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Generalized Linear Models (GLMs), where a random vector $\mathbf{x}$ isobserved through a noisy, possibly nonlinear, function of a linear transform$\mathbf{z}=\mathbf{Ax}$ arise in a range of applications in nonlinearfiltering and regression. Approximate Message Passing (AMP) methods, based onloopy belief propagation, are a promising class of approaches for approximateinference in these models. AMP methods are computationally simple, general, andadmit precise analyses with testable conditions for optimality for large i.i.d.transforms $\mathbf{A}$. However, the algorithms can easily diverge for general$\mathbf{A}$. This paper presents a convergent approach to the generalized AMP(GAMP) algorithm based on direct minimization of a large-system limitapproximation of the Bethe Free Energy (LSL-BFE). The proposed method uses adouble-loop procedure, where the outer loop successively linearizes the LSL-BFEand the inner loop minimizes the linearized LSL-BFE using the AlternatingDirection Method of Multipliers (ADMM). The proposed method, called ADMM-GAMP,is similar in structure to the original GAMP method, but with an additionalleast-squares minimization. It is shown that for strictly convex, smoothpenalties, ADMM-GAMP is guaranteed to converge to a local minima of theLSL-BFE, thus providing a convergent alternative to GAMP that is stable underarbitrary transforms. Simulations are also presented that demonstrate therobustness of the method for non-convex penalties as well.

Inference for Generalized Linear Models via Alternating Directions and Bethe Free Energy Minimization