Optimal Online Discrepancy Minimization

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.