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:本文的目的是通过 Euler 分解来研究取样问题, 其潜在功能被假定为本地平滑分布和微弱散射的混合体。 我们引入了美元/ ALpha+G}$- 当地平滑和美元/ ALpha+H}- 当地混合体。 黑森光滑是新颖的, 通常对分布的混合体感到满意。 在我们的条件下, 我们证明Kullback- Leiber (KL) 与不凝固区域之间的迭代数的趋同。 此外, 我们提供了美元通用高频分布的理论性能, 仅对维度具有多元依赖性。 当海相潜力为$- mooth 和 $/ alpha+H} 美元- 当地平滑和 美元/ 黑森 当地混合体时, 趋同率会得到改善。 我们对于半径球外非凝固的锥形共振的结果是通过对非凝固区域进行混结而获得的。 此外, 我们提供美元通用高频平面高频平滑度测量平滑度平滑度的理论特性, 并证明在 ALsalstealselsteal- settal- settylatestelation 。