Classically, the continuous-time Langevin diffusion converges exponentially fast to its stationary distribution $\pi$ under the sole assumption that $\pi$ satisfies a Poincar\'e inequality. Using this fact to provide guarantees for the discrete-time Langevin Monte Carlo (LMC) algorithm, however, is considerably more challenging due to the need for working with chi-squared or R\'enyi divergences, and prior works have largely focused on strongly log-concave targets. In this work, we provide the first convergence guarantees for LMC assuming that $\pi$ satisfies either a Lata\l{}a--Oleszkiewicz or modified log-Sobolev inequality, which interpolates between the Poincar\'e and log-Sobolev settings. Unlike prior works, our results allow for weak smoothness and do not require convexity or dissipativity conditions.

翻译：经典上，连续时间朗之万扩散在其平稳分布 $\pi$ 仅满足庞加莱不等式的假设下，即以指数速率收敛于 $\pi$。然而，利用这一事实为离散时间朗之万蒙特卡洛算法提供理论保证则更具挑战性，因为这需要处理卡方散度或雷尼散度，且先前工作主要集中于强对数凹目标分布。在本研究中，我们首次在假设 $\pi$ 满足拉塔瓦-奥莱什凯维奇不等式或修正对数索博列夫不等式的条件下，为LMC提供了收敛性保证，这些条件在庞加莱与对数索博列夫情形之间架起了桥梁。与先前工作不同，我们的结果允许弱光滑性假设，且不要求凸性或耗散性条件。