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In stochastic variational inference, use of the reparametrization trick forthe multivariate Gaussian gives rise to efficient updates for the mean andCholesky factor of the covariance matrix, which depend on the first orderderivative of the log joint model density. In this article, we show that analternative unbiased gradient estimate for the Cholesky factor which depends onthe second order derivative of the log joint model density can be derived usingStein's Lemma. This leads to a second order stochastic gradient update for theCholesky factor which is able to improve convergence, as it has variance lowerthan the first order update (almost negligible) when close to the mode. We alsoderive second order update for the Cholesky factor of the precision matrix,which is useful when the precision matrix has a sparse structure reflectingconditional independence in the true posterior distribution. Our results can beused to obtain second order natural gradient updates for the Cholesky factor aswell, which are more robust compared to updates based on Euclidean gradients.

Second order stochastic gradient update for Cholesky factor in Gaussian variational approximation from Stein's Lemma