Optimal Online Discrepancy Minimization in Linear Time

Abstract

We provide an online algorithm with the following guarantee: for any fixed sequence of vectors $v_1,\dots,v_T \in \mathbf{R}^d$ with $\|v_i\|_2\le 1$, the algorithm assigns each arriving vector $v_t$ a random sign $\varepsilon_t$ such that every prefix sum $\sum_{i=1}^t \varepsilon_i v_i $ can be written as the sum of three coupled standard Gaussian vectors. Our algorithm runs in $O(dT)$ time and achieves the optimal prefix discrepancy bound \[ \max_{1 \le t \le T}\left\| \sum_{i=1}^t \varepsilon_i v_i \right\|_\infty = O\left( \sqrt{\log T} \right), \] with high probability. This recovers the optimal bound of Kulkarni, Reis, and Rothvoss, whose algorithm runs in time exponential in $T$ and $d$. The algorithm and main proof were discovered in a GPT-5.5 Pro Extended conversation prompted by the author.