In this work we develop efficient methods for learning random MAP predictors for structured label problems. In particular, we construct posterior distributions over perturbations that can be adjusted via stochastic gradient methods. We show that every smooth posterior distribution would suffice to define a smooth PAC-Bayesian risk bound suitable for gradient methods. In addition, we relate the posterior distributions to computational properties of the MAP predictors. We suggest multiplicative posteriors to learn super-modular potential functions that accompany specialized MAP predictors such as graph-cuts. We also describe label-augmented posterior models that can use efficient MAP approximations, such as those arising from linear program relaxations.