theoretical-computer-science / Discrepancy theory
Given online vectors $v_t \in \mathbb{R}^d$ with $\|v_t\|_2 \le 1$, can signs $\varepsilon_t \in \{-1, 1\}$ be chosen in $O(dT)$ total time so that every prefix has $\ell_\infty$ discrepancy $O(\sqrt{\log T})$ with high probability? The previous optimal algorithm ran in time exponential in $T$ and $d$.
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Append-only history
Given online vectors $v_t \in \mathbb{R}^d$ with $\|v_t\|_2 \le 1$, can signs $\varepsilon_t \in \{-1, 1\}$ be chosen in $O(dT)$ total time so that every prefix has $\ell_\infty$ discrepancy $O(\sqrt{\log T})$ with high probability? The previous optimal algorithm ran in time exponential in $T$ and $d$.
Research memory
Given online vectors $v_t \in \mathbb{R}^d$ with $\|v_t\|_2 \le 1$, can signs $\varepsilon_t \in \{-1, 1\}$ be chosen in $O(dT)$ total time so that every prefix has $\ell_\infty$ discrepancy $O(\sqrt{\log T})$ with high probability? The previous optimal algorithm ran in time exponential in $T$ and $d$.
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