A sharp uniform-in-time error estimate for Stochastic Gradient Langevin Dynamics

The Stochastic Gradient Langevin Dynamics (SGLD) [49], first proposed by Welling and Teh, has drawn great attention of researchers when dealing with optimization or sampling tasks[2, 33, 40]. As a samplingalgorithm, SGLD canbe viewed asa"randombatch"version of the Unadjusted Langevin Algorithm (ULA), which is the Euler-Maruyama discretization of the Langevin diffusion, a stochastic process converging to a target Gibbs' distribution under suitable settings. As an optimization algorithm, SGLD can be viewed as a variant of the classical Stochastic Gradient Descent (SGD) [44], by adding independent Gaussian noise in each iteration of SGD. At recent decades, SGD and its variants [44, 25, 11, 37] have received a great deal of attention when solving high-dimensional tasks, ranging from computer vision, natural language processing, to high dimensional sampling, statistical optimization, etc. Also much theoretical analysis for SGD has been done by former researchers, including loss landscape of SGD iteration [46, 47], its dynamical stability [50] and diffusion approximation [32, 21, 17]. The combination of the SGD algorithm and the Langevin diffusion, can improve the behavior of both methods: for SGD, by taking another independent diffusion term into consideration, though not converging to a fixed point, the algorithm may be able to admit better ergodic properties and obtain better performance near saddle points [26, 52]. Besides, the application of the methodology of random mini-batch to Langevin diffusion could result in some efficient methods that could reduce computational cost while preserving the dynamical and statistical properties. Examples include the SGLD algorithm we study in the paper and the random batch methods for interacting particle systems [22, 23].