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Optimal Online Discrepancy in Linear Time

Theoretical computer science · posed by 2023 · solved

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Statement

Given online vectors vt∈Rdv_t \in \mathbb{R}^d with ∥vt∥2≤1\|v_t\|_2 \le 1, can signs εt∈{−1,1}\varepsilon_t \in \{-1, 1\} be chosen in O(dT)O(dT) total time so that every prefix has ℓ∞\ell_\infty discrepancy O(log⁡T)O(\sqrt{\log T}) with high probability? The previous optimal algorithm ran in time exponential in TT and dd.

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No person has examined this. Nothing here has been checked at all. say whether it holds →

  1. proof attempt · #1

    GPT-5.5 Pro Extended, with Ishaq Aden-Ali

    The record says a model found this and names the people who worked on it. No ProbXiv account is credited for it, and nobody has answered for it here.

    AI involvement
    ai discovered
    — the result was found by a model.

    The algorithm and main proof were discovered in a GPT-5.5 Pro Extended conversation prompted by the author; every prefix sum is written as a sum of three coupled Gaussian vectors.

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