  In recent years, decentralized learning has emerged as a powerful tool not
only for large-scale machine learning, but also for preserving privacy. One of
the key challenges in decentralized learning is that the data distribution held
by each node is statistically heterogeneous. To address this challenge, the
primal-dual algorithm called the Edge-Consensus Learning (ECL) was proposed and
was experimentally shown to be robust to the heterogeneity of data
distributions. However, the convergence rate of the ECL is provided only when
the objective function is convex, and has not been shown in a standard machine
learning setting where the objective function is non-convex. Furthermore, the
intuitive reason why the ECL is robust to the heterogeneity of data
distributions has not been investigated. In this work, we first investigate the
relationship between the ECL and Gossip algorithm and show that the update
formulas of the ECL can be regarded as correcting the local stochastic gradient
in the Gossip algorithm. Then, we propose the Generalized ECL (G-ECL), which
contains the ECL as a special case, and provide the convergence rates of the
G-ECL in both (strongly) convex and non-convex settings, which do not depend on
the heterogeneity of data distributions. Through synthetic experiments, we
demonstrate that the numerical results of both the G-ECL and ECL coincide with
the convergence rate of the G-ECL.
