用平均光滑性改进了Langevin采样误差界，提升采样效率。
Improved Guarantees for Langevin Monte Carlo with Average Smoothness
- 以坐标平均光滑性替代全局光滑性，优化采样误差分析
- 在强对数凹下，误差随维度改善，尤其适用于相关协变量
- 适合关注采样精度与高维统计推断的研究者
我们在强对数凹设定下，针对用水波斯特距离衡量的误差，建立了Langevin Monte Carlo的改进非渐近界。主要结果表明，离散误差由平均坐标光滑性常数决定，而非传统的全局光滑性常数。证明简洁且基于概率方法，利用了精细化的同步耦合技术。进一步表明，相同思路可推广至变步长、拉普拉斯光滑势函数（此时海森伯格利普希茨项被弱化的迹型三阶光滑量取代）以及带固定点控制变量的随机梯度Langevin动力学有限和问题。在有限和情形下，所得SGLD界改进了对分量函数均方光滑性的依赖。在高斯设计的广义线性模型应用中，这些改进可带来显著的、与维度相关的性能提升，尤其在协变量相关时更为明显。
原文摘要 · Abstract (English)
We establish improved nonasymptotic bounds for Langevin Monte Carlo in the strongly log-concave setting, when the error is measured by the Wasserstein distance. The main result shows that the discretization error is governed by an average coordinate-wise smoothness constant, rather than by the usual global smoothness constant. The proof is short and probabilistic, and relies on a refined use of the synchronous coupling. We further show that the same ideas lead to improved bounds for variable step sizes, for potentials whose Laplacian is Lipschitz-continuous, and for finite-sum problems sampled by stochastic-gradient Langevin dynamics with fixed point control variates. In the Laplacian-smooth case, the usual Hessian-Lipschitz contribution is replaced by a weaker trace-type third-order smoothness quantity. In the finite-sum setting, the resulting SGLD bound improves the dependence on the root mean square smoothness of the component functions. Applications to generalized linear models with Gaussian design show that these refinements can yield substantial, dimension-dependent improvements over previously known bounds, especially for correlated covariates.
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