  Generalized Linear Models (GLMs), where a random vector $\mathbf{x}$ is
observed through a noisy, possibly nonlinear, function of a linear transform
$\mathbf{z}=\mathbf{Ax}$ arise in a range of applications in nonlinear
filtering and regression. Approximate Message Passing (AMP) methods, based on
loopy belief propagation, are a promising class of approaches for approximate
inference in these models. AMP methods are computationally simple, general, and
admit 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 limit
approximation of the Bethe Free Energy (LSL-BFE). The proposed method uses a
double-loop procedure, where the outer loop successively linearizes the LSL-BFE
and the inner loop minimizes the linearized LSL-BFE using the Alternating
Direction Method of Multipliers (ADMM). The proposed method, called ADMM-GAMP,
is similar in structure to the original GAMP method, but with an additional
least-squares minimization. It is shown that for strictly convex, smooth
penalties, ADMM-GAMP is guaranteed to converge to a local minima of the
LSL-BFE, thus providing a convergent alternative to GAMP that is stable under
arbitrary transforms. Simulations are also presented that demonstrate the
robustness of the method for non-convex penalties as well.
