  Quantization of the parameters of machine learning models, such as deep
neural networks, requires solving constrained optimization problems, where the
constraint set is formed by the Cartesian product of many simple discrete sets.
For such optimization problems, we study the performance of the Alternating
Direction Method of Multipliers for Quantization ($\texttt{ADMM-Q}$) algorithm,
which is a variant of the widely-used ADMM method applied to our discrete
optimization problem. We establish the convergence of the iterates of
$\texttt{ADMM-Q}$ to certain $\textit{stationary points}$. To the best of our
knowledge, this is the first analysis of an ADMM-type method for problems with
discrete variables/constraints. Based on our theoretical insights, we develop a
few variants of $\texttt{ADMM-Q}$ that can handle inexact update rules, and
have improved performance via the use of "soft projection" and "injecting
randomness to the algorithm". We empirically evaluate the efficacy of our
proposed approaches.
