Appendix A. Proof of Theorem

Here we provide a proof for Theorem 1. Theorem 1. Now, it suffices to prove Eqs. (6), (8) and (9). This completes the proof of Eq. (8). Before we proceed to bound the total regret an prove Eq. This completes the proof of Eq. (9). By applying Hölder's inequality, we have for any a, b R The proof of this subcase is similar to subcase 2.2 and is omitted.