新算法让随机梯度采样能处理更复杂势函数，收敛更快。
Langevin Monte Carlo Beyond Lipschitz Gradient Continuity
- 采用近似邻近算子改进采样过程，放宽光滑性假设。
- 在多项式增长和超二次势函数下仍保持稳定收敛。
- 适用于高维优化与生成模型，尤其适合复杂分布采样。
我们提出了不精确邻近朗之万算法（IPLA），显著拓展了朗之万蒙特卡洛（LMC）方法的应用范围，同时控制计算开销。该算法将LMC的适用性扩展至尾部强凸、具有多项式增长的势函数，突破了传统L-光滑性的限制。此外，IPLA还适用于超二次势函数，并在收敛速度上优于现有算法。我们还为IPLA生成的马尔可夫链的所有矩提供了理论边界，增强了其分析鲁棒性。
原文摘要 · Abstract (English)
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.
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