm of discretization schemes of the underdamped Langevin dynamics. Such algorithms are very popular and commonly used in molecular dynamics and computational statistics to approximatively sample from a target distribution of interest. We show first that, for a very large class of schemes, a minorization condition uniform in the stepsize holds. This class encompasses popular methods such as the Euler-Maruyama scheme and the schemes based on splitting strategies. Second, we provide mild conditions ensuring that the class of schemes that we consider satisfies a geometric Foster--Lyapunov drift condition, again uniform in the stepsize. This allows us to derive geometric convergence bounds, with a convergence rate scaling linearly with the stepsize. This kind of result is of prime interest to obtain estimates on norms of solutions to Poisson equations associated with a given numerical method.

Motivation & Objective

• To establish uniform minorization conditions for time-discretized kinetic Langevin dynamics across a wide class of numerical schemes.
• To derive geometric Foster–Lyapunov drift conditions that are uniform in the timestep, ensuring stability and convergence.
• To provide quantitative convergence bounds in total variation and $V$-norm that scale linearly with the stepsize $\gamma$.
• To support the analysis of Poisson equation solutions arising in statistical inference and molecular dynamics by ensuring uniform ergodicity.
• To extend existing convergence theory to underdamped (kinetic) Langevin dynamics, including non-gradient drifts and non-equilibrium settings.

Proposed method

• Introduces a Lyapunov function $\overline{\mathpzc{W}}_{\gamma,\varpi}$ that combines quadratic and potential terms to capture energy-like behavior.
• Establishes a uniform minorization condition via coupling arguments and hypoellipticity, valid for all $\gamma \leq \bar{\gamma}_{U}$, independent of stepsize.
• Derives a geometric drift condition $R_{\gamma}\overline{\mathpzc{W}}_{\gamma,\varpi} \leq \lambda_{U}^{\gamma}\overline{\mathpzc{W}}_{\gamma,\varpi} + \gamma b_U$ with $\lambda_U^\gamma < 1$, ensuring geometric ergodicity.
• Uses Malliavin calculus and parametrix methods to bound transition densities and support the minorization condition.
• Applies Lipschitz continuity and uniform boundedness of $\phi_\gamma$ and $\Gamma_\gamma$ to control perturbations in the discretization.
• Employs a time-averaged coupling strategy and exponential moment bounds to derive uniform convergence rates.

Experimental results

Research questions

• RQ1 Can a minorization condition be established uniformly in the timestep for a broad class of discretization schemes of kinetic Langevin dynamics?
• RQ2 Do these schemes satisfy a geometric Foster–Lyapunov drift condition uniformly in the stepsize $\gamma$?
• RQ3 What is the dependence of the convergence rate on the timestep $\gamma$ for such schemes?
• RQ4 Can the resulting convergence bounds be used to analyze solutions to Poisson equations in statistical applications?
• RQ5 How do the results extend to non-gradient drifts and non-equilibrium dynamics?

Key findings

• A uniform minorization condition holds for a large class of schemes, including Euler–Maruyama and splitting-based methods, for all $\gamma \leq \bar{\gamma}_{U}$.
• The geometric drift condition is established uniformly in $\gamma$, with $R_{\gamma}\overline{\mathpzc{W}}_{\gamma,\varpi} \leq \lambda_{U}^{\gamma}\overline{\mathpzc{W}}_{\gamma,\varpi} + \gamma b_U$, where $\lambda_U^\gamma < 1$.
• The convergence rate in total variation and $V$-norm scales linearly with the stepsize, with $\log(\rho_\gamma) \propto \gamma$.
• The bound $b_U = (\mathscr{K} - \log \lambda_U) \mathrm{e}^{\mathscr{K} \bar{\gamma}_U + \varpi_U(1 + \mathfrak{C}_\phi K_U)}$ ensures uniform control of the non-geometric term.
• The Lyapunov function $\overline{\mathpzc{W}}_{\gamma,\varpi}$ is shown to be non-negative and uniformly bounded via $\mathfrak{C}_\phi$ and $c_{\mathpzc{W}}$.
• Lipschitz continuity of $\phi_\gamma$ and $\Gamma_\gamma$ is proven with uniform constants, enabling stability analysis under perturbations.