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.

翻译：我们证明存在一种在线算法，对于任意序列的向量 $v_1,\ldots,v_T \in \mathbb{R}^n$ （满足 $\|v_i\|_2 \leq 1$，且按顺序逐个到达），能够决定随机符号 $x_1,\ldots,x_T \in \{ -1,1\}$，使得对于每个 $t \le T$，前缀和 $\sum_{i=1}^t x_iv_i$ 满足 $10$-次高斯性。这一结果改进了 Alweiss、Liu 和 Sawhney 的工作，他们保持前缀和为 $O(\sqrt{\log (nT)})$-次高斯性，并给出了差异 $\max_{t \in T} \|\sum_{i=1}^t x_i v_i\|_\infty$ 的 $O(\sqrt{\log T})$ 上界。我们的证明将 Banaszczyk 前缀平衡结果推广到树形结构，并结合克隆论证以寻找分布而非单一着色。我们还针对一个无对抗性的对手给出了一个匹配的 $\Omega(\sqrt{\log T})$ 策略。