Some remarks on the Gram-Schmidt walk algorithm and consequences for Komlos conjecture

In this paper we improve the best known constant for the discrepancy formulated in the Komlos Conjecture. The result is based on the improvement of the subgaussian bound for the random vector constructed in the Gram-Schmidt Random Walk algorithm. Moreover, we present detailed argument for the smoothed analysis of this random vector. The analysis concerns a modification of a given matrix in the conjecture by a Gaussian type perturbation. Our result improves the recent paper in this direction.