2 research outputs found
A General Analysis of the Convergence of ADMM
We provide a new proof of the linear convergence of the alternating direction
method of multipliers (ADMM) when one of the objective terms is strongly
convex. Our proof is based on a framework for analyzing optimization algorithms
introduced in Lessard et al. (2014), reducing algorithm convergence to
verifying the stability of a dynamical system. This approach generalizes a
number of existing results and obviates any assumptions about specific choices
of algorithm parameters. On a numerical example, we demonstrate that minimizing
the derived bound on the convergence rate provides a practical approach to
selecting algorithm parameters for particular ADMM instances. We complement our
upper bound by constructing a nearly-matching lower bound on the worst-case
rate of convergence.Comment: 10 pages, 6 figure
Linear Approximation to ADMM for MAP inference
Maximum a posteriori (MAP) inference is one of the fundamental inference tasks in graphical models. MAP inference is in general NP-hard, making approximate methods of interest for many problems. One successful class of approximate inference algorithms is based on linear programming (LP) relaxations. The augmented Lagrangian method can be used to overcome a lack of strict convexity in LP relaxations, and the Alternating Direction Method of Multipliers (ADMM) provides an elegant algorithm for finding the saddle point of the augmented Lagrangian. Here we present an ADMM-based algorithm to solve the primal form of the MAP-LP whose closed form updates are based on a linear approximation technique. Our technique gives efficient, closed form updates that converge to the global optimum of the LP relaxation. We compare our algorithm to two existing ADMM-based MAP-LP methods, showing that our technique is faster on general, non-binary or non-pairwise models. 1