On-Line Balancing of Random Inputs

We consider an online vector balancing game where vectors vt, chosen uniformly at random in -1,+1n, arrive over time and a sign xt ∈ -1,+1 must be picked immediately upon the arrival of vt. The goal is to minimize the L^∞ norm of the signed sum sumt xt vt. We give an online strategy for picking the signs xt that has value O(n1/2) with high probability. Up to constants, this is the best possible even when the vectors are given in advance.