Krylov-Safonov theory for Pucci-type extremal inequalities on random data clouds

TL;DR

This paper develops a regularity theory for solutions to discrete equations on random data graphs, showing convergence to PDE solutions as data size grows, with applications to machine learning and game theory.

Contribution

It introduces Krylov-Safonov type regularity results for Pucci-type inequalities on random geometric graphs, bridging discrete stochastic processes and PDE theory.

Findings

Establishes Hölder regularity for solutions on random graphs.

Demonstrates convergence of graph functions to PDE solutions.

Applies to data clouds in machine learning and game scenarios.

Abstract

We establish Krylov-Safonov type H\"older regularity theory for solutions to quite general discrete dynamic programming equations or equivalently discrete stochastic processes on random geometric graphs. Such graphs arise for example from data clouds in graph-based machine learning. The results actually hold to functions satisfying Pucci-type extremal inequalities, and thus we cover many examples including tug-of-war games on random geometric graphs. As an application we show that under suitable assumptions when the number of data points increases, the graph functions converge to a solution of a partial differential equation.