The bigger the corpus, the more topics it can potentially support. To truly make full use of massive text corpora, a topic model inference algorithm must therefore scale efficiently in 1) documents and 2) topics, while 3) achieving accurate inference. Previous methods have achieved two out of three of these criteria simultaneously, but never all three at once. In this paper, we develop an online inference algorithm for topic models which leverages stochasticity to scale well in the number of documents, sparsity to scale well in the number of topics, and which operates in the collapsed representation of the topic model for improved accuracy and run-time performance. We use a Monte Carlo inner loop in the online setting to approximate the collapsed variational Bayes updates in a sparse and efficient way, which we accomplish via the MetropolisHastings Walker method. We showcase our algorithm on LDA and the recently proposed mixed membership skip-gram topic model. Our method requires only amortized O(k_d) computation per word token instead of O(K) operations, where the number of topics occurring for a particular document k_d≪ the total number of topics in the corpus K, to converge to a high-quality solution.