We study the rate at which the initial and current random variables become independent along a Markov chain, focusing on the Langevin diffusion in continuous time and the Unadjusted Langevin Algorithm (ULA) in discrete time. We measure the dependence between random variables via their mutual information. For the Langevin diffusion, we show the mutual information converges to $0$ exponentially fast when the target is strongly log-concave, and at a polynomial rate when the target is weakly log-concave. These rates are analogous to the mixing time of the Langevin diffusion under similar assumptions. For the ULA, we show the mutual information converges to $0$ exponentially fast when the target is strongly log-concave and smooth. We prove our results by developing the mutual version of the mixing time analyses of these Markov chains. We also provide alternative proofs based on strong data processing inequalities for the Langevin diffusion and the ULA, and by showing regularity results for these processes in mutual information.

翻译：我们研究了沿着马尔可夫链的初始随机变量与当前随机变量变得独立的速度，重点关注连续时间下的 Langevin 扩散与离散时间下的未调整 Langevin 算法（ULA）。我们通过互信息度量随机变量之间的依赖关系。对于 Langevin 扩散，我们证明当目标分布为强对数凹时，互信息以指数速度收敛至 $0$；当目标分布为弱对数凹时，互信息以多项式速度收敛。这些速率与类似假设下 Langevin 扩散的混合时间具有类比性。对于 ULA，我们证明当目标分布为强对数凹且光滑时，互信息以指数速度收敛至 $0$。我们通过发展这些马尔可夫链混合时间分析的互信息版本得以证明这些结果。我们还基于 Langevin 扩散和 ULA 的强数据处理不等式，以及通过展示这些过程在互信息中的正则性结果，提供了替代证明。