In this work, we consider the distributed optimization of non-smooth convex
functions using a network of computing units. We investigate this problem under
two regularity assumptions: (1) the Lipschitz continuity of the global
objective function, and (2) the Lipschitz continuity of local individual
functions. Under the local regularity assumption, we provide the first optimal
first-order decentralized algorithm called multi-step primal-dual (MSPD) and
its corresponding optimal convergence rate. A notable aspect of this result is
that, for non-smooth functions, while the dominant term of the error is in
O(1/t), the structure of the communication network only impacts a
second-order term in O(1/t), where t is time. In other words, the error due
to limits in communication resources decreases at a fast rate even in the case
of non-strongly-convex objective functions. Under the global regularity
assumption, we provide a simple yet efficient algorithm called distributed
randomized smoothing (DRS) based on a local smoothing of the objective
function, and show that DRS is within a d1/4 multiplicative factor of the
optimal convergence rate, where d is the underlying dimension.Comment: 17 page