Irreversible perturbations accelerate the convergence of Langevin dynamics, breaking detailed balance while preserving the invariant measure. The design of optimal irreversible perturbations has been studied in the continuous-time Gaussian setting, but extensions to non-Gaussian target distributions, and the impact of time discretization on the design of optimal perturbations, have not been well understood. Numerical discretizations of Langevin dynamics introduce bias, which is typically exacerbated by irreversible perturbations; handling this interaction demands a joint treatment of acceleration and accuracy. This paper develops a systematic framework for optimizing position-independent irreversible perturbations of the unadjusted Langevin algorithm (ULA). We formulate a constrained optimization problem that simultaneously accounts for mixing efficiency and discretization bias, where the former is characterized by a spectral gap analogue and the latter is quantified via a weighted expected squared jump distance. Within this framework, we derive an explicit characterization of the optimal position-independent irreversible perturbation. Extensive numerical experiments demonstrate that our design yields faster convergence with controlled bias, and improves mean squared estimation errors compared to other choices of irreversible perturbation.

翻译：不可逆扰动通过打破细致平衡条件但仍保持不变测度，加速了朗之万动力学的收敛。连续时间高斯框架下最优不可逆扰动的设计已有研究，但针对非高斯目标分布以及时间离散化对最优扰动设计的影响尚未得到深入理解。朗之万动力学的数值离散化会引入偏差，而不可逆扰动通常会加剧这一偏差；处理这种交互作用需要同时考虑加速与精度。本文建立了一个系统化框架，用于优化非调节朗之万算法中与位置无关的不可逆扰动。我们构建了一个约束优化问题，该问题同时考虑混合效率与离散化偏差：前者通过谱间隙的类比特征刻画，后者通过加权期望跳跃距离的平方进行量化。在该框架下，我们推导出最优位置无关不可逆扰动的显式表征。大量数值实验表明，本文设计的扰动可在可控偏差下实现更快的收敛，同时相较于其他不可逆扰动选择，显著降低了均方估计误差。