Stability of MFVI for non-log-concave targets

Determine whether mean-field variational inference (MFVI), defined as minimizing KL(μ || π) over product measures μ ∈ P(R)^⊗d for a target distribution π on R^d, is stable when π is non-log-concave.

Background

Mean-field variational inference (MFVI) approximates a target distribution π by optimizing the Kullback–Leibler divergence over the class of product measures. Stability guarantees (e.g., existence, uniqueness, and continuous dependence on the target) have been established for log-concave targets, and quantitative stability results are known in that regime.

In contrast, for non-log-concave targets, including mixtures and other multimodal distributions, empirical evidence shows potential instability such as mode collapse. The paper highlights that a rigorous understanding of MFVI stability beyond log-concavity is missing, motivating this explicit open question.