  We analyse the properties of an unbiased gradient estimator of the ELBO for
variational inference, based on the score function method with leave-one-out
control variates. We show that this gradient estimator can be obtained using a
new loss, defined as the variance of the log-ratio between the exact posterior
and the variational approximation, which we call the $\textit{log-variance
loss}$. Under certain conditions, the gradient of the log-variance loss equals
the gradient of the (negative) ELBO. We show theoretically that this gradient
estimator, which we call $\textit{VarGrad}$ due to its connection to the
log-variance loss, exhibits lower variance than the score function method in
certain settings, and that the leave-one-out control variate coefficients are
close to the optimal ones. We empirically demonstrate that VarGrad offers a
favourable variance versus computation trade-off compared to other
state-of-the-art estimators on a discrete VAE.
