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The Na\"ive Mean Field (NMF) approximation is widely employed in modernMachine Learning due to the huge computational gains it bestows on thestatistician. Despite its popularity in practice, theoretical guarantees forhigh-dimensional problems are only available under strong structuralassumptions (e.g., sparsity). Moreover, existing theory often does not explainempirical observations noted in the existing literature.  In this paper, we take a step towards addressing these problems by derivingsharp asymptotic characterizations for the NMF approximation inhigh-dimensional linear regression. Our results apply to a wide class ofnatural priors and allow for model mismatch (i.e., the underlying statisticalmodel can be different from the fitted model). We work under an \textit{iid}Gaussian design and the proportional asymptotic regime, where the number offeatures and the number of observations grow at a proportional rate. As aconsequence of our asymptotic characterization, we establish two concretecorollaries: (a) we establish the inaccuracy of the NMF approximation for thelog-normalizing constant in this regime, and (b) we provide theoretical resultsbacking the empirical observation that the NMF approximation can beoverconfident in terms of uncertainty quantification.  Our results utilize recent advances in the theory of Gaussian comparisoninequalities. To the best of our knowledge, this is the first application ofthese ideas to the analysis of Bayesian variational inference problems. Ourtheoretical results are corroborated by numerical experiments. Lastly, webelieve our results can be generalized to non-Gaussian designs and provideempirical evidence to support it.

Sub-optimality of the Naive Mean Field approximation for proportional high-dimensional Linear Regression