Before revision:  Our bounds are optimal up to poly(\log\log n+\log(1/\epsilon)) factors, and improve the 1{\epsilon^3} dependence of Ben-Eliezer , et al.
After revision:  Our bounds are optimal up to poly(\log\log n+\log(1/\epsilon)) factors, and improve the 1{\epsilon^3} dependence of Ben-Eliezer et al.