On the Convergence of a Noisy Gradient Method for Non-convex Distributed Resource Allocation: Saddle Point Escape

TL;DR

This paper introduces a perturbed distributed gradient method that efficiently escapes saddle points in non-convex resource allocation problems, ensuring convergence to near-optimal solutions with high probability.

Contribution

It extends Laplacian-weighted Gradient Descent by adding random perturbations, providing second-order convergence guarantees in non-convex distributed optimization.

Findings

Proposed method effectively escapes saddle points.

Achieves convergence to approximate second-order solutions.

Numerical results show improved performance over standard methods.

Abstract

This paper considers a class of distributed resource allocation problems where each agent privately holds a smooth, potentially non-convex local objective, subject to a globally coupled equality constraint. Built upon the existing method, Laplacian-weighted Gradient Descent, we propose to add random perturbations to the gradient iteration to enable efficient escape from saddle points and achieve second-order convergence guarantees. We show that, with a sufficiently small fixed step size, the iterates of all agents converge to an approximate second-order optimal solution with high probability. Numerical experiments confirm the effectiveness of the proposed approach, demonstrating improved performance over standard weighted gradient descent in non-convex scenarios.