Variational inference (VI) is a central tool in modern machine learning, used to approximate an intractable target density by optimising over a tractable family of distributions. As the variational family cannot typically represent the target exactly, guarantees on the quality of the resulting approximation are crucial for understanding which of its properties VI can faithfully capture. Recent work has identified instances in which symmetries of the target and the variational family enable the recovery of certain statistics, even under model misspecification. However, these guarantees are inherently problem-specific and offer little insight into the fundamental mechanism by which symmetry forces statistic recovery. In this paper, we overcome this limitation by developing a general theory of symmetry-induced statistic recovery in variational inference. First, we characterise when variational minimisers inherit the symmetries of the target and establish conditions under which these pin down identifiable statistics. Second, we unify existing results by showing that previously known statistic recovery guarantees in location-scale families arise as special cases of our theory. Third, we apply our framework to distributions on the sphere to obtain novel guarantees for directional statistics in von Mises-Fisher families. Together, these results provide a modular blueprint for deriving new recovery guarantees for VI in a broad range of symmetry settings.

翻译：变分推断是现代机器学习中的核心工具，用于通过在可处理的分布族上优化来近似难以处理的目标密度。由于变分族通常无法精确表示目标，因此对所得近似质量的保证至关重要，这有助于理解变分推断能忠实捕获目标的哪些性质。近期研究已发现，在某些实例中，目标和变分族的对称性能够在模型误设情况下恢复特定统计量。然而，这些保证本质上依赖于具体问题，对对称性迫使统计量恢复的基本机制缺乏深入见解。本文通过发展变分推断中对称性诱导统计量恢复的一般理论克服了这一局限。首先，我们刻画了变分极小化过程何时继承目标对称性，并建立了这些对称性确定可识别统计量的条件。其次，我们统一了现有结果，证明先前在位置-尺度族中已知的统计量恢复保证是我们理论的特例。第三，我们将框架应用于球面分布，在von Mises-Fisher族中获得了方向统计学的新保证。综合这些结果，我们为在广泛对称性设置中推导变分推断的新恢复保证提供了模块化蓝图。