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# Bethe Learning of Graphical Models via MAP Decoding

*Proceedings of the 19th International Conference on Artificial Intelligence and Statistics*, PMLR 51:1096-1104, 2016.

#### Abstract

Many machine learning tasks require fitting probabilistic models over structured objects, such as pixel grids, matchings, and graph edges. Maximum likelihood estimation (MLE) for such domains is challenging due to the intractability of computing partition functions. One can resort to approximate marginal inference in conjunction with gradient descent, but such algorithms require careful tuning. Alternatively, in frameworks such as the structured support vector machine (SVM-Struct), discriminative functions are learned by iteratively applying efficient maximum a posteriori (MAP) decoders. We introduce MLE-Struct, a method for learning discrete exponential family models using the Bethe approximation to the partition function. Remarkably, this problem can also be reduced to iterative (MAP) decoding. This connection emerges by combining the Bethe approximation with the Frank-Wolfe (FW) algorithm on a convex dual objective, which circumvents the intractable partition function. Our method can learn both generative and conditional models and is substantially faster and easier to implement than existing MLE approaches while still relying on the same black-box interface to MAP decoding as SVM-Struct. We perform competitively on problems in denoising, segmentation, matching, and new datasets of roommate assignments and news and financial time series.