  We consider the "all-for-one" decentralized learning problem for generalized
linear models. The features of each sample are partitioned among several
collaborating agents in a connected network, but only one agent observes the
response variables. To solve the regularized empirical risk minimization in
this distributed setting, we apply the Chambolle--Pock primal--dual algorithm
to an equivalent saddle-point formulation of the problem. The primal and dual
iterations are either in closed-form or reduce to coordinate-wise minimization
of scalar convex functions. We establish convergence rates for the empirical
risk minimization under two different assumptions on the loss function
(Lipschitz and square root Lipschitz), and show how they depend on the
characteristics of the design matrix and the Laplacian of the network.
