LeanMachineLearning

Bandits.UCB.isAlgEnvSeqUntil_roundRobinAlgorithm๐Ÿ”—

Lemma

Until round K - 1, the UCB algorithm behaves like the Round-Robin algorithm.

๐Ÿ”—theorem
Bandits.UCB.isAlgEnvSeqUntil_roundRobinAlgorithm.{u_1} {K : โ„•}
  {hK : 0 < K} {c : โ„} {ฮฝ : ProbabilityTheory.Kernel (Fin K) โ„}
  [ProbabilityTheory.IsMarkovKernel ฮฝ] {ฮฉ : Type u_1}
  {mฮฉ : MeasurableSpace ฮฉ} {P : MeasureTheory.Measure ฮฉ}
  [MeasureTheory.IsProbabilityMeasure P] {A : โ„• โ†’ ฮฉ โ†’ Fin K}
  {R : โ„• โ†’ ฮฉ โ†’ โ„}
  (h :
    Learning.IsAlgEnvSeq A R (ucbAlgorithm hK c)
      (Learning.stationaryEnv ฮฝ) P) :
  Learning.IsAlgEnvSeqUntil A R (Learning.roundRobinAlgorithm hK)
    (Learning.stationaryEnv ฮฝ) P (K - 1)
Bandits.UCB.isAlgEnvSeqUntil_roundRobinAlgorithm.{u_1}
  {K : โ„•} {hK : 0 < K} {c : โ„}
  {ฮฝ : ProbabilityTheory.Kernel (Fin K) โ„}
  [ProbabilityTheory.IsMarkovKernel ฮฝ]
  {ฮฉ : Type u_1} {mฮฉ : MeasurableSpace ฮฉ}
  {P : MeasureTheory.Measure ฮฉ}
  [MeasureTheory.IsProbabilityMeasure P]
  {A : โ„• โ†’ ฮฉ โ†’ Fin K} {R : โ„• โ†’ ฮฉ โ†’ โ„}
  (h :
    Learning.IsAlgEnvSeq A R
      (ucbAlgorithm hK c)
      (Learning.stationaryEnv ฮฝ) P) :
  Learning.IsAlgEnvSeqUntil A R
    (Learning.roundRobinAlgorithm hK)
    (Learning.stationaryEnv ฮฝ) P (K - 1)

Code

lemma isAlgEnvSeqUntil_roundRobinAlgorithm
    (h : IsAlgEnvSeq A R (ucbAlgorithm hK c) (stationaryEnv ฮฝ) P) :
    IsAlgEnvSeqUntil A R (roundRobinAlgorithm hK) (stationaryEnv ฮฝ) P (K - 1) where
  measurable_action
Proof
h.measurable_action
  measurable_feedback := h.measurable_feedback
  hasLaw_action_zero := h.hasLaw_action_zero
  hasCondDistrib_feedback_zero := h.hasCondDistrib_feedback_zero
  hasCondDistrib_action n hn := by
    convert h.hasCondDistrib_action n using 1
    simp only [roundRobinAlgorithm, detAlgorithm_policy, ucbAlgorithm]
    congr 1 with h
    simp [UCB.nextArm, hn]
  hasCondDistrib_feedback n _ := h.hasCondDistrib_feedback n

Actions: Source ยท Open Issue

Meaning last changed in v4.34.0-rc2-14-gf86702d (2026-08-25), the 5th recorded change.

Self-contained, with its dependencies inlined and proofs replaced by sorry: download the raw file ยท open it in the Lean web editor.

Dependency graph

Audit surface: 19 project declarations, 105 external constants

โœ“ Proved: no sorry anywhere in its closure

This is the tool's own reading of one build's recorded axioms, and it is not robust against an author who wants it to pass. Checking meant to be relied on should go through Comparator, which replays the proof through the kernel from an export against an explicit list of permitted axioms.