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This paper extends theoretical guarantees for variational inference (VI) when approximating symmetric target densities with misspecified location-scale variational families. It derives sufficient conditions for exact mean recovery using forward KL divergence and $\alpha$-divergences, even when the variational family doesn't contain the true target. The analysis also identifies failure modes in the absence of these conditions, offering guidance for selecting variational families and $\alpha$-values.
Even when your variational approximation is wrong, symmetries in the target distribution can guarantee you still get the mean right.
When approximating an intractable density via variational inference (VI) the variational family is typically chosen as a simple parametric family that very likely does not contain the target. This raises the question: Under which conditions can we recover characteristics of the target despite misspecification? In this work, we extend previous results on robust VI with location-scale families under target symmetries. We derive sufficient conditions guaranteeing exact recovery of the mean when using the forward Kullback-Leibler divergence and $\alpha$-divergences. We further show how and why optimization can fail to recover the target mean in the absence of our sufficient conditions, providing initial guidelines on the choice of the variational family and $\alpha$-value.
