Optimal Online Discrepancy Minimization

08/02/2023

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by   Janardhan Kulkarni, et al.

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We prove that there exists an online algorithm that for any sequence of vectors v_1,…,v_T ∈ℝ^n with v_i_2 ≤ 1, arriving one at a time, decides random signs x_1,…,x_T ∈{ -1,1} so that for every t ≤ T, the prefix 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(√(log (nT)))-subgaussian, and gives a O(√(log T)) bound on the discrepancy max_t ∈ T∑_i=1^t x_i v_i_∞. 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(√(log T)) strategy for an oblivious adversary.