We prove rates of convergence and robustness to prior misspecification within a Generalised Variational Inference (GVI) framework with bounded divergences. This addresses a significant open challenge for GVI and Federated GVI that employ a different divergence to the Kullback-Leibler under prior misspecification, operate within a subset of possible probability measures, and result in intractable posteriors. Our theoretical contributions extend to misspecified priors that lead to inconsistent Bayes posteriors. In particular, we are able to establish sufficient conditions for existence and uniqueness of GVI posteriors on arbitrary Polish spaces, prove that the GVI posterior measure concentrates on a neighbourhood of loss minimisers, and extend this to rates of convergence regardless of the prior measure.

翻译：我们在采用有界散度的广义变分推断（GVI）框架内，证明了先验误设条件下的收敛速率与稳健性。这解决了GVI及联邦GVI领域一个重要的开放性挑战：这些方法在先验误设时使用不同于Kullback-Leibler散度的度量，在可能概率测度的子集上操作，并导致后验分布难以处理。我们的理论贡献进一步扩展到会导致贝叶斯后验不一致的误设先验。具体而言，我们能够在任意波兰空间上建立GVI后验存在性与唯一性的充分条件，证明GVI后验测度集中于损失最小化点的邻域，并将此结论推广到与先验测度无关的收敛速率分析。