Greedy Vector Balancing (opens in new tab)

In online vector balancing, vectors $t_1,\dots,t_n$ arrive one by one from a given set $T$ and the goal is to assign signs $s_1,\dots,s_n\in\{\pm1\}$ in an online manner so as to minimize the largest norm of any signed prefix sum $\sum_{i=1}^ks_i t_i$, $k \in [n]$. In this paper, we analyze the natural Euclidean greedy vector balancing algorithm for this problem: at each step $k$, the sign $s_k\in\{\pm1\}$ is chosen so that $s_k t_k$ has non-pos...