Variational families with full-rank covariance approximations are known not to work well in black-box variational inference (BBVI), both empirically and theoretically. In fact, recent computational complexity results for BBVI have established that full-rank variational families scale poorly with the dimensionality of the problem compared to e.g. mean-field families. This is particularly critical to hierarchical Bayesian models with local variables; their dimensionality increases with the size of the datasets. Consequently, one gets an iteration complexity with an explicit $\mathcal{O}(N^2)$ dependence on the dataset size $N$. In this paper, we explore a theoretical middle ground between mean-field variational families and full-rank families: structured variational families. We rigorously prove that certain scale matrix structures can achieve a better iteration complexity of $\mathcal{O}\left(N\right)$, implying better scaling with respect to $N$. We empirically verify our theoretical results on large-scale hierarchical models.

翻译：已知具有满秩协方差近似的变分族在黑盒变分推断（BBVI）中效果不佳，这已得到经验与理论的双重验证。事实上，BBVI的最新计算复杂性结果表明，与均值场族等方法相比，满秩变分族随问题维度的增加而扩展性显著下降。这对于具有局部变量的分层贝叶斯模型尤为关键：其维度随数据集规模增大而增加。因此，迭代复杂度会显式地呈现与数据集规模N的$\mathcal{O}(N^2)$依赖关系。本文在均值场变分族与满秩变分族之间探索了一种理论上的中间路径：结构化变分族。我们严格证明了特定尺度矩阵结构能够实现$\mathcal{O}\left(N\right)$的迭代复杂度，这意味着其关于N的扩展性更优。我们在大规模分层模型上实证验证了理论结果。