  Motivated by the need for decentralized learning, this paper aims at
designing a distributed algorithm for solving nonconvex problems with general
linear constraints over a multi-agent network. In the considered problem, each
agent owns some local information and a local variable for jointly minimizing a
cost function, but local variables are coupled by linear constraints. Most of
the existing methods for such problems are only applicable for convex problems
or problems with specific linear constraints. There still lacks a distributed
algorithm for such problems with general linear constraints and under nonconvex
setting. In this paper, to tackle this problem, we propose a new algorithm,
called "proximal dual consensus" (PDC) algorithm, which combines a proximal
technique and a dual consensus method. We build the theoretical convergence
conditions and show that the proposed PDC algorithm can converge to an
$\epsilon$-Karush-Kuhn-Tucker solution within $\mathcal{O}(1/\epsilon)$
iterations. For computation reduction, the PDC algorithm can choose to perform
cheap gradient descent per iteration while preserving the same order of
$\mathcal{O}(1/\epsilon)$ iteration complexity. Numerical results are presented
to demonstrate the good performance of the proposed algorithms for solving a
regression problem and a classification problem over a network where agents
have only partial observations of data features.
