Distributed Optimization of Bivariate Polynomial Graph Spectral Functions via Subgraph Optimization

TL;DR

This work tackles edge-weight design under fixed topology to shape the entire Laplacian spectrum, moving beyond single-eigenvalue objectives to whole-spectrum costs. It introduces a distributed, iterate-and-embed scheme that solves local subgraph problems whose gradients align with the global descent direction, validated via a SVD-based ZC test. A warm-start via degree regularization using gossiping, along with a learning-based one-shot edge-update baseline on maximal subgraph embeddings, offers practical strategies for scalable, constraint-preserving spectrum-aware design on large graphs. Empirical results on geometric graphs show the warm-start and distributed subgraph methods approach centralized performance, while the learning-based approach provides a competitive, decentralized alternative under certain conditions.

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 \(h(λ_i-λ_j)\), 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.

Figures (8)

• Figure 1: Visual illustration of a geometric graph $G$, an induced subgraph of the graph $H$ (highlighted in red), and the subgraph's 4-hop core $H'$ (highlighted in green). If the degree of $g(\lambda_i, \lambda_j)$ is 4, then the weights of green edges will be tuned to minimize the local cost function supported on the subgraph consisting of the green and red edges.
• Figure 2: Learning-based Method: Schematic illustrating the model prediction pipeline, where input graph data undergoes a 1-hop maximal subgraph embedding of graph edge weights for node $u$, which is then fed into a DNN with 4 layers. The initial weights inputted ($w_1$ to $w_6$) are the edges connected to the center node(indicated by a red edge, where bright red are the present edges and the lighter red are the edges with a weight of 0). The following edges inputted are the edges within the 1-hop neighborhood that are not connected to the center node (indicated by a gray edge, where again the dark gray represents present edges and the lighter gray are the edges with a weight of 0). The model outputs predicted edge weights for the edges connected to the center node to perform spectrum optimization, and accuracy is evaluated using Mean Squared Error (MSE) by comparing predictions against the ground truth edge weights, which consist of the initial weights after undergoing centralized optimization.
• Figure 3: Example of 1-hop and 2-hop maximal embeddings of a graph with 12 1-hop subgraph nodes and 2 2-hop subgraph nodes. The left graph is the original graph. The center graph is the 1-hop maximal embedding, where the 1-hop maximal subgraph has 13 nodes. The right graph is the 2-hop maximal embedding, where the maximal subgraph has 13 1-hop nodes and 3 2-hop nodes. The red edges are 1-hop edges connected to the center node, the green edges are the 1-hop edges that are not connected to the center node, the blue edges are 2-hop edges(excluding the 1-hop edges), and the black edges are edges that do not exist in the graph but are present in the maximal embedding.
• Figure 4: Descent curves of warm start versus cold start on the same graph. The $x$-axis marks the optimization epochs, whereas the $y$-axis records the percentage of decrease with respect to centralized method (namely, $1 - \rm DOPR$). The value $1$ corresponds to the fact that the objective value does not decrease at all, where as the value $0$ indicates that the objective value reaches the level after centralized optimization.
• Figure 5: Histograms for DOPRs of cold start and warm start methods. It can be noticed from the figure that the DOPR of warm-start may reach $\sim95\%{}$ in average, whereas cold-start may only reach $\sim90\%{}$.

Theorems & Definitions (4)

• Definition 1
• Proposition 2
• Proposition 3
• Proposition 4