# Revisiting Graph Convolutional Network on Semi-Supervised Node Classification from an Optimization Perspective

arXiv: https://arxiv.org/abs/2009.11469 · HTML: https://mlsanity.com/paper/2009.11469/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Sep 2020.

## Built on by (most cited first, 17 of 17)

- [Graph Neural Networks Inspired by Classical Iterative Algorithms](https://mlsanity.com/paper/2103.06064/) (2021)
- [Cold Brew: Distilling Graph Node Representations with Incomplete or Missing Neighborhoods](https://mlsanity.com/paper/2111.04840/) (2021)
- [Bag of Tricks for Training Deeper Graph Neural Networks: A Comprehensive Benchmark Study](https://mlsanity.com/paper/2108.10521/) (2021)
- [Transformers from an Optimization Perspective](https://mlsanity.com/paper/2205.13891/) (2022)
- [From Hypergraph Energy Functions to Hypergraph Neural Networks](https://mlsanity.com/paper/2306.09623/) (2023)
- [Implicit vs Unfolded Graph Neural Networks](https://mlsanity.com/paper/2111.06592/) (2021)
- [Does your graph need a confidence boost? Convergent boosted smoothing on graphs with tabular node features](https://mlsanity.com/paper/2110.13413/) (2021)
- [Descent Steps of a Relation-Aware Energy Produce Heterogeneous Graph Neural Networks](https://mlsanity.com/paper/2206.11081/) (2022)
- [Training-Free Message Passing for Learning on Hypergraphs](https://mlsanity.com/paper/2402.05569/) (2024)
- [Efficient Link Prediction via GNN Layers Induced by Negative Sampling](https://mlsanity.com/paper/2310.09516/) (2023)
- [Towards Understanding Graph Neural Networks: An Algorithm Unrolling Perspective](https://mlsanity.com/paper/2206.04471/) (2022)
- [ASGNN: Graph Neural Networks with Adaptive Structure](https://mlsanity.com/paper/2210.01002/) (2022)
- [MuseGNN: Forming Scalable, Convergent GNN Layers that Minimize a Sampling-Based Energy](https://mlsanity.com/paper/2310.12457/) (2023)
- [A Robust Stacking Framework for Training Deep Graph Models with Multifaceted Node Features](https://mlsanity.com/paper/2206.08473/) (2022)
- [BloomGML: Graph Machine Learning through the Lens of Bilevel Optimization](https://mlsanity.com/paper/2403.04763/) (2024)
- [HERTA: A High-Efficiency and Rigorous Training Algorithm for Unfolded Graph Neural Networks](https://mlsanity.com/paper/2403.18142/) (2024)
- [Aggregate-then-Calibrate for Human-centered Assessment with Theoretical Guarantees](https://mlsanity.com/paper/2608.02455/) (2026)

## Abstract (arXiv)

Graph convolutional networks (GCNs) have achieved promising performance on various graph-based tasks. However they suffer from over-smoothing when stacking more layers. In this paper, we present a quantitative study on this observation and develop novel insights towards the deeper GCN. First, we interpret the current graph convolutional operations from an optimization perspective and argue that over-smoothing is mainly caused by the naive first-order approximation of the solution to the optimization problem. Subsequently, we introduce two metrics to measure the over-smoothing on node-level tasks. Specifically, we calculate the fraction of the pairwise distance between connected and disconnected nodes to the overall distance respectively. Based on our theoretical and empirical analysis, we establish a universal theoretical framework of GCN from an optimization perspective and derive a novel convolutional kernel named GCN+ which has lower parameter amount while relieving the over-smoothing inherently. Extensive experiments on real-world datasets demonstrate the superior performance of GCN+ over state-of-the-art baseline methods on the node classification tasks.