Nonlinear Perturbation-based Non-Convex Optimization over Time-Varying Networks

TL;DR

This paper introduces a novel decentralized optimization algorithm that efficiently handles non-convex objectives and nonlinear data transmission over dynamic networks, ensuring optimal convergence and stability without extra consensus steps.

Contribution

It presents a single-time scale, gradient-based distributed algorithm capable of solving non-convex problems over time-varying networks with nonlinear link effects, a significant advancement over existing methods.

Findings

Proves exact convergence despite nonlinear link effects.

Demonstrates stability over switching, time-varying networks.

Shows effectiveness through numerical simulations.

Abstract

Decentralized optimization strategies are helpful for various applications, from networked estimation to distributed machine learning. This paper studies finite-sum minimization problems described over a network of nodes and proposes a computationally efficient algorithm that solves distributed convex problems and optimally finds the solution to locally non-convex objective functions. In contrast to batch gradient optimization in some literature, our algorithm is on a single-time scale with no extra inner consensus loop. It evaluates one gradient entry per node per time. Further, the algorithm addresses link-level nonlinearity representing, for example, logarithmic quantization of the exchanged data or clipping of the exchanged data bits. Leveraging perturbation-based theory and algebraic Laplacian network analysis proves optimal convergence and dynamics stability over time-varying and switching networks. The time-varying network setup might be due to packet drops or link failures. Despite the nonlinear nature of the dynamics, we prove exact convergence in the face of odd sign-preserving sector-bound nonlinear data transmission over the links. Illustrative numerical simulations further highlight our contributions.