#dual-norm

Question 1: Single choice

The negative entropy is 1-strongly convex on the simplex with respect to both the $\ell_2$ and the $\ell_1$ norm. The regret bound of (optimistic) multiplicative weights involves the strong-convexity norm and its dual norm applied to the utility vectors. Which choice gives the stronger bound?

$\ell_1$ : the dual norm $\ell_\infty$ of the utilities is never larger than their $\ell_2$ norm1
$\ell_2$ , because the $\ell_2$ norm of a utility vector is never larger than its $\ell_\infty$ norm2
Both give the same bound, because the strong-convexity constant is 1 in both cases3
$\ell_2$ , because its unit ball is smaller than the $\ell_1$ unit ball4

Timed test