Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Compressed Gradient Tracking Algorithms for Distributed Nonconvex Optimization

Lei Xu, Xinlei Yi, Guanghui Wen, Yang Shi, Karl H. Johansson, Tao Yang

TL;DR

This work tackles distributed nonconvex optimization over directed graphs under communication constraints by integrating gradient-tracking with three general compressor models. It develops three compressed gradient-tracking algorithms, including an error-feedback variant, and establishes convergence via Lyapunov functions: sublinear rates $O(1/T)$ in general and linear rates under the Polyak–Łojasiewicz condition, with several results not requiring prior knowledge of the P–Ł constant. For compressors with bounded relative, globally bounded absolute, and locally bounded absolute errors, the authors provide dedicated analyses and show linear convergence under P–Ł without needing the constant to tune parameters in many cases. Numerical experiments reveal significant reductions in communication while maintaining convergence, outperforming state-of-the-art compressed methods across multiple compressor types. The findings advance scalable distributed optimization by enabling efficient, provably convergent operation under realistic communication constraints.

Abstract

In this paper, we study the distributed nonconvex optimization problem, which aims to minimize the average value of the local nonconvex cost functions using local information exchange. To reduce the communication overhead, we introduce three general classes of compressors, i.e., compressors with bounded relative compression error, compressors with globally bounded absolute compression error, and compressors with locally bounded absolute compression error. By integrating them with distributed gradient tracking algorithm, we then propose three compressed distributed nonconvex optimization algorithms. For each algorithm, we design a novel Lyapunov function to demonstrate its sublinear convergence to a stationary point if the local cost functions are smooth. Furthermore, when the global cost function satisfies the Polyak--Łojasiewicz (P--Ł) condition, we show that our proposed algorithms linearly converge to a global optimal point. It is worth noting that, for compressors with bounded relative compression error and globally bounded absolute compression error, our proposed algorithms' parameters do not require prior knowledge of the P--Ł constant. The theoretical results are illustrated by numerical examples, which demonstrate the effectiveness of the proposed algorithms in significantly reducing the communication burden while maintaining the convergence performance. Moreover, simulation results show that the proposed algorithms outperform state-of-the-art compressed distributed nonconvex optimization algorithms.

Compressed Gradient Tracking Algorithms for Distributed Nonconvex Optimization

TL;DR

This work tackles distributed nonconvex optimization over directed graphs under communication constraints by integrating gradient-tracking with three general compressor models. It develops three compressed gradient-tracking algorithms, including an error-feedback variant, and establishes convergence via Lyapunov functions: sublinear rates in general and linear rates under the Polyak–Łojasiewicz condition, with several results not requiring prior knowledge of the P–Ł constant. For compressors with bounded relative, globally bounded absolute, and locally bounded absolute errors, the authors provide dedicated analyses and show linear convergence under P–Ł without needing the constant to tune parameters in many cases. Numerical experiments reveal significant reductions in communication while maintaining convergence, outperforming state-of-the-art compressed methods across multiple compressor types. The findings advance scalable distributed optimization by enabling efficient, provably convergent operation under realistic communication constraints.

Abstract

In this paper, we study the distributed nonconvex optimization problem, which aims to minimize the average value of the local nonconvex cost functions using local information exchange. To reduce the communication overhead, we introduce three general classes of compressors, i.e., compressors with bounded relative compression error, compressors with globally bounded absolute compression error, and compressors with locally bounded absolute compression error. By integrating them with distributed gradient tracking algorithm, we then propose three compressed distributed nonconvex optimization algorithms. For each algorithm, we design a novel Lyapunov function to demonstrate its sublinear convergence to a stationary point if the local cost functions are smooth. Furthermore, when the global cost function satisfies the Polyak--Łojasiewicz (P--Ł) condition, we show that our proposed algorithms linearly converge to a global optimal point. It is worth noting that, for compressors with bounded relative compression error and globally bounded absolute compression error, our proposed algorithms' parameters do not require prior knowledge of the P--Ł constant. The theoretical results are illustrated by numerical examples, which demonstrate the effectiveness of the proposed algorithms in significantly reducing the communication burden while maintaining the convergence performance. Moreover, simulation results show that the proposed algorithms outperform state-of-the-art compressed distributed nonconvex optimization algorithms.
Paper Structure (10 sections, 9 theorems, 164 equations, 3 figures, 1 table, 3 algorithms)

This paper contains 10 sections, 9 theorems, 164 equations, 3 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Compressed Distributed Nonconvex Algorithms: Bounded Relative Compression Error
  4. Compressed Gradient Tracking Algorithm: Bounded Relative Compression Error
  5. Error Feedback Based Compressed Gradient Tracking Algorithm: Bounded Relative Compression Error
  6. Compressed Distributed Nonconvex Algorithm: Bounded Absolute Compression Error
  7. Compressed Gradient Tracking Algorithm: Globally Bounded Absolute Compression Error
  8. Compressed Gradient Tracking Algorithm: Locally Bounded Absolute Compression Error
  9. Numerical Examples
  10. Conclusions

Key Result

Theorem 1

Suppose that Assumptions assum-4--assum-6, and assum-1 hold. Let each agent $i\in\mathcal{V}$ run Algorithm alg:1 with algorithm parameters $\varphi_{X}$, $\varphi_{Y}\in(0,\frac{1}{r})$, $\eta$ and $\gamma$ such that where $c_{1}=\frac{\varphi_{X}\psi r}{2}$, $c_{2}=\frac{\varphi_{Y}\psi r}{2}$, and $C=2r^{2}(1-\psi)+2(1-r)^{2}$. Then, we have and where

Figures (3)

  • Figure 1: Evolutions of $\Upsilon(T)$ with respect to the number of iterations of different compressed distributed algorithms with the norm-sign compressor.
  • Figure 2: Evolutions of $\Upsilon(T)$ with respect to the number of iterations of different compressed distributed algorithms with the standard uniform quantizer.
  • Figure 3: Evolutions of $\Upsilon(T)$ with respect to the number of iterations of different compressed distributed algorithms with the 1-bit binary quantizer.

Theorems & Definitions (13)

  • Remark 1
  • Theorem 1
  • Remark 2
  • Theorem 2
  • Remark 3
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • ...and 3 more