# Consistency of semi-supervised learning, stochastic tug-of-war games, and the p-Laplacian

arXiv: https://arxiv.org/abs/2401.07463 · HTML: https://mlsanity.com/paper/2401.07463/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 34 of its 122 references that are in the map.
- First posted on arXiv 15 Jan 2024.

## Builds on (most cited first, 30 of 34)

- [A Simple Framework for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2002.05709/) (2020)
- [Predict then Propagate: Graph Neural Networks meet Personalized PageRank](https://mlsanity.com/paper/1810.05997/) (2018)
- [A Survey on Deep Semi-supervised Learning](https://mlsanity.com/paper/2103.00550/) (2021)
- [Combining Label Propagation and Simple Models Out-performs Graph Neural Networks](https://mlsanity.com/paper/2010.13993/) (2020)
- [Error estimates for spectral convergence of the graph Laplacian on random geometric graphs towards the Laplace--Beltrami operator](https://mlsanity.com/paper/1801.10108/) (2018)
- [Poisson Learning: Graph Based Semi-Supervised Learning At Very Low Label Rates](https://mlsanity.com/paper/2006.11184/) (2020)
- [On the Equivalence of Decoupled Graph Convolution Network and Label Propagation](https://mlsanity.com/paper/2010.12408/) (2020)
- [Analysis of p-Laplacian Regularization in Semi-Supervised Learning](https://mlsanity.com/paper/1707.06213/) (2017)
- [Lipschitz regularity of graph Laplacians on random data clouds](https://mlsanity.com/paper/2007.06679/) (2020)
- [Continuum limit of total variation on point clouds](https://mlsanity.com/paper/1403.6355/) (2014)
- [The game theoretic p-Laplacian and semi-supervised learning with few labels](https://mlsanity.com/paper/1711.10144/) (2017)
- [Large Data and Zero Noise Limits of Graph-Based Semi-Supervised Learning Algorithms](https://mlsanity.com/paper/1805.09450/) (2018)
- [Properly-weighted graph Laplacian for semi-supervised learning](https://mlsanity.com/paper/1810.04351/) (2018)
- [Rates of Convergence for Laplacian Semi-Supervised Learning with Low Labeling Rates](https://mlsanity.com/paper/2006.02765/) (2020)
- [Algorithms for Lipschitz Learning on Graphs](https://mlsanity.com/paper/1505.00290/) (2015)
- [Asymptotic behavior of ℓp-based Laplacian regularization in semi-supervised learning](https://mlsanity.com/paper/1603.00564/) (2016)
- [Multiclass Data Segmentation using Diffuse Interface Methods on Graphs](https://mlsanity.com/paper/1302.3913/) (2013)
- [Consistency of Lipschitz learning with infinite unlabeled data and finite labeled data](https://mlsanity.com/paper/1710.10364/) (2017)
- [Analysis and algorithms for ℓp-based semi-supervised learning on graphs](https://mlsanity.com/paper/1901.05031/) (2019)
- [Spectral Analysis Of Weighted Laplacians Arising In Data Clustering](https://mlsanity.com/paper/1909.06389/) (2019)
- [A random matrix analysis and improvement of semi-supervised learning for large dimensional data](https://mlsanity.com/paper/1711.03404/) (2017)
- [Geometric structure of graph Laplacian embeddings](https://mlsanity.com/paper/1901.10651/) (2019)
- [Consistent Semi-Supervised Graph Regularization for High Dimensional Data](https://mlsanity.com/paper/2006.07575/) (2020)
- [Uniform Convergence Rates for Lipschitz Learning on Graphs](https://mlsanity.com/paper/2111.12370/) (2021)
- [Model-Change Active Learning in Graph-Based Semi-Supervised Learning](https://mlsanity.com/paper/2110.07739/) (2021)
- [Hamilton-Jacobi equations on graphs with applications to semi-supervised learning and data depth](https://mlsanity.com/paper/2202.08789/) (2022)
- [Graph-based Active Learning for Semi-supervised Classification of SAR Data](https://mlsanity.com/paper/2204.00005/) (2022)
- [Poisson Reweighted Laplacian Uncertainty Sampling for Graph-based Active Learning](https://mlsanity.com/paper/2210.15786/) (2022)
- [LaplaceNet: A Hybrid Graph-Energy Neural Network for Deep Semi-Supervised Classification](https://mlsanity.com/paper/2106.04527/) (2021)
- [A Method Based on Total Variation for Network Modularity Optimization using the MBO Scheme](https://mlsanity.com/paper/1304.4679/) (2013)

## Built on by (most cited first, 3 of 3)

- [Numerical solution of a PDE arising from prediction with expert advice](https://mlsanity.com/paper/2406.05754/) (2024)
- [Predicting Anthropometric Body Composition Variables Using 3D Optical Imaging and Machine Learning](https://mlsanity.com/paper/2506.14815/) (2025)
- [Target-Aware State-Adaptive p-Dirichlet Graph Neural Regression for Non-Invasive Body-Composition Estimation](https://mlsanity.com/paper/2608.29496/) (2026)

## Abstract (arXiv)

In this paper we give a broad overview of the intersection of partial differential equations (PDEs) and graph-based semi-supervised learning. The overview is focused on a large body of recent work on PDE continuum limits of graph-based learning, which have been used to prove well-posedness of semi-supervised learning algorithms in the large data limit. We highlight some interesting research directions revolving around consistency of graph-based semi-supervised learning, and present some new results on the consistency of p-Laplacian semi-supervised learning using the stochastic tug-of-war game interpretation of the p-Laplacian. We also present the results of some numerical experiments that illustrate our results and suggest directions for future work.