On the Komlós Conjecture

A conjecture of Komlós states that the discrepancy of any collection of unit vectors is $O(1)$ — i.e., for any matrix $A$ with $n$ unit length columns, there is a vector $x$ with $-1,1$ entries such that $\|Ax\|_\infty = O(1)$. In his talk from the workshop on The Role of TCS in Modern Machine Learning, Nikhil Bansal (University of Michigan) describes an $O((\log n)^{1/4})$ bound for the Komlós problem, improving upon the long-standing $O((\log n)^{1/2})$ bound due to Banaszczyk.