Variational approaches are often used to approximate intractable posteriors or normalization constants in hierarchical latent variable models. While often effective in practice, it is known that the approximation error can be arbitrarily large. We propose a new class of bounds on the marginal log-likelihood of directed latent variable models. Our approach relies on random projections to simplify the posterior. In contrast to standard variational methods, our bounds are guaranteed to be tight with high probability. We provide a new approach for learning latent variable models based on optimizing our new bounds on the log-likelihood. We demonstrate empirical improvements on benchmark datasets in vision and language for sigmoid belief networks, where a neural network is used to approximate the posterior.

This paper provides theoretical bounds that are tighter than existing variational bounds for the problem of learning latent variable models. The authors extend applied existing theory of hash-based learning and amortized inference to design a black-box learning algorithm. They later applied it to learning a Sigmoid Belief Network. The main advantage to this approach seems to be the partitioning of the search space for posterior distributions into buckets/subsets that are faster to search than with a typical sampling method. The proposed inference scheme then leverages mean-field inference (used heavily in the context of variational inference) within each subset. One of the main technical contributions is the tighter bound on the likelihood using two aggregate estimators which was an extension of an existing work (specific to undirected graphical models) to the directed models setting.

Variational approaches are often used to approximate intractable posteriors or normalization constants in hierarchical latent variable models. While often effective in practice, it is known that the approximation error can be arbitrarily large. We propose a new class of bounds on the marginal log-likelihood of directed latent variable models. Our approach relies on random projections to simplify the posterior. In contrast to standard variational methods, our bounds are guaranteed to be tight with high probability.