  We design differentially private algorithms for the problem of online linear
optimization in the full information and bandit settings with optimal
$\tilde{O}(\sqrt{T})$ regret bounds. In the full-information setting, our
results demonstrate that $\epsilon$-differential privacy may be ensured for
free -- in particular, the regret bounds scale as
$O(\sqrt{T})+\tilde{O}\left(\frac{1}{\epsilon}\right)$. For bandit linear
optimization, and as a special case, for non-stochastic multi-armed bandits,
the proposed algorithm achieves a regret of
$\tilde{O}\left(\frac{1}{\epsilon}\sqrt{T}\right)$, while the previously known
best regret bound was
$\tilde{O}\left(\frac{1}{\epsilon}T^{\frac{2}{3}}\right)$.
