Poisson Midpoint Method for Log Concave Sampling: Beyond the Strong Error Lower Bounds
Rishikesh Srinivasan, Dheeraj Nagaraj
TL;DR
This paper introduces a Poisson midpoint discretization method for sampling from strongly log-concave distributions, achieving faster convergence in Wasserstein distance than traditional methods, and reveals new complexity insights for underdamped Langevin dynamics.
Contribution
It proves convergence of the Poisson midpoint method in Wasserstein distance with a cubic speedup, surpassing existing randomized midpoint bounds, and compares complexity bounds for underdamped Langevin dynamics.
Findings
Achieves cubic speedup in convergence rate over Euler-Maruyama.
Surpasses existing bounds for randomized midpoint methods.
Shows complexity of Wasserstein convergence is lower than $L^2$ error bounds for underdamped Langevin.
Abstract
We study the problem of sampling from strongly log-concave distributions over using the Poisson midpoint discretization (a variant of the randomized midpoint method) for overdamped/underdamped Langevin dynamics. We prove its convergence in the 2-Wasserstein distance (), achieving a cubic speedup in dependence on the target accuracy () over the Euler-Maruyama discretization, surpassing existing bounds for randomized midpoint methods. Notably, in the case of underdamped Langevin dynamics, we demonstrate the complexity of convergence is much smaller than the complexity lower bounds for convergence in strong error established in the literature.
Peer Reviews
Decision·ICLR 2026 Poster
Important question, clear positioning: The introduction and §1.1 make a precise case that strong (L2) lower bounds for ULD do not preclude faster W2 rates, and the results indeed obtain $\tilde O(ε^{-1/3})$ for the underdamped case (Theorem 2, Cor. 2). Clear presentation on technical novelty and algorithmic efficiency.
Typo: In Eq (2), the coefficient on Brownian term should be $\sqrt{2\gamma}d$, in order to achieve right invariant distribution. No empirical study: The paper is purely theoretical. There are no experiments illustrating constants, stability, or the practical effect of hyper-parameters.
The paper obtains surprising low accuracy sampling guarantees with oracle complexity scaling as $\varepsilon^{-1/3}$. This defeats a conceptual lower bound. The paper obtains these new rates in a conceptually orthogonal way to any prior analysis, which potentially opens the doors to successive works adapting these techniques to other settings. I do not want to overelaborate, but I think these are clear and very salient contributions, and I believe the techniques in this paper deserve more expo
In principle, improved rates for low accuracy algorithms in $\varepsilon$ are not useful if they come at the expense of dimension dependence, due to the existence of high accuracy algorithms. In particular, if we take $\varepsilon = \tilde{\varepsilon}/d^{1/2}$ here, then we observe worse dimension dependence $d^{3/8}$ in the second term (as $p \to \infty$). However, I do not see this as a severe issue. This paper explores only the simplest principled setting; many settings still fall outside t
he paper is very well-written. It provides a clear description of the problem, a thorough comparison of its theoretical results with existing works, and insightful proof techniques that illustrate the reasons behind the achieved convergence rates.
It is recommended to exchange the positions of Sections 2.2 and 2.3 for a more logical flow. Additionally, the origin of the Poisson midpoint method should be clarified: is it a novel proposal of this work, or is it adopted from prior literature?
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Taxonomy
TopicsMarkov Chains and Monte Carlo Methods · Stochastic Gradient Optimization Techniques · Random Matrices and Applications