Stochastic Gradient Langevin Dynamics (SGLD) is a popular variant of Stochastic Gradient Descent, where properly scaled isotropic Gaussian noise is added to an unbiased estimate of the gradient at each iteration. This modest change allows SGLD to escape local minima and suffices to guarantee asymptotic convergence to global minimizers for sufficiently regular non-convex objectives (Gelfand and Mitter, 1991). The present work provides a nonasymptotic analysis in the context of non-convex learning problems: SGLD requires $\tilde{O}(\varepsilon^{-4})$ iterations to sample $\tilde{O}(\varepsilon)$-approximate minimizers of both empirical and population risk, where $\tilde{O}(\cdot)$ hides polynomial dependence on a temperature parameter, the model dimension, and a certain spectral gap parameter. As in the asymptotic setting, our analysis relates the discrete-time SGLD Markov chain to a continuous-time diffusion process. A new tool that drives the results is the use of weighted transportation cost inequalities to quantify the rate of convergence of SGLD to a stationary distribution in the Euclidean $2$-Wasserstein distance.

使用加权运输成本不等式来量化SGLD在欧几里得2-瓦瑟斯坦距离下收敛到随机分布的速率，并在非凸学习问题的背景下提供有限时间保证来找到两种风险的近似最小化器。

随机梯度朗逊动力学的非凸学习: 非渐近分析