Variational Inference in high-dimensional linear regression
This work addresses theoretical challenges in variational inference for statisticians and machine learning researchers, but it appears incremental as it builds on existing non-linear large deviations theory.
The paper tackles the problem of high-dimensional Bayesian linear regression by deriving conditions for the correctness of the naive mean-field approximation to the posterior's log-normalizing constant, and establishes a limiting variational formula with a unique optimizer under certain separation conditions.
We study high-dimensional Bayesian linear regression with product priors. Using the nascent theory of non-linear large deviations (Chatterjee and Dembo,2016), we derive sufficient conditions for the leading-order correctness of the naive mean-field approximation to the log-normalizing constant of the posterior distribution. Subsequently, assuming a true linear model for the observed data, we derive a limiting infinite dimensional variational formula for the log normalizing constant of the posterior. Furthermore, we establish that under an additional "separation" condition, the variational problem has a unique optimizer, and this optimizer governs the probabilistic properties of the posterior distribution. We provide intuitive sufficient conditions for the validity of this "separation" condition. Finally, we illustrate our results on concrete examples with specific design matrices.