Optimal Online Discrepancy Minimization

We prove that there exists an online algorithm that for any sequence of vectors v1,ldots,vT ∈ ℝn with |vi|2 ≤ 1, arriving one at a time, decides random signs x1,ldots,xT ∈ -1,1 so that for every t ≤ T, the prefix sum sumi=1t xivi is 10-subgaussian. This improves over the work of Alweiss, Liu and Sawhney who kept prefix sums O(√log (nT))-subgaussian, and gives a O(√log T) bound on the discrepancy maxt ∈ T |sumi=1t xi vi|_∞. 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(√log T) strategy for an oblivious adversary.