Bayesian approaches for single-variable and group-structured sparsity outperform L1 regularization, but are challenging to apply to large, potentially intractable models. Here we show how noncentered parameterizations, a common trick for improving the efficiency of exact inference in hierarchical models, can similarly improve the accuracy of variational approximations. We develop this with two contributions: First, we introduce Fadeout, an approach for variational inference that uses noncentered parameterizations to capture a posteriori correlations between parameters and hyperparameters. Second, we extend stochastic variational inference to undirected models, enabling efficient hierarchical Bayes without approximations of intractable normalizing constants. We find that this framework substantially improves inferences of undirected graphical models under both sparse and group-sparse priors.