It is well known in many settings that reversible Langevin diffusions in confining potentials converge to equilibrium exponentially fast. Adding irreversible perturbations to the drift of a Langevin diffusion that maintain the same invariant measure accelerates its convergence to stationarity. Many existing works thus advocate the use of such non-reversible dynamics for sampling. When implementing Markov Chain Monte Carlo algorithms (MCMC) using time discretisations of such Stochastic Differential Equations (SDEs), one can append the discretization with the usual Metropolis–Hastings accept–reject step and this is often done in practice because the accept–reject step eliminates bias. On the other hand, such a step makes the resulting chain reversible. It is not known whether adding the accept–reject step preserves the faster mixing properties of the non-reversible dynamics. In this paper, we address this gap between theory and practice by analyzing the optimal scaling of MCMC algorithms constructed from proposal moves that are time-step Euler discretisations of an irreversible SDE, for high dimensional Gaussian target measures. We call the resulting algorithm the ipMALA , in comparison to the classical MALA algorithm (here ip is for irreversible proposal). In order to quantify how the cost of the algorithm scales with the dimension N, we prove invariance principles for the appropriately rescaled chain. In contrast to the usual MALA algorithm, we show that there could be two regimes asymptotically: (i) a diffusive regime, as in the MALA algorithm and (ii) a “fluid” regime where the limit is an ordinary differential equation. We provide concrete examples where the limit is a diffusion, as in the standard MALA, but with provably higher limiting acceptance probabilities. Numerical results are also given corroborating the theory.

中文翻译：

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具有不可逆提议的高斯目标的MALA算法的最佳缩放

众所周知，在许多情况下，在限制势中可逆的兰格文扩散迅速地以指数形式收敛到平衡。在保持相同不变性度量的Langevin扩散的漂移中添加不可逆的扰动会加速其趋于平稳。因此，许多现有的工作提倡使用这种不可逆的动力学进行采样。当使用此类随机微分方程（SDE）的时间离散化实现马尔可夫链蒙特卡罗算法（MCMC）时，可以将离散化附加到通常的Metropolis-Hastings接受-拒绝步骤，并且在实践中通常会这样做，因为接受-拒绝步骤消除偏见。另一方面，这样的步骤使所得的链可逆。不知道增加接受-拒绝步骤是否会保留不可逆动力学的更快混合特性。在本文中，我们通过分析从提议动作构建的MCMC算法的最佳缩放比例来解决理论与实践之间的这种差距，该提议动作是不可逆SDE的时步Euler离散化，用于高维高斯目标测度。与经典的MALA算法相比，我们将结果算法称为ipMALAip是不可逆转的提案）。为了量化算法的成本如何以维度N缩放，我们证明了适当重新缩放的链的不变性原理。与通常的MALA算法相反，我们表明可能有两种渐近形式：（i）像MALA算法中那样的扩散形式，以及（ii）极限是一个常微分方程的“流体”形式。我们提供了一些具体示例，其中极限是扩散，如标准的MALA中一样，但是极限接受概率更高。数值结果也证实了该理论。