Quantitative Error Feedback for Quantization Noise Reduction of Filtering Over Graphs

This paper introduces an innovative error feedback framework designed to mitigate quantization noise in distributed graph filtering, where communications are constrained to quantized messages. It comes from error spectrum shaping techniques from state-space digital filters, and therefore establishes connections between quantized filtering processes over different domains. In contrast to existing error compensation methods, our framework quantitatively feeds back the quantization noise for exact compensation. We examine the framework under three key scenarios: (i) deterministic graph filtering, (ii) graph filtering over random graphs, and (iii) graph filtering with random node-asynchronous updates. Rigorous theoretical analysis demonstrates that the proposed framework significantly reduces the effect of quantization noise, and we provide closed-form solutions for the optimal error feedback coefficients. Moreover, this quantitative error feedback mechanism can be seamlessly integrated into communication-efficient decentralized optimization frameworks, enabling lower error floors. Numerical experiments validate the theoretical results, consistently showing that our method outperforms conventional quantization strategies in terms of both accuracy and robustness.