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 $v_1,\u2026,v_T \\in R^n$ 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 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