With a growing need to understand large-scale networks, factorial relational models, such as binary matrix factorization models (BMFs), have become important in many applications. Although BMFs have a natural capability to uncover overlapping group structures behind network data, existing inference techniques have issues of either high computational cost or lack of model selection capability, and this limits their applicability. For scalable model selection of BMFs, this paper proposes stochastic factorized asymptotic Bayesian (sFAB) inference that combines concepts in two recently-developed techniques: stochastic variational inference (SVI) and FAB inference. sFAB is a highly-efficient algorithm, having both scalability and an inherent model selection capability in a single inference framework. Empirical results show the superiority of sFAB/BMF in both accuracy and scalability over state-of-the-art inference methods for overlapping relational models.