We study the variational inference problem of minimizing a regularized R\'enyi divergence over an exponential family, and propose a relaxed moment-matching algorithm, which includes a proximal-like step. Using the information-geometric link between Bregman divergences and the Kullback-Leibler divergence, this algorithm is shown to be equivalent to a Bregman proximal gradient algorithm. This novel perspective allows us to exploit the geometry of our approximate model while using stochastic black-box updates. We use this point of view to prove strong convergence guarantees including monotonic decrease of the objective, convergence to a stationary point or to the minimizer, and geometric convergence rates. These new theoretical insights lead to a versatile, robust, and competitive method, as illustrated by numerical experiments.

翻译：我们研究在指数族上最小化正则化Rényi散度的变分推断问题，并提出一种包含近端步的松弛矩匹配算法。通过利用Bregman散度与Kullback-Leibler散度之间的信息几何联系，该算法被证明等价于Bregman近端梯度算法。这一新视角使我们能够在采用随机黑盒更新的同时，充分利用近似模型的几何结构。基于此观点，我们证明了强收敛性保证，包括目标函数的单调递减、收敛至驻点或最小值点，以及几何收敛速率。数值实验表明，这些新的理论洞见催生了一种通用、稳健且具有竞争力的方法。