Classically, the continuous-time Langevin diffusion converges exponentially fast to its stationary distribution \(\pi \) under the sole assumption that \(\pi \) satisfies a Poincaré 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 Rényi 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 Latała–Oleszkiewicz or modified log-Sobolev inequality, which interpolates between the Poincaré and log-Sobolev settings. Unlike prior works, our results allow for weak smoothness and do not require convexity or dissipativity conditions.

经典地，连续时间朗之万扩散在满足庞加莱不等式的唯一假设下以指数方式快速收敛到其平稳分布 \(\pi \)。然而，由于需要处理卡方或 Rényi 散度，因此利用这一事实为离散时间 Langevin Monte Carlo (LMC) 算法提供保证更具挑战性，并且之前的工作主要集中在强对数凹上目标。在这项工作中，我们为 LMC 提供了第一个收敛保证，假设 \(\pi \) 满足 Latała–Oleszkiewicz 或修改后的 log-Sobolev 不等式，该不等式在 Poincaré 和 log-Sobolev 设置之间进行插值。与之前的工作不同，我们的结果允许弱平滑度，并且不需要凸性或耗散性条件。