We derive first-order (in the stepsize) bounds on the bias in Wasserstein distances of the invariant measure of stochastic gradient kinetic Langevin dynamics with minimal assumptions on the stochastic gradient noise. These bounds sharpen existing non-asymptotic guarantees for stochastic-gradient MCMC methods and provide a quantitative resolution of a previously open problem on invariant measure accuracy. The main technical ingredients are new Gaussian convolution inequalities controlling the Wasserstein-$p$ distance between a Gaussian convolved with a mean-zero perturbation and the Gaussian itself. We anticipate that these inequalities will be of independent interest beyond the present application.

翻译：我们推导了在步长参数下，随机梯度动能朗之万动力学不变测度在Wasserstein距离中偏差的一阶（关于步长）界。该结果在随机梯度噪声的最小假设下成立。这些界优化了现有随机梯度MCMC方法的非渐近保证，并为先前关于不变测度精度的开放问题提供了定量解答。主要技术工具是新的高斯卷积不等式，这些不等式控制了高斯分布与带零均值扰动的卷积结果之间的Wasserstein-p距离。我们预期这些不等式将在当前应用之外具有独立的研究价值。