  Parameter estimation in Markov random fields (MRFs) is a difficult task, in
which inference over the network is run in the inner loop of a gradient descent
procedure. Replacing exact inference with approximate methods such as loopy
belief propagation (LBP) can suffer from poor convergence. In this paper, we
provide a different approach for combining MRF learning and Bethe
approximation. We consider the dual of maximum likelihood Markov network
learning - maximizing entropy with moment matching constraints - and then
approximate both the objective and the constraints in the resulting
optimization problem. Unlike previous work along these lines (Teh & Welling,
2003), our formulation allows parameter sharing between features in a general
log-linear model, parameter regularization and conditional training. We show
that piecewise training (Sutton & McCallum, 2005) is a very restricted special
case of this formulation. We study two optimization strategies: one based on a
single convex approximation and one that uses repeated convex approximations.
We show results on several real-world networks that demonstrate that these
algorithms can significantly outperform learning with loopy and piecewise. Our
results also provide a framework for analyzing the trade-offs of different
relaxations of the entropy objective and of the constraints.
