Gradient and Lipschitz Estimates for Tug-of-War Type Games

Abstract

We define a random step size tug-of-war game, and show that the gradient of a value function exists almost everywhere. We also prove that the gradients of value functions are uniformly bounded and converge weakly to the gradient of the corresponding p-harmonic function. Moreover, we establish an improved Lipschitz estimate when boundary values are close to a plane. Such estimates are known to play a key role in higher regularity theory of partial differential equations. The proofs are based on cancellation and coupling methods as well as improved version of the cylinder walk argument.

Properly-Weighted Graph Laplacian for Semi-supervised Learning

“…The Python code to reproduce the results in this section is available on GitHub. 5 Our first experiment is with a stochastic block model graph to illustrate Theorem 4.22. We take a graph with n = 3000 vertices and consider the case of equal block sizes N 0 = N 1 = 1500 and unbalanced block sizes N 0 = 2000, N 1 = 1000.…”

“…This work motivated a study of the game-theoretic p-Laplacian on graphs [85] as well as finite difference approaches for numerically approximating solutions [3,97,98]. The tug-of-war interpretation of the p-Laplacian has led to simple alternative proofs of regularity for p-harmonic functions, including the Harnack inequality and gradient estimates [5,78]. In addition, many variants of tug-of-war games have been introduced, including games with bias [100], mixed Neumann/Dirichlet boundary conditions [30], obstacle problems [74], nonlocal tug-of-war for the fractional p-Laplacian [72], time dependent equations [52], and variants on the core structure of the game [70].…”