The Coordinate Ascent Variational Inference scheme is a popular algorithm used to compute the mean-field approximation of a probability distribution of interest. We analyze its random scan version, under log-concavity assumptions on the target density. Our approach builds on the recent work of M. Arnese and D. Lacker, \emph{Convergence of coordinate ascent variational inference for log-concave measures via optimal transport} [arXiv:2404.08792] which studies the deterministic scan version of the algorithm, phrasing it as a block-coordinate descent algorithm in the space of probability distributions endowed with the geometry of optimal transport. We obtain tight rates for the random scan version, which imply that the total number of factor updates required to converge scales linearly with the condition number and the number of blocks of the target distribution. By contrast, available bounds for the deterministic scan case scale quadratically in the same quantities, which is analogue to what happens for optimization of convex functions in Euclidean spaces.

翻译：坐标上升变分推断方案是一种用于计算目标概率分布均值场近似的流行算法。我们在目标密度满足对数凹性假设下，分析其随机扫描版本。我们的方法建立在 M. Arnese 和 D. Lacker 近期工作《通过最优传输研究对数凹测度下坐标上升变分推断的收敛性》[arXiv:2404.08792] 的基础上，该工作研究了算法的确定性扫描版本，将其表述为在赋予最优传输几何结构的概率分布空间中的块坐标下降算法。我们获得了随机扫描版本的紧致收敛速率，该结果表明收敛所需的总因子更新次数与目标分布的条件数及块数呈线性比例关系。相比之下，确定性扫描情形的现有收敛界在相同量上呈二次比例关系，这与欧几里得空间中凸函数优化的情况类似。