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

TL;DR

This work investigates the consistency of graph-based semi-supervised learning (SSL) through the lens of PDE theory, focusing on the p-Laplacian on graphs and its stochastic tug-of-war interpretation. It develops a martingale-based framework and a tug-of-war lemma to derive consistency bounds for the solution $u$ of the game-theoretic $p$-Laplacian on general, geometric, and SBM graphs, with explicit dependence on the graph length scale $\varepsilon_W$, Lipschitz regularity of the label function $g$, and labeling rate $\beta$. The authors show that under mild conditions (notably $\delta \ge c\beta$ and appropriate graph constructions), the discrepancy $|u-g|$ remains controlled, and vertex classification is accurate away from the decision boundary; they also provide numerical evidence on toy problems and real datasets demonstrating improved performance for larger $p$ at very low label rates. By connecting stochastic processes, martingale techniques, and PDE-inspired graph learning, the paper highlights when and how p-Laplacian SSL can be consistent across diverse graph structures and outlines key directions for future work and broader applications.

Abstract

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.

Figures (6)

• Figure 1: A toy example comparing fully and semi-supervised learning. In (a) we show a data set with 2 labeled examples --- the blue circle and green square --- along with $98$ unlabeled data points --- the black dots. In (b) we show the decision regions from training a fully supervised classification algorithm, while in (c) we show the decision regions for a semi-supervised learning algorithm, which uses the unlabeled data to inform the decision boundary.
• Figure 2: Comparison of different semi-supervised learning methods on a toy example with two labeled data points with labels $+1$ and $-1$. The graph consists of $n=20,000$ uniformly distributed random variables on the unit box $[0,1]^2$ with geometric Gaussian kernel weights with $\varepsilon=0.05$. Here WNLL stands for Weighted NonLocal Laplacian shi2017weighted, and PWLL stands for Poisson Weighted Laplace Learning miller2023activecalder2023poisson.
• Figure 3: Classification results using different graph-based semi-supervised learning algorithms on the two-moons data set. There are two classes, the upper moon and the lower moon, and the label function is binary $g(x)\in \{0,1\}$. The red points are the only labeled data points. The $p$-Laplace method used $p=3$, while high order Laplace learning used order $m=4$, both of which gave the best results over a reasonable search.
• Figure 4: Example of the effect of $r/q$ on the classification performance on stochastic block model graphs for various values of $p$.
• Figure 5: Sample of MNIST 4s and 9s, and Cifar-10 deer and dogs.

Theorems & Definitions (32)

• Proposition 4.1: Dynamic Programming Principle
• Lemma 4.2
• proof
• Lemma 4.3
• proof
• Definition 4.4
• Theorem 4.5
• proof
• Theorem 4.6
• proof