改进粒子采样算法，实现更精准的分布逼近与可调参数设计
Finite-Particle Rates for Regularized Stein Variational Gradient Descent
- 引入预条件核修正偏差，提升粒子系统收敛精度
- 给出非渐近误差界，证明真实费舍尔信息收敛与W1距离收敛
- 提供正则化、步长、平均窗口的理论调参方法，适合算法优化者
我们推导了由He等（2024）提出的正则化斯坦因变分梯度下降（R-SVGD）算法的有限粒子速率。该算法通过施加类再生核预条件器，修正了经典SVGD中的常数阶偏差。对于由此产生的$N$-粒子相互作用系统，我们建立了时间平均（退火）经验测度的显式非渐近界，展示了在真实（非核化）费舍尔信息下的收敛性；在目标分布满足$ m{W}_1 m{I}$条件下，对一大类光滑核实现了对应的$ m{W}_1$收敛。分析涵盖连续与离散时间动态，并给出了正则化参数、步长和平均时长的合理调节规则，量化了逼近沃瑟斯坦梯度流与控制有限粒子估计误差之间的权衡。
原文摘要 · Abstract (English)
We derive finite-particle rates for the regularized Stein variational gradient descent (R-SVGD) algorithm introduced by He et al. (2024) that corrects the constant-order bias of the SVGD by applying a resolvent-type preconditioner to the kernelized Wasserstein gradient. For the resulting interacting $N$-particle system, we establish explicit non-asymptotic bounds for time-averaged (annealed) empirical measures, illustrating convergence in the \emph{true} (non-kernelized) Fisher information and, under a $\mathrm{W}_1\mathrm{I}$ condition on the target, corresponding $\mathrm{W}_1$ convergence for a large class of smooth kernels. Our analysis covers both continuous- and discrete-time dynamics and yields principled tuning rules for the regularization parameter, step size, and averaging horizon that quantify the trade-off between approximating the Wasserstein gradient flow and controlling finite-particle estimation error.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。