Online Koml\'os converges to mean curvature flow

Abstract

We determine the asymptotics of a game inspired by classic vector balancing problems in combinatorial discrepancy theory. In this game, which we call the online Koml\'os game, two players, Paul and Carol, update the state vector $y$ in $\mathbb{R}^m$, initially placed at $0$. At each round, Paul chooses freely a set of $n$ vectors in the Euclidean unit ball, and Carol chooses, for each such vector, whether to leave it unchanged or reverse its sign. The resulting vectors are all added to $y$, and the game proceeds to a new round. After $T$ rounds, the game ends, and the $\ell_\infty$ norm of the state vector $y$ is determined. Paul's objective throughout the game is to maximize this norm, and Carol's objective is to minimize it. As $T$ gets large, we establish that the leading order term of the value of this game is $\sqrt{T/2\tau}$, where $\tau$ is the extinction time of the unit cube in $\mathbb{R}^m$ under a curvature-based flow characterized by the values of $m$ and $n$. When $n\geq m-1$, this flow is the mean curvature flow, and we show that $1/\sqrt{2\tau} =\Theta(\sqrt{\log m})$. Our results build upon the work of Kohn and Serfaty on deterministic games and mean curvature flow, combined with Banaszczyk's $\ell^2$ analogue of the Beck-Fiala theorem. As the large $T$ limit of the online Koml\'os game amounts to a localization of the classic Koml\'os problem, we hope this work can shed light on this and other vector balancing problems. Our results generalize to the version of the online Koml\'os game with the final value given by an arbitrary norm in $\mathbb{R}^m$.