Rising Rested MAB with Linear Drift

We consider non-stationary multi-arm bandit (MAB) where the expected reward of each action follows a linear function of the number of times we executed the action. Our main result is a tight regret bound of $\tilde{\mathrm{\Theta }}(T^{4\mathrm{/}5}{K}^{3\mathrm{/}5})$, by providing both upper and lower bounds. We extend our results to derive instance dependent regret bounds, which depend on the unknown parametrization of the linear drift of the rewards.