Average Regret Bounds for Linear Stochastic Bandits
Using the "doubling trick," you can turn your favorite high probability algorithm (e.g. LinUCB) to be an anytime algorithm with expected regret as a function of $T$, usually giving a $\tilde{O}(\sqrt{T})$ dependence without knowing $T$ in advance.

This turns out to be tight, i.e. in Chu et al. (2011), we give lower bounds on expected regret of $\Theta(\sqrt{dT})$ for the $d$-dimensional linear stochastic bandit problem.  Hence, the expected regret, in the worst case, will indeed still be a function of $T$.