We extend several recent results providing symmetry-based guarantees for variational inference (VI) with location-scale families. VI approximates a target density~$p$ by the best match $q^*$ in a family $Q$ of tractable distributions that in general does not contain $p$. It is known that VI can recover key properties of $p$, such as its mean and correlation matrix, when $p$ and $Q$ exhibit certain symmetries and $q^*$ is found by minimizing the reverse Kullback-Leibler divergence. We extend these guarantees in two important directions. First, we provide symmetry-based guarantees for a broader family of divergences, highlighting the properties of variational objectives under which VI provably recovers the mean and correlation matrix. Second, we obtain further guarantees for VI when the target density $p$ exhibits even and elliptical symmetries in some but not all of its coordinates. These partial symmetries arise naturally in Bayesian hierarchical models, where the prior induces a challenging geometry but still possesses axes of symmetry. We illustrate these theoretical results in a number of experimental settings.

翻译：我们扩展了若干近期关于基于对称性为变分推断（VI）提供保证的研究结果，这些结果主要针对位置-尺度分布族。VI通过在一族易处理的分布Q中寻找最佳匹配q*来近似目标密度p，而Q通常不包含p。已知当p和Q表现出特定对称性且q*通过最小化反向Kullback-Leibler散度得到时，VI能够恢复p的关键性质，如其均值与相关矩阵。我们在两个重要方向上扩展了这些保证。首先，我们为更广泛的散度族提供了基于对称性的保证，阐明了变分目标在何种条件下可证明地恢复均值与相关矩阵。其次，当目标密度p在其部分（而非全部）坐标上呈现偶对称与椭圆对称时，我们为VI获得了进一步的保证。这种部分对称性在贝叶斯层次模型中自然出现，其中先验分布引入了复杂的几何结构但仍保持对称轴。我们在多个实验场景中验证了这些理论结果。