This technical note presents proof of the convergence of the Alternating Direction Method of Multipliers (ADMM) for addressing the sharing problem when applied in conjunction with two algorithms: 1) the stochastic STEP-GP algorithm and 2) its variant named LGP, which includes adaptive uniform quantization. For the case using LGP, the coordinator can assign different quantization resolutions at each iteration and we assume that the number of bits that can be assigned is unrestricted and can go to infinity. This document describes and analyzes the two methods for integrating learning and uniform quantization into the ADMM to reduce its communication overhead and a general formulation of their communication decision method. The problems are formulated for a multi-agent setting.