Optimal Online Discrepancy in Linear Time

Given online vectors $v_t \in \mathbb{R}^d$ with $\|v_t\|_2 \le 1$, can signs $\varepsilon_t \in \{-1, 1\}$ be chosen in $O(dT)$ total time so that every prefix has $\ell_\infty$ discrepancy $O(\sqrt{\log T})$ with high probability? The previous optimal algorithm ran in time exponential in $T$ and $d$.

Result

Proved

Status

Resolved

AI contribution

AI-discovered

Method

Argument

Field

Discrepancy theory

Posed by

—

Year posed

2023

Years open

3y

Solved

2026-07-06

Model

GPT-5.5 Pro Extended

Vendor

OpenAI

Collaborators

Ishaq Aden-Ali

Verification

Unreviewed

Publication

Preprint

Significance

10 / 100*

Disclosed cost

—

Wikipedia

No dedicated article