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