The PAC-Bayesian approach is a powerful set of techniques to derive non-
asymptotic risk bounds for random estimators. The corresponding optimal
distribution of estimators, usually called the Gibbs posterior, is
unfortunately intractable. One may sample from it using Markov chain Monte
Carlo, but this is often too slow for big datasets. We consider instead
variational approximations of the Gibbs posterior, which are fast to compute.
We undertake a general study of the properties of such approximations. Our main
finding is that such a variational approximation has often the same rate of
convergence as the original PAC-Bayesian procedure it approximates. We
specialise our results to several learning tasks (classification, ranking,
matrix completion),discuss how to implement a variational approximation in each
case, and illustrate the good properties of said approximation on real
datasets.

本文通过变分逼近 Gibbs posterior 的优化分布，从而实现和原始 PAC-Bayesian 程序同样的收敛速度，以替代通常过慢的 Markov chain Monte Carlo 方法，在多个学习任务中（分类、排名、矩阵完成）取得了良好结果。

关于 Gibbs 后验变分逼近的性质

On the properties of variational approximations of Gibbs posteriors

This paper aims at developing a quasi-Bayesian analysis of the nonparametric
instrumental variables model, with a focus on the asymptotic properties of
quasi-posterior distributions. In this paper, instead of assuming a
distributional assumption on the data generating process, we consider a
quasi-likelihood induced from the conditional moment restriction, and put
priors on the function-valued parameter. We call the resulting posterior
quasi-posterior, which corresponds to ``Gibbs posterior'' in the literature.
Here we focus on priors constructed on slowly growing finite-dimensional
sieves. We derive rates of contraction and a nonparametric Bernstein-von Mises
type result for the quasi-posterior distribution, and rates of convergence for
the quasi-Bayes estimator defined by the posterior expectation. We show that,
with priors suitably chosen, the quasi-posterior distribution (the quasi-Bayes
estimator) attains the minimax optimal rate of contraction (convergence,
resp.). These results greatly sharpen the previous related work.

本文着眼于以准 - Bayesian 分析非参数工具变量模型为主题，并重点关注准后验分布的渐近特性。