Langevin Dynamics has been extensively employed in global non-convex optimization due to the concentration of its stationary distribution around the global minimum of the potential function at low temperatures. In this paper, we propose to utilize a more comprehensive class of stochastic processes, known as reversible diffusion, and apply the Euler-Maruyama discretization for global non-convex optimization. We design the diffusion coefficient to be larger when distant from the optimum and smaller when near, thus enabling accelerated convergence while regulating discretization error, a strategy inspired by landscape modifications. Our proposed method can also be seen as a time change of Langevin Dynamics, and we prove convergence with respect to KL divergence, investigating the trade-off between convergence speed and discretization error. The efficacy of our proposed method is demonstrated through numerical experiments.

翻译：朗之万动力学因其稳态分布在低温下集中于势函数全局最小值附近，已被广泛用于全局非凸优化。本文提出利用更广泛的随机过程类别——可逆扩散，并采用欧拉-丸山离散化方法进行全局非凸优化。我们设计扩散系数在远离最优值时较大、接近最优值时较小，从而在调控离散化误差的同时加速收敛，这一策略受景观修正思想的启发。所提方法也可视为朗之万动力学的时间变换，我们从KL散度角度证明其收敛性，并探究收敛速度与离散化误差之间的权衡关系。数值实验验证了所提方法的有效性。