Regret Minimization in Heavy-Tailed Bandits with Unknown Moment Parameters

In a $K$-armed bandit run for $T$ rounds, arm $i$ has reward distribution $\nu_i$ satisfying

$$\mathbb E_{X\sim\nu_i}|X|^{1+\epsilon}\leq u$$

for common but unknown $\epsilon\in(0,1]$ and $u>0$. Write $\mu_i=\mathbb E X$, $\mu^\star=\max_i\mu_i$, $\Delta_i=\mu^\star-\mu_i$, and

$$R_T=\sum_{t=1}^T(\mu^\star-\mu_{I_t}).$$

1. What is the best regret rate that can be incurred in the heavy-tailed multi-armed bandit problem without any knowledge of $\epsilon$ and $u$ and without any additional assumption? This requires a minimax lower bound and a matching upper bound.

2. Does there exist an algorithm that can achieve such a bound without knowing $\epsilon$ and $u$ and without any additional assumption?

3. Is there an assumption that is better than the others, or that encompasses all of the distributions for which adaptation comes at no additional cost?