Mean field approximations via log-concavity

Abstract

We propose a new approach to deriving quantitative mean field approximations for any probability measure $P$ on ${\mathbb{R}}^n$ with density proportional to $e^{f(x)}$, for $f$ strongly concave. We bound the mean field approximation for the log partition function $\log \int e^{f(x)}dx$ in terms of ${\sum }_{i\ne j}{\mathbb{E}}_{Q^{\ast }}|{\partial }_{ij}f{|}^2$, for a semi-explicit probability measure $Q^{\ast }$ characterized as the unique mean field optimizer, or equivalently as the minimizer of the relative entropy $H(\cdot |P)$ over product measures. This notably does not involve metric-entropy or gradient-complexity concepts which are common in prior work on nonlinear large deviations. Three implications are discussed, in the contexts of continuous Gibbs measures on large graphs, high-dimensional Bayesian linear regression, and the construction of decentralized near-optimizers in high-dimensional stochastic control problems. Our arguments are based primarily on functional inequalities and the notion of displacement convexity from optimal transport.