We consider stochastic approximations of sampling algorithms, such as Stochastic Gradient Langevin Dynamics (SGLD) and the Random Batch Method (RBM) for Interacting Particle Dynamcs (IPD). We observe that the noise introduced by the stochastic approximation is nearly Gaussian due to the Central Limit Theorem (CLT) while the driving Brownian motion is exactly Gaussian. We harness this structure to absorb the stochastic approximation error inside the diffusion process, and obtain improved convergence guarantees for these algorithms. For SGLD, we prove the first stable convergence rate in KL divergence without requiring uniform warm start, assuming the target density satisfies a Log-Sobolev Inequality. Our result implies superior first-order oracle complexity compared to prior works, under significantly milder assumptions. We also prove the first guarantees for SGLD under even weaker conditions such as H\"{o}lder smoothness and Poincare Inequality, thus bridging the gap between the state-of-the-art guarantees for LMC and SGLD. Our analysis motivates a new algorithm called covariance correction, which corrects for the additional noise introduced by the stochastic approximation by rescaling the strength of the diffusion. Finally, we apply our techniques to analyze RBM, and significantly improve upon the guarantees in prior works (such as removing exponential dependence on horizon), under minimal assumptions.

翻译：我们考虑采样算法的随机近似方法，如随机梯度朗之万动力学（SGLD）和相互作用粒子动力学（IPD）的随机批处理方法（RBM）。我们发现，由于中心极限定理（CLT），随机近似引入的噪声近似于高斯分布，而驱动布朗运动恰好是高斯分布。我们利用这一结构将随机近似误差吸收到扩散过程中，从而获得了这些算法更优的收敛保证。对于SGLD，在假设目标密度满足对数索博列夫不等式的情况下，我们首次证明了KL散度下的稳定收敛率，且无需均匀热启动。与先前工作相比，我们的结果在显著更弱的假设下实现了更优的一阶预言复杂度。我们还证明了SGLD在更弱条件（如赫尔德光滑性和庞加莱不等式）下的首个保证，从而弥合了LMC与SGLD当前最优保证之间的差距。我们的分析提出了一种称为协方差修正的新算法，通过重新调整扩散强度来修正随机近似引入的额外噪声。最后，我们将技术应用于分析RBM，在最小假设下显著改进了先前工作的保证（例如移除对时间跨度的指数依赖）。