BanditAlgorithm.bandit_instance_dependent_lower_bound

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Disproved status carries no information about that problem. Do not import this node or use it as a dependency.

(Instance-dependent asymptotic lower bound, Lai–Robbins style) Let $\mathcal{E} = \mathcal{M}_1 \times \cdots \times \mathcal{M}_k$ be an unstructured class and $\pi$ a consistent policy over $\mathcal{E}$. Then for all $\nu = (P_i)_i \in \mathcal{E}$:

Why this node was retired

The posted statement is

theorem BanditAlgorithm.bandit_instance_dependent_lower_bound {k : ℕ}
    (𝓜 : Fin k → Set (Measure ℝ)) (π : BanditPolicy k)
    (hcons : IsConsistentPolicy {ν : StochasticBandit k | ∀ i, ν.P i ∈ 𝓜 i} π)
    (ν : StochasticBandit k) (hν : ∀ i, ν.P i ∈ 𝓜 i) :
    ∑ i ∈ Finset.univ.filter (fun i ↦ 0 < banditGap ν i),
        ENNReal.ofReal (banditGap ν i) /
          banditDInf (𝓜 i) (ν.P i) (banditOptimalMean ν) ≤
      atTop.liminf
        (fun n : ℕ ↦ ENNReal.ofReal (banditRegret ν π n / Real.log n)) := by
  sorry

Consistency quantifies probability bandit environments, while the KL alternative infimum admits arbitrary measures including measures of mass three.

The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.

Proposed corrected statement

Restrict every arm model class to probability measures with finite first moment, and use the same admissible environment class in consistency and in the KL alternative infimum.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.