I am reading this paper on linear stochastic bandits :
http://papers.nips.cc/paper/4417-improved-algorithms-for-linear-stochastic-bandits.pdf
All the results are stated in a high-probability setting, that is $\delta$ is a parameter to the algorithm , the regret at time $T$ is such that ,
\begin{align*} \mathbb{P}(R_T > f(T) ) \geq 1 - \delta \end{align*}
Is there any resource where the bounds are obtained in the form $\mathbb{E}[R_T] \leq f(T) $ ?
Trivially replacing $\delta = \frac{1}{T}$, does not give an anytime algorithm, that is the bounds don't hold for all $t > 0$.