[Paper Review] Generalization bounds for graph convolutional neural networks via Rademacher complexity

Shaogao Lv|arXiv (Cornell University)|Feb 20, 2021

Advanced Graph Neural Networks28 references4 citations

TL;DR

This paper establishes tight generalization bounds for one-hidden-layer graph convolutional networks (GCNs) using Rademacher complexity, showing that the bounds depend explicitly on the largest eigenvalue of the graph convolution filter and the graph's degree distribution. It proves both upper and matching lower bounds, demonstrating the optimality of the derived complexity estimates and providing theoretical justification for filter and graph structure design in GCNs.

ABSTRACT

This paper aims at studying the sample complexity of graph convolutional networks (GCNs), by providing tight upper bounds of Rademacher complexity for GCN models with a single hidden layer. Under regularity conditions, theses derived complexity bounds explicitly depend on the largest eigenvalue of graph convolution filter and the degree distribution of the graph. Again, we provide a lower bound of Rademacher complexity for GCNs to show optimality of our derived upper bounds. Taking two commonly used examples as representatives, we discuss the implications of our results in designing graph convolution filters an graph distribution.

Motivation & Objective

• To provide sharp generalization bounds for one-hidden-layer GCNs using Rademacher complexity.
• To analyze how graph structure—specifically the largest eigenvalue of the graph convolution filter and degree distribution—affects generalization performance.
• To establish both upper and lower bounds on Rademacher complexity to demonstrate the optimality of the derived bounds.
• To guide the design of graph convolution filters and graph distributions by quantifying their impact on generalization.
• To extend theoretical understanding of GCN generalization beyond standard deep learning settings, accounting for graph-specific structural dependencies.

Proposed method

• Derives an upper bound on the empirical Rademacher complexity of GCNs under parameter norm constraints using spectral graph theory.
• Uses the spectral properties of the graph Laplacian and the graph convolution filter to characterize the complexity of the hypothesis class.
• Applies symmetrization and contraction arguments to bound the expected Rademacher complexity via the largest eigenvalue of the filter and the degree distribution.
• Constructs a lower bound on Rademacher complexity by carefully choosing input features and exploiting the structure of the filter matrix.
• Demonstrates that the upper and lower bounds match in order of magnitude, proving the optimality of the derived complexity bounds.
• Analyzes two common graph convolution filters—unnormalized (A+I) and normalized (D⁻¹A+I)—to illustrate the implications of the general results.

Experimental results

Research questions

• RQ1 How does the largest eigenvalue of the graph convolution filter influence the generalization performance of GCNs?
• RQ2 In what way does the degree distribution of the graph affect the Rademacher complexity and generalization bounds of GCNs?
• RQ3 Can tight upper and lower bounds on Rademacher complexity be derived for one-hidden-layer GCNs to establish the optimality of the bounds?
• RQ4 How do different graph convolution filters (unnormalized vs. normalized) affect the generalization gap in GCNs?
• RQ5 To what extent do graph structure properties such as regularity or random graph models (e.g., Erdős–Rényi) impact the generalization bound?

Key findings

• The generalization bound for GCNs scales as O(λ_max(g(L)) q^{3/2} m^{-1/2}), where λ_max is the largest eigenvalue of the graph filter and q is the maximum degree.
• For Erdős–Rényi graphs with edge probability Ω(log n / n), the bound is O(log^{5/2}(n) m^{-1/2}) under the unnormalized filter (A+I).
• For regular graphs with bounded degree (q=O(1)), the bound reduces to O(m^{-1/2}), independent of graph size n, matching standard Euclidean deep learning models.
• With the normalized filter (D⁻¹A+I), the bound improves to O(log^{3/2}(n) m^{-1/2}) for Erdős–Rényi graphs, showing a log(n) improvement over the unnormalized case.
• The lower bound on Rademacher complexity matches the upper bound in order of magnitude, proving that the derived upper bound is optimal.
• The results show that spectral properties of the graph filter and degree distribution are critical factors in controlling generalization error in GCNs.