Distributed Optimization of Pairwise Polynomial Graph Spectral Functions via Subgraph Optimization

2025-11-14

AI Summary

This work addresses distributed optimization over fixed-topology graphs under a global edge-weight budget constraint, targeting finite-degree polynomial functions of the Laplacian spectrum—emphasizing coordinated control over the entire spectrum, not merely extremal eigenvalues. We propose an iterative embedding framework based on 1-hop subgraph decomposition: the global objective is reformulated into a bilinear form, and singular value decomposition (SVD) of the zero-centered (ZC) matrix enables local gradient approximation aligned with the global descent direction. A learnable edge-update proposer supports efficient, one-shot structural adjustments. Degree regularization with warm-start initialization, coupled with randomized gossip-based average-degree estimation, ensures decentralization, edge-weight positivity, and strict budget feasibility. The method scales effectively to large geometric graphs: warm-start performance reaches 95% of centralized optimization, significantly accelerating convergence and reducing the spectral objective value.

Technology Category

• Search and Optimization: Distributed Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We study distributed optimization of finite-degree polynomial Laplacian spectral objectives under fixed topology and a global weight budget, targeting the collective behavior of the entire spectrum rather than a few extremal eigenvalues. By re-formulating the global cost in a bilinear form, we derive local subgraph problems whose gradients approximately align with the global descent direction via an SVD-based test on the $ZC$ matrix. This leads to an iterate-and-embed scheme over disjoint 1-hop neighborhoods that preserves feasibility by construction (positivity and budget) and scales to large geometric graphs. For objectives that depend on pairwise eigenvalue differences $h(lambda_i-lambda_j)$, we obtain a quadratic upper bound in the degree vector, which motivates a ``warm-start''by degree-regularization. The warm start uses randomized gossip to estimate global average degree, accelerating subsequent local descent while maintaining decentralization, and realizing $sim95%{}$ of the performance with respect to centralized optimization. We further introduce a learning-based proposer that predicts one-shot edge updates on maximal 1-hop embeddings, yielding immediate objective reductions. Together, these components form a practical, modular pipeline for spectrum-aware weight tuning that preserves constraints and applies across a broader class of whole-spectrum costs.

Problem

• Optimize Laplacian spectral functions under topology and budget constraints
• Align local subgraph gradients with global descent via SVD decomposition
• Develop decentralized pipeline for whole-spectrum weight tuning in large graphs

Innovation

• Distributed optimization via local subgraph problems
• Warm-start using randomized gossip for degree regularization
• Learning-based proposer predicts one-shot edge updates