Optimal Online Discrepancy Minimization

TL;DR

This paper introduces an online algorithm that assigns signs to vectors to keep prefix sums highly subgaussian, significantly improving discrepancy bounds over previous methods and matching lower bounds for oblivious adversaries.

Contribution

The authors develop a novel online discrepancy minimization algorithm with improved subgaussian guarantees, extending Banaszczyk's result to trees and providing tight bounds.

Findings

Achieves $10$-subgaussian prefix sums for any vector sequence.

Improves discrepancy bounds to $O(\sqrt{\log T})$ for the infinity norm.

Provides a matching lower bound for oblivious adversaries.

Abstract

We prove that there exists an online algorithm that for any sequence of vectors $v_1,\ldots,v_T \in \mathbb{R}^n$ with $\|v_i\|_2 \leq 1$, arriving one at a time, decides random signs $x_1,\ldots,x_T \in \{ -1,1\}$ so that for every $t \le T$, the prefix sum $\sum_{i=1}^t x_iv_i$ is $10$-subgaussian. This improves over the work of Alweiss, Liu and Sawhney who kept prefix sums $O(\sqrt{\log (nT)})$-subgaussian, and gives a $O(\sqrt{\log T})$ bound on the discrepancy $\max_{t \in T} \|\sum_{i=1}^t x_i v_i\|_\infty$. Our proof combines a generalization of Banaszczyk's prefix balancing result to trees with a cloning argument to find distributions rather than single colorings. We also show a matching $\Omega(\sqrt{\log T})$ strategy for an oblivious adversary.