论文标题
在较弱的平滑度条件下，Stein变异梯度下降的收敛性
Convergence of Stein Variational Gradient Descent under a Weaker Smoothness Condition
论文作者
论文摘要
Stein变异梯度下降（SVGD）是langevin型算法的重要替代方法，用于从$π（x）\ propto \ propto \ exp（-v（x）$的概率分布中采样。在现有的langevin型算法和SVGD的理论中，潜在函数$ v $通常被认为是$ l $ -smooth。但是，这种限制性条件不包括大量潜在功能，例如多项式的学位大于$ 2 $。我们的论文研究了SVGD算法的收敛性，用于具有$（l_0，l_1）$ - 平滑电位的分布。张等人引入了这种松弛的平滑度假设。 [2019a]用于分析梯度剪辑算法。借助轨迹无关的辅助条件，我们提供了下降引理，确保该算法在每次迭代时都会降低$ \ mathrm {kl} $ divergence，并证明在Stein Fisher信息方面，SVGD在人群中限制了SVGD的复杂性。
Stein Variational Gradient Descent (SVGD) is an important alternative to the Langevin-type algorithms for sampling from probability distributions of the form $π(x) \propto \exp(-V(x))$. In the existing theory of Langevin-type algorithms and SVGD, the potential function $V$ is often assumed to be $L$-smooth. However, this restrictive condition excludes a large class of potential functions such as polynomials of degree greater than $2$. Our paper studies the convergence of the SVGD algorithm for distributions with $(L_0,L_1)$-smooth potentials. This relaxed smoothness assumption was introduced by Zhang et al. [2019a] for the analysis of gradient clipping algorithms. With the help of trajectory-independent auxiliary conditions, we provide a descent lemma establishing that the algorithm decreases the $\mathrm{KL}$ divergence at each iteration and prove a complexity bound for SVGD in the population limit in terms of the Stein Fisher information.
