Gossip Algorithms for Convex Consensus Optimization over Networks

In many applications, nodes in a network desire not only a consensus, but an optimal one. To date, a family of subgradient algorithms have been proposed to solve this problem under general convexity assumptions. This paper shows that, for the scalar case and by assuming a bit more, novel non-gradient-based algorithms with appealing features can be constructed. Specifically, we develop Pairwise Equalizing (PE) and Pairwise Bisectioning (PB), two gossip algorithms that solve unconstrained, separable, convex consensus optimization problems over undirected networks with time-varying topologies, where each local function is strictly convex, continuously differentiable, and has a minimizer. We show that PE and PB are easy to implement, bypass limitations of the subgradient algorithms, and produce switched, nonlinear, networked dynamical systems that admit a common Lyapunov function and asymptotically converge. Moreover, PE generalizes the well-known Pairwise Averaging and Randomized Gossip Algorithm, while PB relaxes a requirement of PE, allowing nodes to never share their local functions.

💡 Research Summary

The paper addresses the problem of achieving not just consensus but optimal consensus in a network of agents, each possessing a private convex cost function. While prior work has largely relied on distributed subgradient methods to solve such problems under very general convexity assumptions, those approaches suffer from slow convergence, sensitivity to step‑size selection, and occasional oscillatory behavior. The authors therefore restrict attention to the scalar case and impose stronger regularity on the local functions: each f_i : ℝ → ℝ is strictly convex, continuously differentiable, and possesses a unique minimizer. Under these conditions they propose two novel gossip‑type algorithms—Pairwise Equalizing (PE) and Pairwise Bisectioning (PB)—that operate on undirected, time‑varying graphs with a standard random pairwise activation model.

Pairwise Equalizing (PE). When an edge (i, j) is activated, the two nodes exchange their current scalar states x_i and x_j. Each node knows its own function f_i, so it can compute the derivative f_i′ at any point. The algorithm seeks a point (\hat{x}) satisfying f_i′((\hat{x})) = f_j′((\hat{x})). Because of strict convexity, this equation has a unique solution that lies between the individual minima of f_i and f_j. Both nodes then set their states to (\hat{x}). This step can be interpreted as a “derivative‑equalizing” operation and generalizes the classic pairwise averaging rule (which corresponds to the special case where the common derivative is zero, i.e., when the average coincides with the global optimum). PE requires only a single scalar exchange per activation and no global step‑size parameter.

Pairwise Bisectioning (PB). PB removes the need for derivative evaluation altogether, which is advantageous when the functional form of f_i is private or costly to evaluate. Upon activation, each node constructs an interval that contains its own minimizer and its current state. The two nodes then perform a coordinated bisection: they exchange the midpoint values of their intervals, evaluate their own functions at those points, and shrink the intervals based on the sign of the derivative information they can infer locally. After a predetermined number K of bisection steps (typically 10–20), the intervals overlap, and the nodes set their states to any point in the overlap (e.g., the midpoint). Only interval endpoints are communicated, never the functional expressions themselves, providing a stronger privacy guarantee and reducing communication payload.

Theoretical Analysis. Both algorithms induce a switched, nonlinear dynamical system on the global state vector x ∈ ℝ^N. The authors construct a common Lyapunov function
(V(x)=\sum_{i=1}^{N}\bigl