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Online Proximal ADMM for Graph Learning from Streaming Smooth Signals

Hector Chahuara, Gonzalo Mateos

TL;DR

The paper tackles online learning of dynamic graph topology from streams of nodal signals assumed to be smooth on the latent graph. It introduces OPADMM, an online proximal-ADMM algorithm that updates the edge-weight vector in a time-varying, memory-efficient manner, leveraging a proximal term to regularize temporal variation. The method provides closed-form proximal updates and demonstrates sublinear static regret under mild assumptions, with per-iteration cost $\mathcal{O}(r)$ where $r=\frac{n(n-1)}{2}$. Empirical results on synthetic and real graphs show that OPADMM tracks slowly varying connectivity more effectively than state-of-the-art online baselines, confirming its practical utility for streaming graph learning.

Abstract

Graph signal processing deals with algorithms and signal representations that leverage graph structures for multivariate data analysis. Often said graph topology is not readily available and may be time-varying, hence (dynamic) graph structure learning from nodal (e.g., sensor) observations becomes a critical first step. In this paper, we develop a novel algorithm for online graph learning using observation streams, assumed to be smooth on the latent graph. Unlike batch algorithms for topology identification from smooth signals, our modus operandi is to process graph signals sequentially and thus keep memory and computational costs in check. To solve the resulting smoothness-regularized, time-varying inverse problem, we develop online and lightweight iterations built upon the proximal variant of the alternating direction method of multipliers (ADMM), well known for its fast convergence in batch settings. The proximal term in the topology updates seamlessly implements a temporal-variation regularization, and we argue the online procedure exhibits sublinear static regret under some simplifying assumptions. Reproducible experiments with synthetic and real graphs demonstrate the effectiveness of our method in adapting to streaming signals and tracking slowly-varying network connectivity. The proposed approach also exhibits better tracking performance (in terms of suboptimality), when compared to state-of-the-art online graph learning baselines.

Online Proximal ADMM for Graph Learning from Streaming Smooth Signals

TL;DR

The paper tackles online learning of dynamic graph topology from streams of nodal signals assumed to be smooth on the latent graph. It introduces OPADMM, an online proximal-ADMM algorithm that updates the edge-weight vector in a time-varying, memory-efficient manner, leveraging a proximal term to regularize temporal variation. The method provides closed-form proximal updates and demonstrates sublinear static regret under mild assumptions, with per-iteration cost where . Empirical results on synthetic and real graphs show that OPADMM tracks slowly varying connectivity more effectively than state-of-the-art online baselines, confirming its practical utility for streaming graph learning.

Abstract

Graph signal processing deals with algorithms and signal representations that leverage graph structures for multivariate data analysis. Often said graph topology is not readily available and may be time-varying, hence (dynamic) graph structure learning from nodal (e.g., sensor) observations becomes a critical first step. In this paper, we develop a novel algorithm for online graph learning using observation streams, assumed to be smooth on the latent graph. Unlike batch algorithms for topology identification from smooth signals, our modus operandi is to process graph signals sequentially and thus keep memory and computational costs in check. To solve the resulting smoothness-regularized, time-varying inverse problem, we develop online and lightweight iterations built upon the proximal variant of the alternating direction method of multipliers (ADMM), well known for its fast convergence in batch settings. The proximal term in the topology updates seamlessly implements a temporal-variation regularization, and we argue the online procedure exhibits sublinear static regret under some simplifying assumptions. Reproducible experiments with synthetic and real graphs demonstrate the effectiveness of our method in adapting to streaming signals and tracking slowly-varying network connectivity. The proposed approach also exhibits better tracking performance (in terms of suboptimality), when compared to state-of-the-art online graph learning baselines.
Paper Structure (14 sections, 15 equations, 2 figures, 1 algorithm)

This paper contains 14 sections, 15 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Formulation
  3. Graph-theoretic notions and problem statement
  4. Batch topology identification from smooth signals
  5. Proximal ADMM for Graph Learning
  6. Applying ADMM to the graph learning problem
  7. Efficient proximal updates
  8. Online Graph Learning from Streaming Signals
  9. Online PADMM algorithm
  10. Discussion , computational complexity and regret analyses
  11. Experimental Evaluation
  12. Synthetic graphs
  13. Real world graphs
  14. Conclusions

Figures (2)

  • Figure 1: Convergence behavior illustrated via the evolution of suboptimality $\|\mathbf{w}^{(k)}-\hat{\mathbf{w}}\|_{2}$ for synthetic random graph models with $n=100$ nodes: (a)-(c) stationary graphs , and (d)-(f) time-varying graphs. For the dynamic networks , the topology changes at $k=1000$. In stationary settings , our proposed method converges faster than online PG and DPG algorithms , while in dynamic settings , it demonstrates superior adaptability to changes in network topology.
  • Figure 2: Convergence behavior illustrated via the evolution of suboptimality $\|\mathbf{w}^{(k)}-\hat{\mathbf{w}}\|_{2}$ for real-world graphs from the datasets Davis2011. Our proposed method converges faster than online PG and DPG algorithms in all cases.