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Distributed Optimization with Finite Bit Adaptive Quantization for Efficient Communication and Precision Enhancement

Apostolos I. Rikos, Wei Jiang, Themistoklis Charalambous, Karl H. Johansson

TL;DR

This paper addresses the challenge of unconstrained distributed optimization by leveraging zoom-in and zoom-out operations to adjust quantizer parameters dynamically and shows that during the algorithm’s operation nodes are able to converge to the exact optimal solution.

Abstract

In realistic distributed optimization scenarios, individual nodes possess only partial information and communicate over bandwidth constrained channels. For this reason, the development of efficient distributed algorithms is essential. In our paper we addresses the challenge of unconstrained distributed optimization. In our scenario each node's local function exhibits strong convexity with Lipschitz continuous gradients. The exchange of information between nodes occurs through $3$-bit bandwidth-limited channels (i.e., nodes exchange messages represented by a only $3$-bits). Our proposed algorithm respects the network's bandwidth constraints by leveraging zoom-in and zoom-out operations to adjust quantizer parameters dynamically. We show that during our algorithm's operation nodes are able to converge to the exact optimal solution. Furthermore, we show that our algorithm achieves a linear convergence rate to the optimal solution. We conclude the paper with simulations that highlight our algorithm's unique characteristics.

Distributed Optimization with Finite Bit Adaptive Quantization for Efficient Communication and Precision Enhancement

TL;DR

This paper addresses the challenge of unconstrained distributed optimization by leveraging zoom-in and zoom-out operations to adjust quantizer parameters dynamically and shows that during the algorithm’s operation nodes are able to converge to the exact optimal solution.

Abstract

In realistic distributed optimization scenarios, individual nodes possess only partial information and communicate over bandwidth constrained channels. For this reason, the development of efficient distributed algorithms is essential. In our paper we addresses the challenge of unconstrained distributed optimization. In our scenario each node's local function exhibits strong convexity with Lipschitz continuous gradients. The exchange of information between nodes occurs through -bit bandwidth-limited channels (i.e., nodes exchange messages represented by a only -bits). Our proposed algorithm respects the network's bandwidth constraints by leveraging zoom-in and zoom-out operations to adjust quantizer parameters dynamically. We show that during our algorithm's operation nodes are able to converge to the exact optimal solution. Furthermore, we show that our algorithm achieves a linear convergence rate to the optimal solution. We conclude the paper with simulations that highlight our algorithm's unique characteristics.
Paper Structure (8 sections, 2 theorems, 6 equations, 2 figures)

This paper contains 8 sections, 2 theorems, 6 equations, 2 figures.

Table of Contents

  1. Introduction
  2. NOTATION AND BACKGROUND
  3. Problem Formulation
  4. Distributed Optimization with Finite Bit Adaptive Quantization
  5. Distributed Optimization Algorithm with Finite Bit Adaptive Quantization
  6. Convergence of Algorithm \ref{['alg1']}
  7. Simulation Results
  8. Conclusions

Key Result

Proposition 1

Let us assume that during the operation of Algorithm alg1, $x^* \notin (b_q - 3 \Delta^{[0]}, b_q + 3\Delta^{[0]})$. Then, after a finite number of time steps $\nu_0$ for which nodes are able to calculate $\Delta^{[\nu_0]}$ such that $x^* \in (b_q - 3 \Delta^{[\nu_0]}, b_q + 3\Delta^{[\nu_0]})$.

Figures (2)

  • Figure 1: A $3$-bit mid-rise uniform quantizer, $Q^{\text{$3$MRU}}(b_q, \xi,\bar{\Delta})$.
  • Figure 2: Execution of Algorithm \ref{['alg1']} over a random digraph of $20$ nodes.

Theorems & Definitions (4)

  • Remark 1: Challenges of Optimization Problem
  • Proposition 1: Finite Zoom-out Instances
  • Theorem 1: Linear Convergence Rate 2023:Rikos_Johan_IFAC
  • Remark 2: Possible $c^{\text{in}}$ Values