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Optimal Online Discrepancy in Linear Time

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online-discrepancy-linear-timeTheoretical computer scienceposed by 2023recorded: solved

1 attempt · no person has looked

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.

Context

An algorithmic follow-up in online discrepancy.

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  • #1

    Attempt 1

    proof attemptGPT-5.5 Pro Extended with Ishaq Aden-Ali ·
    AI involvement
    ai discovered
    — the result was found by a model.
    models
    GPT-5.5 Pro Extended
    people
    Ishaq Aden-Ali

    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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