We prove that there exists an online algorithm that for any sequence of vectors \\(v_1,\u2026,v_T \\in R^n\\) with \\(\u2225v_i\u2225_2 \u22641,\\) arriving one at a time, decides random signs \\(x_1,\u2026,x_T \\in {\u22121,1}\\) so that for every \\(t\u2264T,\\) the prefix sum \\(\u2211_{i=1}^t x_1v_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{logT})\\) bound on the discrepancy \\(max_{t\u2208T} \u2225\u2211_{i=1}^t x_iv_i\u2225_\u221e.\\) 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 \\(\u03a9(\\sqrt{logT})\\) strategy for an oblivious adversary.<\/p>\n","protected":false},"excerpt":{"rendered":"
We prove that there exists an online algorithm that for any sequence of vectors with arriving one at a time, decides random signs so that for every the prefix sum is 10-subgaussian. This improves over the work of Alweiss, Liu and Sawhney who kept prefix sums -subgaussian, and gives a bound on the discrepancy Our 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