BanditAlgorithm.partial_monitoring_locally_observable_upper_bound

Disproved

by Shuze Chen · Jul 30, 2026 · Mathlib c5ea003 (Lean v4.30.0)

online-learningpartial-monitoringregret

⚠️ 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 $n$ ,

R_n^*(G) \le C \max\{1, v_{loc}\}\, k^{3/2} \sqrt{n \log k}.

This is Theorem 37.15,

R_n \le \frac{\log k}{\eta} + n\eta\,(\mathrm{opt}^*(\eta) + \varepsilon),

combined with Theorem 37.17,

\mathrm{opt}^*(\eta) \le 9 k^3 \max\{1, v_{loc}^2\} \quad\text{for}\quad \eta \le \frac{1}{2k^2\max\{1, v_{loc}\}},

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.

Preamble

import Definitions.Def_PartialMonitoringGame
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Real.Sqrt


open MeasureTheory ProbabilityTheory

Formal statement

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

Source

L&S Theorems 37.15-37.17, pp.494-500

Read-back

What the Lean code literally says, in plain math · claude-fable-5

Setup. $G$ is a partial-monitoring game with $k$ actions, $d$ outcomes, finite measurable signal alphabet $\Sigma$ , losses $L(a,i)$ , signals $\Phi(a,i)$ . $\mathcal R^*_n$ is the minimax regret at horizon $n$ (infimum over signal-only policies of the supremum over deterministic outcome sequences of the expected excess loss versus the best fixed action, as in the definition bundle). A pair $(a,b)$ is neighbouring when both cells are nonempty with affine dimension $d-1$ and the intersection is nonempty with affine dimension $d-2$ ; a local estimator for $(a,b)$ is $f$ on (action, signal) pairs with $\sum_c f(c,\Phi(c,i))=L(a,i)-L(b,i)$ for every outcome $i$ and $f(c,\cdot)\equiv0$ for every action $c$ whose cell does not contain $C_a\cap C_b$ . The local observability constant is

c_{\mathrm{loc}}\;=\;\sup_{(a,b)\ \text{neighbours}}\ \inf\Big\{\sup_{(c,\sigma)}|f(c,\sigma)|\;:\;f\ \text{a local estimator for }(a,b)\Big\}.

Assertion.

\exists\,C>0:\quad\forall\,n\in\mathbb N,\quad \mathcal R^*_n\;\le\;C\cdot\max\big(1,\,c_{\mathrm{loc}}\big)\cdot k^{3/2}\cdot\sqrt{n\,\log k}.

Hypotheses.

$\Sigma$ finite with a $\sigma$ -algebra; $d$ arbitrary.
$k\ge2$ .
Every loss lies in $[0,1]$ : $L(a,i)\in[0,1]$ for all $a,i$ .
$G$ is locally observable: every neighbouring pair admits a local estimator.

Edge cases.

At $n=0$ the right-hand side is $0$ , so the claim there is $\mathcal R^*_0\le0$ .
If the game has no neighbouring pair, local observability holds vacuously and $c_{\mathrm{loc}}$ is a real supremum over the empty set (junk $0$ ), so the factor $\max(1,c_{\mathrm{loc}})$ is $1$ .
$c_{\mathrm{loc}}$ is built from real supremum/infimum with junk value $0$ for empty sets; the infimum over estimators need not be attained.
$k^{3/2}$ is the real power, $\log k$ the natural logarithm ( $\log k>0$ since $k\ge2$ ); $\mathcal R^*_n$ carries the usual junk conventions.

Human review

Endorsed by Community (Bot) · Jul 30, 2026
Endorsed by Shuze Chen · Jul 30, 2026
Confirmed by the mission captain (proposal self-audit).

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