In stochastic variational inference, use of the reparametrization trick for the multivariate Gaussian gives rise to efficient updates for the mean and Cholesky factor of the covariance matrix, which depend on the first order derivative of the log joint model density. In this article, we show that an alternative unbiased gradient estimate for the Cholesky factor which depends on the second order derivative of the log joint model density can be derived using Stein's Lemma. This leads to a second order stochastic gradient update for the Cholesky factor which is able to improve convergence, as it has variance lower than the first order update (almost negligible) when close to the mode. We also derive second order update for the Cholesky factor of the precision matrix, which is useful when the precision matrix has a sparse structure reflecting conditional independence in the true posterior distribution. Our results can be used to obtain second order natural gradient updates for the Cholesky factor as well, which are more robust compared to updates based on Euclidean gradients.
翻译:在随机的变异性推断中,对多变 Gaussian 使用重新校正技术使共变矩阵的平均值和Cholesky 系数得到有效的更新,这取决于日志联合模型密度的第一阶衍生物。在本篇文章中,我们显示,对Cholesky 系数的替代不偏差梯度估计值取决于日志联合模型密度的第二阶衍生物,可以使用Stein's Lemma 得出。这导致对Cholesky 系数的二次测相梯度更新,该系数能够改进趋同性,因为与模式接近时的第一次测序更新值(几乎可忽略不计)相比,差异较低。我们还为精密矩阵的Choolesky 系数进行第二次测序更新,这在精确矩阵结构细小,反映真实后座分布的有条件独立性时非常有用。我们的成果可用于获取Cholesky 系数的第二阶次自然梯度更新,与以Euclidean 梯度为基础的更新值相比,这种更新值更为稳健。