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，例如随机梯度朗之万动力学（SGLD）以及用于相互作用粒子动力学（IPD）的随机批方法（RBM）。我们观察到，由于中心极限定理（CLT），随机近似引入的噪声近似服从高斯分布，而驱动布朗运动恰好是高斯分布的。我们利用这一结构将随机近似误差吸收到扩散过程中，并为这些算法获得了改进的收敛保证。对于SGLD，我们证明了在KL散度下的首个稳定收敛率，且无需均匀热启动，前提是目标密度满足对数索博列夫不等式。我们的结果表明，在显著更宽松的假设下，其一阶预言复杂度优于先前工作。我们还证明了在更弱条件下（如赫尔德光滑性和庞加莱不等式）SGLD的首个保证，从而弥合了LMC与SGLD之间现有最优保证的差距。我们的分析提出了一种称为协方差校正的新算法，通过重新缩放扩散强度来校正随机近似引入的额外噪声。最后，我们将我们的技术应用于分析RBM，并在最小假设下显著改进了先前工作的保证（例如去除了对时间范围的指数依赖性）。