Mean field approximations via log-concavity

TL;DR

This paper introduces a novel method for deriving quantitative mean field approximations for strongly log-concave probability measures, using functional inequalities and optimal transport, with applications in Gibbs measures, Bayesian regression, and stochastic control.

Contribution

It presents a new approach to mean field approximation bounds that avoids metric-entropy and gradient-complexity, using displacement convexity and functional inequalities.

Findings

Bound on mean field approximation for log partition function.

Characterization of the mean field optimizer as a unique measure.

Applications to Gibbs measures, Bayesian linear regression, and stochastic control.

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 $\sum_{i \neq j}\mathbb{E}_{Q^*}|\partial_{ij}f|^2$, for a semi-explicit probability measure $Q^*$ 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.