Mean Field Approximations via Log-Concavity
Mean Field Approximations via Log-Concavity
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DOI：
10.1093/imrn/rnad302
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发表时间：
2022-06
影响因子：
1
通讯作者：
D. Lacker;S. Mukherjee;Lane Chun Yeung
中科院分区：
文献类型：
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作者：
D. Lacker;S. Mukherjee;Lane Chun Yeung
我们提出了一种新的方法来推导任意概率度量$P$的定量平均场近似，其中密度与$e^{f(X)}$成正比，且$f$是强凹的。对于刻画唯一平均场优化器的半显式概率度量$q^*$，我们用$\sum_{i\neq j}\mathbb{E}_{q^{*}}|\Partial_{ij}f|^{2}$给出了对数分拆函数$\log\int e^{f(X)}dx$的平均场近似，或者等价地定义为乘积测度上相对熵$H(\CDOT\，|\P)$的最小化.值得注意的是，这不涉及度量-熵或梯度复杂性概念，这些概念在以前关于非线性大偏差的工作中很常见。在大图上的连续Gibbs测度、高维Bayesian线性回归和高维随机控制问题中分散近优化器的构造的背景下，讨论了三个方面的含义。我们的论点主要基于泛函不等式和最优传输的位移凸性的概念。
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.