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Quantized Distributed Nonconvex Optimization Algorithms with Linear Convergence under the Polyak--$Ł$ojasiewicz Condition

Lei Xu, Xinlei Yi, Jiayue Sun, Yang Shi, Karl H. Johansson, Tao Yang

TL;DR

This work addresses distributed nonconvex optimization over undirected networks with quantized communications under the Polyak–Łojasiewicz condition. It proposes two algorithms that integrate an encoder–decoder quantization scheme with (i) distributed gradient tracking and (ii) distributed PI control, and proves linear convergence to a global optimum with a quantization level above a threshold. Remarkably, the analysis shows linear convergence even at extremely low data rates, including 1-bit transmissions, provided algorithm parameters are properly chosen. Numerical experiments on networks with 10 and 100 agents validate the theory and demonstrate favorable bit-efficiency versus existing quantized methods. The results extend linear-convergence guarantees beyond strong convexity to the PŁ setting and offer practically bandwidth-efficient distributed nonconvex optimization solutions.

Abstract

This paper considers distributed optimization for minimizing the average of local nonconvex cost functions, by using local information exchange over undirected communication networks. To reduce the required communication capacity, we introduce an encoder--decoder scheme. By integrating them with distributed gradient tracking and proportional integral algorithms, respectively, we then propose two quantized distributed nonconvex optimization algorithms. Assuming the global cost function satisfies the Polyak--Łojasiewicz condition, which does not require the global cost function to be convex and the global minimizer is not necessarily unique, we show that our proposed algorithms linearly converge to a global optimal point and that larger quantization level leads to faster convergence speed. Moreover, we show that a low data rate is sufficient to guarantee linear convergence when the algorithm parameters are properly chosen. The theoretical results are illustrated by numerical examples.

Quantized Distributed Nonconvex Optimization Algorithms with Linear Convergence under the Polyak--$Ł$ojasiewicz Condition

TL;DR

This work addresses distributed nonconvex optimization over undirected networks with quantized communications under the Polyak–Łojasiewicz condition. It proposes two algorithms that integrate an encoder–decoder quantization scheme with (i) distributed gradient tracking and (ii) distributed PI control, and proves linear convergence to a global optimum with a quantization level above a threshold. Remarkably, the analysis shows linear convergence even at extremely low data rates, including 1-bit transmissions, provided algorithm parameters are properly chosen. Numerical experiments on networks with 10 and 100 agents validate the theory and demonstrate favorable bit-efficiency versus existing quantized methods. The results extend linear-convergence guarantees beyond strong convexity to the PŁ setting and offer practically bandwidth-efficient distributed nonconvex optimization solutions.

Abstract

This paper considers distributed optimization for minimizing the average of local nonconvex cost functions, by using local information exchange over undirected communication networks. To reduce the required communication capacity, we introduce an encoder--decoder scheme. By integrating them with distributed gradient tracking and proportional integral algorithms, respectively, we then propose two quantized distributed nonconvex optimization algorithms. Assuming the global cost function satisfies the Polyak--Łojasiewicz condition, which does not require the global cost function to be convex and the global minimizer is not necessarily unique, we show that our proposed algorithms linearly converge to a global optimal point and that larger quantization level leads to faster convergence speed. Moreover, we show that a low data rate is sufficient to guarantee linear convergence when the algorithm parameters are properly chosen. The theoretical results are illustrated by numerical examples.
Paper Structure (16 sections, 9 theorems, 127 equations, 5 figures, 3 tables, 2 algorithms)

This paper contains 16 sections, 9 theorems, 127 equations, 5 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Encoder--Decoder Scheme for Quantized Communication
  4. Distributed Gradient Tracking Algorithm with Finite Data Rates
  5. Distributed Proportional Integral Algorithm with Finite Data Rates
  6. Numerical Examples
  7. Conclusions
  8. Useful Lemma
  9. Proof of Lemma \ref{['lem-1']}
  10. Proof of Lemma \ref{['lem-3']}
  11. Proof of Proposition \ref{['prop2']}
  12. Proof of Theorem \ref{['thm-3']}
  13. Proof of Theorem \ref{['thm-4']}
  14. Proof of Proposition \ref{['prop']}
  15. Proof of Theorem \ref{['thm-1']}
  16. ...and 1 more sections

Key Result

Lemma 1

Suppose that the parameters $\beta$ and $\delta$ satisfy where Then, the following linear matrix inequality holds: where $\iota=\min\{\frac{{\color{black} c}_{2}}{2},\frac{\delta}{4}\nu\}$, $\Theta=[\Theta_{1},1,\Theta_{2}]^{T}$ and the nonnegative matrix $\Phi$ is given by where

Figures (5)

  • Figure 1: Random connected network of 100 agents.
  • Figure 2: Evolutions of ${\color{black} \Upsilon(k)}$ with respect to the number of iterations for different algorithms.
  • Figure 3: Evolutions of ${\color{black} \Upsilon(k)}$ with respect to the number of transmitted bits for different distributed algorithms.
  • Figure 4: Transmitted bits for different algorithms and quantization levels to reach ${\color{black} \Upsilon(k)}\leq10^{-5}$.
  • Figure 5: Evolutions of $\Upsilon(k)$ with respect to the number of iterations for different quantized distributed algorithms.

Theorems & Definitions (16)

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