Given a target function $H$ to minimize or a target Gibbs distribution $\pi_{\beta}^0 \propto e^{-\beta H}$ to sample from in the low temperature, in this paper we propose and analyze Langevin Monte Carlo (LMC) algorithms that run on an alternative landscape as specified by $H^f_{\beta,c,1}$ and target a modified Gibbs distribution $\pi^f_{\beta,c,1} \propto e^{-\beta H^f_{\beta,c,1}}$, where the landscape of $H^f_{\beta,c,1}$ is a transformed version of that of $H$ which depends on the parameters $f,\beta$ and $c$. While the original Log-Sobolev constant affiliated with $\pi^0_{\beta}$ exhibits exponential dependence on both $\beta$ and the energy barrier $M$ in the low temperature regime, with appropriate tuning of these parameters and subject to assumptions on $H$, we prove that the energy barrier of the transformed landscape is reduced which consequently leads to polynomial dependence on both $\beta$ and $M$ in the modified Log-Sobolev constant associated with $\pi^f_{\beta,c,1}$. This yield improved total variation mixing time bounds and improved convergence toward a global minimum of $H$. We stress that the technique developed in this paper is not only limited to LMC and is broadly applicable to other gradient-based optimization or sampling algorithms.

翻译：针对在低温环境下需最小化的目标函数$H$或需从中采样的目标吉布斯分布$\pi_{\beta}^0 \propto e^{-\beta H}$，本文提出并分析了运行于由$H^f_{\beta,c,1}$定义的替代地形上的朗之万蒙特卡洛（LMC）算法，该算法以修正的吉布斯分布$\pi^f_{\beta,c,1} \propto e^{-\beta H^f_{\beta,c,1}}$为目标。其中$H^f_{\beta,c,1}$的地形是$H$经参数$f$、$\beta$和$c$变换后的版本。在低温条件下，原始$\pi^0_{\beta}$的对数索博列夫常数对$\beta$和能垒$M$呈指数依赖关系。通过适当调节这些参数并基于对$H$的假设，我们证明变换后地形的能垒降低，从而使得与$\pi^f_{\beta,c,1}$相关的修正对数索博列夫常数对$\beta$和$M$呈多项式依赖关系。这改进了全变差混合时间界，并加速向$H$全局最小值的收敛。需强调的是，本文发展的技术不仅限于LMC，还可广泛适用于其他基于梯度的优化或采样算法。