Dimension-Independent Convergence of Underdamped Langevin Monte Carlo in KL Divergence
arXiv:2603.02429v1 Announce Type: new Abstract: Underdamped Langevin dynamics (ULD) is a widely-used sampler for Gibbs distributions $\pi\propto e^{-V}$, and is often empirically effective in high dimensions. However, existing non-asymptotic convergence guarantees for discretized ULD typically scale polynomially with the ambient dimension $d$, leading to vacuous bounds when $d$ is large. The main known dimension-free result concerns the randomized midpoint discretization in Wasserstein-2 distance (Liu et al.,2023), while dimension-independent guarantees for ULD discretizations in KL divergence have remained open. We close this gap by proving the first dimension-free KL divergence bounds for discretized ULD. Our analysis refines the KL local error framework (Altschuler et al., 2025) to a dimension-free setting and yields bounds that depend on $\mathrm{tr}(\mathbf{H})$, where $\mathbf{H}$ upper bounds the Hessian of $V$, rather than on $d$. As a consequence, we obtain improved iteration
arXiv:2603.02429v1 Announce Type: new Abstract: Underdamped Langevin dynamics (ULD) is a widely-used sampler for Gibbs distributions $\pi\propto e^{-V}$, and is often empirically effective in high dimensions. However, existing non-asymptotic convergence guarantees for discretized ULD typically scale polynomially with the ambient dimension $d$, leading to vacuous bounds when $d$ is large. The main known dimension-free result concerns the randomized midpoint discretization in Wasserstein-2 distance (Liu et al.,2023), while dimension-independent guarantees for ULD discretizations in KL divergence have remained open. We close this gap by proving the first dimension-free KL divergence bounds for discretized ULD. Our analysis refines the KL local error framework (Altschuler et al., 2025) to a dimension-free setting and yields bounds that depend on $\mathrm{tr}(\mathbf{H})$, where $\mathbf{H}$ upper bounds the Hessian of $V$, rather than on $d$. As a consequence, we obtain improved iteration complexity for underdamped Langevin Monte Carlo relative to overdamped Langevin methods in regimes where $\mathrm{tr}(\mathbf{H})\ll d$.
Executive Summary
This article closes a long-standing gap in the convergence guarantees of underdamped Langevin Monte Carlo (ULD) in KL divergence. By refining the KL local error framework, the authors provide dimension-free bounds for discretized ULD, relying on the trace of the Hessian matrix rather than the ambient dimension. This breakthrough has significant implications for high-dimensional sampling and improves the iteration complexity of ULD compared to overdamped methods. The results are particularly relevant in regimes where the trace of the Hessian is small relative to the dimension. Overall, this article contributes a crucial step towards more accurate and efficient sampling methods for complex distributions.
Key Points
- ▸ The article provides the first dimension-free KL divergence bounds for discretized ULD.
- ▸ The bounds rely on the trace of the Hessian matrix rather than the ambient dimension.
- ▸ The results improve the iteration complexity of ULD compared to overdamped methods.
Merits
Strength of Convergence Guarantees
The article provides tight and dimension-free convergence guarantees for ULD, addressing a long-standing open problem in the field.
Improved Iteration Complexity
The results demonstrate that ULD can achieve better iteration complexity than overdamped methods in certain regimes, making it a more attractive choice for high-dimensional sampling.
Demerits
Assumption on Hessian Matrix
The bounds rely on the assumption that the Hessian matrix is upper-bounded by a matrix with a small trace, which may not always be the case in practice.
Limited Scope of Application
The results are specific to KL divergence and may not generalize to other divergences or sampling problems.
Expert Commentary
This article marks a significant breakthrough in the convergence guarantees of ULD and has far-reaching implications for high-dimensional sampling. The dimension-free bounds provided in the article are a crucial step towards more accurate and efficient sampling methods. However, it is essential to acknowledge the limitations of the results, particularly the assumption on the Hessian matrix. Future research should focus on relaxing this assumption and extending the results to other divergences and sampling problems. Additionally, the implications of this work extend beyond the field of Monte Carlo methods, with potential applications in optimization and machine learning.
Recommendations
- ✓ Future research should focus on relaxing the assumption on the Hessian matrix and extending the results to other divergences and sampling problems.
- ✓ The results should be tested and validated in practice to demonstrate their efficiency and accuracy in high-dimensional sampling.