We present a significant advancement in the field of Langevin Monte Carlo (LMC) methods by introducing the Inexact Proximal Langevin Algorithm (IPLA). This novel algorithm broadens the scope of problems that LMC can effectively address while maintaining controlled computational costs. IPLA extends LMC's applicability to potentials that are convex, strongly convex in the tails, and exhibit polynomial growth, beyond the conventional $L$-smoothness assumption. Moreover, we extend LMC's applicability to super-quadratic potentials and offer improved convergence rates over existing algorithms. Additionally, we provide bounds on all moments of the Markov chain generated by IPLA, enhancing its analytical robustness.

翻译：我们通过提出不精确近端朗之万算法（IPLA），在朗之万蒙特卡洛（LMC）方法领域取得了一项重要进展。这种新颖的算法在保持可控计算成本的同时，拓宽了LMC能够有效处理的问题范围。IPLA将LMC的适用性扩展到势函数具有凸性、尾部强凸性并呈现多项式增长的情形，超越了传统的$L$-光滑性假设。此外，我们将LMC的适用性扩展到超二次势函数，并提供了优于现有算法的收敛速率。另外，我们还给出了IPLA生成的马尔可夫链所有矩的界，从而增强了其分析鲁棒性。