论文标题
通过log-concavity的平均字段近似
Mean field approximations via log-concavity
论文作者
论文摘要
我们提出了一种新的方法，用于在$ \ mathbb {r}^n $上获得任何概率度量$ p $的定量平均近似值，其密度与$ e^{f（x）} $成正比，$ f $ for $ f $ for $ f $ contly lassly凹入。 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作为相对熵$ h（\ cdot \，| \，p）$的最小化器，产品量。显着的不涉及公制或梯度复杂性概念，这些概念在非线性大偏差的先前工作中很常见。在大图，高维贝叶斯线性回归以及在高维随机控制问题中的分散式近距离临近器的构建的情况下，讨论了三个含义。我们的论点主要基于功能不平等和最佳运输中位移凸的概念。
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.
