论文标题
Langevin模拟的退火算法与乘法噪声II：总变化
Convergence of Langevin-Simulated Annealing algorithms with multiplicative noise II: Total Variation
论文作者
论文摘要
我们研究了Langevin模拟的退火类型算法的收敛性，即$ V：\ Mathbb {r}^d \ to \ Mathbb {r} $一个潜在的功能可以最小化，我们考虑了随机微分方程$ dy_t $ dy_t = -dt = -dt = -dt \ top \ nabla V（y__________________________________________________t + dt dt dt dt dt dt dt dt dt dt dt dt dt dt dt dt dt dt dt dt dt dt dt dt dt dt（y__t） a（t）σ（y_t）dw_t + a（t）^2υ（y_t）dt $，其中$（w_t）$是一个布朗运动，其中$σ：\ Mathbb {r}^d \ to \ to \ nathcal {mathcal {m} _d（m} _d _d（\ mathbb {r}） \ Mathbb {r}^+ \ to \ mathbb {r}^+ $是一个函数降低到$ 0 $，而$υ$是一个更正术语。与经典的langevin方程相比，允许$σ$取决于位置会带来更快的收敛性。在上一篇论文中，我们在$ l^1 $ -wasserstein中建立了$ y_t $的距离及其相关的Euler方案$ \ bar {y} _t _t $ to $ \ text {argmin}（argmin}（v）$，带有经典附表$ a（t）= a（t）= a \ log^a \ log^a \ log^{ - 1/2}（ - 1/2}（t）$。在本文中，我们证明了总变化距离的收敛性。总变化案例似乎更需要处理，需要正则化引理。
We study the convergence of Langevin-Simulated Annealing type algorithms with multiplicative noise, i.e. for $V : \mathbb{R}^d \to \mathbb{R}$ a potential function to minimize, we consider the stochastic differential equation $dY_t = - σσ^\top \nabla V(Y_t) dt + a(t)σ(Y_t)dW_t + a(t)^2Υ(Y_t)dt$, where $(W_t)$ is a Brownian motion, where $σ: \mathbb{R}^d \to \mathcal{M}_d(\mathbb{R})$ is an adaptive (multiplicative) noise, where $a : \mathbb{R}^+ \to \mathbb{R}^+$ is a function decreasing to $0$ and where $Υ$ is a correction term. Allowing $σ$ to depend on the position brings faster convergence in comparison with the classical Langevin equation $dY_t = -\nabla V(Y_t)dt + σdW_t$. In a previous paper we established the convergence in $L^1$-Wasserstein distance of $Y_t$ and of its associated Euler scheme $\bar{Y}_t$ to $\text{argmin}(V)$ with the classical schedule $a(t) = A\log^{-1/2}(t)$. In the present paper we prove the convergence in total variation distance. The total variation case appears more demanding to deal with and requires regularization lemmas.
