Two popular approaches to forming principled bounds in approximate Bayesian inference are local variational methods and minimal Kullback-Leibler divergence methods. For a large class of models, we explicitly relate the two approaches, showing that the local variational method is equivalent to a weakened form of Kullback-Leibler Gaussian approximation. This gives a strong motivation to develop efficient methods for KL minimisation. An important and previously unproven property of the KL variational Gaussian bound is that it is a concave function in the parameters of the Gaussian for log concave sites. This observation, along with compact concave parameterisations of the covariance, enables us to develop fast scalable optimisation procedures to obtain lower bounds on the marginal likelihood in large scale Bayesian linear models.