Zero-Gradient-Sum Algorithms for Distributed Convex Optimization: The Continuous-Time Case

This paper presents a set of continuous-time distributed algorithms that solve unconstrained, separable, convex optimization problems over undirected networks with fixed topologies. The algorithms are developed using a Lyapunov function candidate that exploits convexity, and are called Zero-Gradient-Sum (ZGS) algorithms as they yield nonlinear networked dynamical systems that evolve invariantly on a zero-gradient-sum manifold and converge asymptotically to the unknown optimizer. We also describe a systematic way to construct ZGS algorithms, show that a subset of them actually converge exponentially, and obtain lower and upper bounds on their convergence rates in terms of the network topologies, problem characteristics, and algorithm parameters, including the algebraic connectivity, Laplacian spectral radius, and function curvatures. The findings of this paper may be regarded as a natural generalization of several well-known algorithms and results for distributed consensus, to distributed convex optimization.

💡 Research Summary

The paper addresses the problem of solving an unconstrained, separable convex optimization problem in a distributed manner over a fixed, undirected communication network. Each node i holds a private convex function f_i : ℝ^d → ℝ that is μ‑strongly convex and has an L‑Lipschitz continuous gradient. The global objective is F(x)=∑_{i=1}^N f_i(x) and the goal is to find the unique minimizer x^* without a central coordinator.

The authors introduce a new class of continuous‑time algorithms called Zero‑Gradient‑Sum (ZGS) algorithms. The central idea is to enforce that the sum of all local gradients remains zero throughout the evolution, thereby confining the system to a zero‑gradient‑sum manifold that is invariant under the dynamics. To achieve this, each node maintains a state variable x_i and an auxiliary variable λ_i. The dynamics are defined as

  \dot{x}i = –∇f_i(x_i) – Σ{j∈N_i}(λ_i – λ_j),
  \dot{λ}i = Σ{j∈N_i}(x_i – x_j),

where N_i denotes the set of neighbors of node i. The first equation drives the state toward the negative local gradient while correcting it with the disagreement in the λ‑variables; the second equation updates λ based on the disagreement in the states. By construction, the aggregate gradient Σ_i ∇f_i(x_i) and the sum of λ_i’s remain zero for all time if they are zero initially.

A Lyapunov function V is proposed:

 V = Σ_i