High-dimensional normal approximations for sums of Langevin Markov chains

TL;DR

The paper derives dimension-explicit, nonasymptotic normal approximation rates for normalized sums of Langevin Monte Carlo iterates in high dimensions. By combining a martingale decomposition with Stein's method and an exchangeable-pair construction, it obtains 1-Wasserstein bounds between the scaled partial sums and a d-dimensional standard normal, with explicit dependence on the dimension d, step size η, and iteration count n. A linear-case warm-up shows faster convergence when ∇U is linear, while the general case relies on a Jacobi-flow-based regularity analysis and careful remainder control. These results provide theoretical guarantees for Gaussian approximations in high-dimensional Langevin sampling, informing step-size and iteration choices for accurate finite-sample behavior.

Abstract

Consider the well-known Langevin diffusion on $\mathbb{R}^d$ $$\mathrm{d} X_t = -\nabla U(X_t)\,\mathrm{d} t + \sqrt{2}\mathrm{d} B_t, $$ and its Euler-Maruyama discretization given by $$X_{k+1}=X_k-η\nabla U(X_k)+\sqrt{2η}ξ_{k+1},$$ where $η$ is the step size. Under mild conditions, the Langevin diffusion admits $π(\mathrm{d} x)\propto \exp(-U(x))\mathrm{d} x$ as its unique stationary distribution. In this paper, we mainly study the normal approximation of the normalized partial sum $$ W_n = η^{1/2} n^{-1/2} \left( \sum_{i=0}^{n-1} X_i- \int_{\mathbb{R}^d} x\,π(\mathrm{d} x) \right).$$ To the best of our knowledge, this work provides the first dimension-explicit convergence rates in high-dimensional settings. Our main tool is a novel upper bound for the 1-Wasserstein distance $W_1(W,γ)$ via the exchange pair approach, where $W$ is any random vector of interest and $γ$ is a $d$-dimensional standard normal random vector.

Theorems & Definitions (32)

• Theorem 1.2
• Remark 1.3
• Remark 1.4
• Corollary 1.5
• Remark 1.6
• Lemma 1.7
• Theorem 2.1
• Lemma 2.2
• proof
• proof : Proof of Theorem \ref{['T1']}