Conj3.2 [partially_resolved] Conjecture 3.2 (almost instance-wise optimality): there is a δ-correct algorithm for Best-1-Arm (n unit-variance Gaussian arms, unique best arm) whose expected sample complexity is O(L(I,δ) + Δ_2^{-2} ln ln Δ_2^{-1}) for every instance I and every δ < 0.1, where L(I,δ) is the order-oblivious instance-wise lower bound of Definition 3.1 (infimum over δ-correct algorithms of the average of T_A(π∘I) o | remains open: Construct a δ-correct algorithm whose expected sample complexity is O(L(I,δ) + Δ_[2]^{-2} ln ln Δ_[2]^{-1}) for every instance (removing the polylog(n,1/δ) factor on the additive term in Chen-Li-Qiao's Theorem 1.11), and certify the main term against L(I,δ) for arbitrary (non-monotone) δ-correct alg
Conj3.5 [partially_resolved] Conjecture 3.5 (gap-entropy conjecture): for every Best-1-Arm instance I (n unit-variance Gaussian arms, unique best arm) and every δ < 0.1, L(I,δ) = Θ( Σ_{i=2}^n Δ_[i]^{-2} · (ln δ^{-1} + Ent(I)) ), where Ent(I) = Σ_{G_k≠∅} p_k log p_k^{-1} is the Shannon entropy of p_k = H_k/Σ_j H_j, H_k = Σ_{i∈G_k} Δ_[i]^{-2}, G_k = {i ∈ [2,n] : 2^{-k} ≤ Δ_[i] < 2^{-k+1}}; the authors single out the lower-bound | remains open: Prove (or refute) L(I,δ) = Ω(H(I)·(ln δ^{-1} + Ent(I))) for every instance I and every δ-correct algorithm on I itself (not merely on a sub-instance and not merely for monotone algorithms), including non-dyadic gap profiles and the range 0.01 ≤ δ < 0.1; on the upper side, show L(I,δ) = O(H(I)(ln δ^{