Victor Reis, Institute for Advanced Study

Abstract

We prove that there exists an online algorithm that for any sequence of vectors v_1, ..., v_T in R^n of Euclidean norm at most 1, arriving one at a time, decides random signs x_1,..., x_T ∈ {−1,1} so that for every t ≤ T, their signed prefix sum 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 le 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 Ω(sqrt(log T)) strategy for an oblivious adversary.

See: https://theory.cs.rutgers.edu/theory_seminar