# Linear Convergence of First- and Zeroth-Order Primal-Dual Algorithms for Distributed Nonconvex Optimization

arXiv: https://arxiv.org/abs/1912.12110 · HTML: https://mlsanity.com/paper/1912.12110/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Dec 2019.

## Built on by (most cited first, 2 of 2)

- [Distributed Online Bandit Nonconvex Optimization with One-Point Residual Feedback via Dynamic Regret](https://mlsanity.com/paper/2409.15680/) (2024)
- [A Stochastic Approximation Approach for Efficient Decentralized Optimization on Random Networks](https://mlsanity.com/paper/2410.18774/) (2024)

## Abstract (arXiv)

This paper considers the distributed nonconvex optimization problem of minimizing a global cost function formed by a sum of local cost functions by using local information exchange. We first consider a distributed first-order primal-dual algorithm. We show that it converges sublinearly to a stationary point if each local cost function is smooth and linearly to a global optimum under an additional condition that the global cost function satisfies the Polyak-Lojasiewicz condition. This condition is weaker than strong convexity, which is a standard condition for proving linear convergence of distributed optimization algorithms, and the global minimizer is not necessarily unique. Motivated by the situations where the gradients are unavailable, we then propose a distributed zeroth-order algorithm, derived from the considered first-order algorithm by using a deterministic gradient estimator, and show that it has the same convergence properties as the considered first-order algorithm under the same conditions. The theoretical results are illustrated by numerical simulations.