Optimal Best Arm Identification with Fixed Budget

Let $\mathcal S$ be a class of $k$-armed bandit instances with a unique best arm. A fixed-budget algorithm adaptively selects $n$ samples and outputs $\widehat I_n$; write $p_{\mu,n}=\Pr_\mu(\widehat I_n\neq I^*(\mu))$.

Problem 1.

Do there exist a natural algorithm class $\mathcal A$ and a function $\Gamma^*_{\mathrm{fb}}\colon\mathcal S\to\mathbb R$ such that every algorithm in $\mathcal A$ satisfies

$$\liminf_{n\to\infty}\frac{n}{\log(1/p_{\mu,n})} \geq\Gamma^*_{\mathrm{fb}}(\mu) \qquad(\mu\in\mathcal S),$$

and some algorithm in $\mathcal A$ satisfies the matching upper bound with $\limsup$? The class should be more informative than mere consistency; one candidate contains algorithms that are uniformly no worse than uniform sampling.

Problem 2.

Is there an algorithm other than uniform sampling itself that performs uniformly no worse than uniform sampling in the fixed-budget setting, for every $\mu\in\mathcal S$, in terms of the asymptotic error exponent $\limsup_{n\to\infty}n/\log(1/p_{\mu,n})$?