BanditAlgorithm.partial_monitoring_locally_observable_upper_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.

(Easy games, Theorems 37.15–37.17 combined) If $G$ is locally observable with losses in $[0,1]$ and $k \ge 2$ actions, then the exponential-weights policy of Algorithm 26 with tuned learning rate gives a game-dependent constant $C > 0$ such that for every horizon $n$,

This is Theorem 37.15,

combined with Theorem 37.17,

the $\Theta(\sqrt{n})$ upper half of the classification.

Why this node was retired

The posted statement is

theorem BanditAlgorithm.partial_monitoring_locally_observable_upper_bound {k d : ℕ}
    {𝕊 : Type*} [Fintype 𝕊] [MeasurableSpace 𝕊]
    (G : PartialMonitoringGame k d 𝕊) (hk : 2 ≤ k)
    (hL : ∀ a i, G.L a i ∈ Set.Icc (0 : ℝ) 1)
    (hloc : LocallyObservable G) :
    ∃ C : ℝ, 0 < C ∧ ∀ n : ℕ,
      pmMinimaxRegret G n ≤
        C * max 1 (pmLocObsConst G) * (k : ℝ) ^ ((3 : ℝ) / 2) *
          Real.sqrt (n * Real.log k) := by
  sorry

The finite feedback alphabet has an arbitrary sigma-algebra; with the indiscrete one, the policy cannot observe the nominally informative signal.

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

Proposed corrected statement

Add [MeasurableSingletonClass 𝕊] (equivalently the discrete sigma-algebra for this finite signal type), and retain the loss, policy, and local-observability hypotheses.

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.