We study the problem of non-convex optimization using Stochastic Gradient Langevin Dynamics (SGLD). SGLD is a natural and popular variation of stochastic gradient descent where at each step, appropriately scaled Gaussian noise is added. To our knowledge, the only strategy for showing global convergence of SGLD on the loss function is to show that SGLD can sample from a stationary distribution which assigns larger mass when the function is small (the Gibbs measure), and then to convert these guarantees to optimization results. We employ a new strategy to analyze the convergence of SGLD to global minima, based on Lyapunov potentials and optimization. We convert the same mild conditions from previous works on SGLD into geometric properties based on Lyapunov potentials. This adapts well to the case with a stochastic gradient oracle, which is natural for machine learning applications where one wants to minimize population loss but only has access to stochastic gradients via minibatch training samples. Here we provide 1) improved rates in the setting of previous works studying SGLD for optimization, 2) the first finite gradient complexity guarantee for SGLD where the function is Lipschitz and the Gibbs measure defined by the function satisfies a Poincar\'e Inequality, and 3) prove if continuous-time Langevin Dynamics succeeds for optimization, then discrete-time SGLD succeeds under mild regularity assumptions.

我们研究了使用随机梯度朗之万动力学（SGLD）进行非凸优化的问题。我们采用了一种基于李雅普诺夫势函数和优化的新策略来分析SGLD收敛到全局最小值的情况，将以前关于SGLD的轻微条件转化为基于李雅普诺夫势函数的几何属性。我们提供了在以前研究SGLD进行优化的设置中的改进速度，SGLD的第一个有界梯度复杂性保证以及连续时间朗之万动力学在满足一些适度正则性假设时，离散时间SGLD也能成功的证明。

Langevin动力学：通过李雅普诺夫势函数统一优化的视角