Year: 2026
Type: preprint
Abstract: We study online vector balancing with $N$ random vectors in $\mathbb{R}^M$ revealed sequentially, where each vector must be assigned an irrevocable sign upon arrival. The goal is to minimize the expected $\ell^\infty$ norm of the final signed sum. For i.i.d. entries with mean zero, variance one, and a finite fourth moment, we prove that, as $M/N\toα\in(0,\infty)$, the optimal value divided by $\sqrt N$ converges to a limit $R_α$ independent of the entry distribution. This limit is the stochastic control value identified for Gaussian inputs by Fiedler, Jackson, Lacker, and Niles-Weed. In particular, it determines the exact asymptotic optimum for Rademacher inputs. For every $κ>R_α$, we construct a randomized online algorithm whose final signed sum has $\ell^\infty$ norm at most $κ\sqrt N$ with high probability; for $κ
Source: arXiv (Cornell University)
Authors: Jo, Sunghyeon, Lee, Taekyun
Language: English
Subfield: Theoretical Computer Science
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7216042543