# On-Line Balancing of Random Inputs

## TL;DR

This paper introduces an online strategy for vector balancing with random inputs that achieves an $O(
oot n)$ bound on the maximum coordinate sum, matching the best possible even with full knowledge of the vectors.

## Contribution

The authors develop an online sign assignment method for random vectors that attains near-optimal bounds, advancing understanding of online vector balancing.

## Key findings

- Achieves $O(
oot n)$ bound with high probability
- Optimal up to constant factors for random vectors
- Provides a strategy matching offline best possible bounds

## Abstract

We consider an online vector balancing game where vectors $v_t$, chosen uniformly at random in $\{-1,+1\}^n$, arrive over time and a sign $x_t \in \{-1,+1\}$ must be picked immediately upon the arrival of $v_t$. The goal is to minimize the $L^\infty$ norm of the signed sum $\sum_t x_t v_t$. We give an online strategy for picking the signs $x_t$ that has value $O(n^{1/2})$ with high probability. Up to constants, this is the best possible even when the vectors are given in advance.