Exact Universality of Online Discrepancy

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 $M/N\to\alpha\in(0,\infty)$, the optimal value divided by $\sqrt N$ converges to a limit $R_\alpha$ 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 $\kappa>R_\alpha$, we construct a randomized online algorithm whose final signed sum has $\ell^\infty$ norm at most $\kappa\sqrt N$ with high probability; for $\kappa<R_\alpha$, every online algorithm has vanishing success probability. Consequently, the online threshold of the symmetric binary perceptron is universal at every positive margin. The main step is a coupling that transfers Brownian controls to non-Gaussian inputs, while truncation controls rare large entries.