## AD3: Alternating Directions Dual Decomposition for MAP Inference in Graphical Models

André F. T. Martins, Mário A. T. Figueiredo, Pedro M. Q. Aguiar, Noah A. Smith, Eric P. Xing; 16(16):495−545, 2015.

### Abstract

We present AD$^3$, a new algorithm for approximate maximum a posteriori (MAP) inference on factor graphs, based on the alternating directions method of multipliers. Like other dual decomposition algorithms, AD$^3$ has a modular architecture, where local subproblems are solved independently, and their solutions are gathered to compute a global update. The key characteristic of AD$^3$ is that each local subproblem has a quadratic regularizer, leading to faster convergence, both theoretically and in practice. We provide closed-form solutions for these AD$^3$ subproblems for binary pairwise factors and factors imposing first-order logic constraints. For arbitrary factors (large or combinatorial), we introduce an active set method which requires only an oracle for computing a local MAP configuration, making AD$^3$ applicable to a wide range of problems. Experiments on synthetic and real-world problems show that AD$^3$ compares favorably with the state-of-the-art.

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