The purpose of this paper is to examine the sampling problem through Euler discretization, where the potential function is assumed to be a mixture of locally smooth distributions and weakly dissipative. We introduce $\alpha_{G}$-mixture locally smooth and $\alpha_{H}$-mixture locally Hessian smooth, which are novel and typically satisfied with a mixture of distributions. Under our conditions, we prove the convergence in Kullback-Leibler (KL) divergence with the number of iterations to reach $\epsilon$-neighborhood of a target distribution in only polynomial dependence on the dimension. The convergence rate is improved when the potential is $1$-smooth and $\alpha_{H}$-mixture locally Hessian smooth. Our result for the non-strongly convex outside the ball of radius $R$ is obtained by convexifying the non-convex domains. In addition, we provide some nice theoretical properties of $p$-generalized Gaussian smoothing and prove the convergence in the $L_{\beta}$-Wasserstein distance for stochastic gradients in a general setting.

翻译：本文旨在研究通过欧拉离散化进行采样的问题，其中势函数被假设为局部光滑分布与弱耗散性的混合。我们引入了$\alpha_{G}$-混合局部光滑性和$\alpha_{H}$-混合局部海森光滑性，这些概念是新颖的，并且通常适用于分布混合情形。在此条件下，我们证明了Kullback-Leibler (KL)散度的收敛性，达到目标分布的$\epsilon$邻域所需的迭代次数仅与维度呈多项式关系。当势能为$1$-光滑且满足$\alpha_{H}$-混合局部海森光滑性时，收敛速度得以提升。对于半径$R$球外非强凸的情形，我们通过将非凸域凸化来获得结果。此外，我们还提供了$p$-广义高斯平滑的良好理论性质，并证明了在一般设置下随机梯度的$L_{\beta}$-Wasserstein距离收敛性。