Also this beautiful paper by J. Domke proves that you can replace Gaussian approximations by almost any location-scale family https://arxiv.org/abs/1901.08431
reference request: are there nontrivial examples of problems in variational inference where one can provably solve the KL minimisation problem? i.e. find global minimiser in polynomial time. the only one i'm currently aware of is fitting a Gaussian to something log-concave.
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Thanks Pierre - exactly the kind of thing I was looking for. Much appreciated.
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Cool question! What’s the ref for the log-concave case?
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It's mentioned in http://proceedings.mlr.press/v32/titsias14.html … (Doubly Stochastic Variational Bayes for non-Conjugate Inference, Michalis Titsias, Miguel Lázaro-Gredilla), but I think they say it's been noted earlier.
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Our paper on VI to appear in AoS and the previous one in JMLR have both such examples http://www.jmlr.org/papers/v17/15-290.html … https://arxiv.org/abs/1706.09293
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When viewed as a nonconvex optimization problem, you can sometimes say something, although it got a bit complicated :-) https://arxiv.org/abs/1910.02008
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I'd wager you may find something along those lines in the literature that connects VI to message-passing algorithms, but I'm unaware of any specific references
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In discrete case, we saw that KL from factorial Bernoulli to a Boltzman machine has a closed form (up to the normalization constant). This is heavily used for mean field approximation. When BM is fixed, mean field updates converge quickly to a local optimum.
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