  Variational inference is a very efficient and popular heuristic used in
various forms in the context of latent variable models. It's closely related to
Expectation Maximization (EM), and is applied when exact EM is computationally
infeasible. Despite being immensely popular, current theoretical understanding
of the effectiveness of variaitonal inference based algorithms is very limited.
In this work we provide the first analysis of instances where variational
inference algorithms converge to the global optimum, in the setting of topic
models.
  More specifically, we show that variational inference provably learns the
optimal parameters of a topic model under natural assumptions on the topic-word
matrix and the topic priors. The properties that the topic word matrix must
satisfy in our setting are related to the topic expansion assumption introduced
in (Anandkumar et al., 2013), as well as the anchor words assumption in (Arora
et al., 2012c). The assumptions on the topic priors are related to the well
known Dirichlet prior, introduced to the area of topic modeling by (Blei et
al., 2003).
  It is well known that initialization plays a crucial role in how well
variational based algorithms perform in practice. The initializations that we
use are fairly natural. One of them is similar to what is currently used in
LDA-c, the most popular implementation of variational inference for topic
models. The other one is an overlapping clustering algorithm, inspired by a
work by (Arora et al., 2014) on dictionary learning, which is very simple and
efficient.
  While our primary goal is to provide insights into when variational inference
might work in practice, the multiplicative, rather than the additive nature of
the variational inference updates forces us to use fairly non-standard proof
arguments, which we believe will be of general interest.
