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Variational inference is a popular alternative to Markov chain Monte Carlo methodsthat constructs a Bayesian posterior approximation by minimizing a discrepancy tothe true posterior within a pre-specified family. This converts Bayesian inferenceinto an optimization problem, enabling the use of simple and scalable stochas-tic optimization algorithms. However, a key limitation of variational inferenceis that the optimal approximation is typically not tractable to compute; even insimple settings the problem is nonconvex. Thus, recently developed statisticalguarantees—which all involve the (data) asymptotic properties of the optimal varia-tional distribution—are not reliably obtained in practice. In this work, we providetwo major contributions: a theoretical analysis of the asymptotic convexity prop-erties of variational inference in the popular setting with a Gaussian family; andconsistent stochastic variational inference (CSVI), an algorithm that exploits theseproperties to find the optimal approximation in the asymptotic regime. CSVI con-sists of a tractable initialization procedure that finds the local basin of the optimalsolution, and a scaled gradient descent algorithm that stays locally confined to thatbasin. Experiments on nonconvex synthetic examples show that compared withstandard stochastic gradient descent, CSVI improves the likelihood of obtaining theglobally optimal posterior approximation.