LeanMachineLearning exposition

Bandits.gap_nonneg_of_le🔗

This page has the declaration's own card below, then its dependency graph, then a card for each dependency (type dependencies first, then the rest of the transitive closure). For a theorem, the graph and the dependency cards only follow its statement's dependencies (its proof is replaced by sorry, so what it proves doesn't depend on how); for everything else, both the type and the body/value are followed, since their content is part of what later declarations build on.

Minimal Lean file

gap_nonneg_of_le🔗

LemmaBandits.gap_nonneg_of_le
Details

The gap is non-negative if the means are bounded by u : ℝ (even if 𝓐 is not Finite).

🔗theorem
Bandits.gap_nonneg_of_le.{u_1} {𝓐 : Type u_1} {m𝓐 : MeasurableSpace 𝓐}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ} {a : 𝓐} {u : ℝ}
  (h : ∀ (a : 𝓐), ∫ (x : ℝ), id x ∂ν a ≤ u) : 0 ≤ gap ν a
Bandits.gap_nonneg_of_le.{u_1}
  {𝓐 : Type u_1} {m𝓐 : MeasurableSpace 𝓐}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ}
  {a : 𝓐} {u : ℝ}
  (h :
    ∀ (a : 𝓐), ∫ (x : ℝ), id x ∂ν a ≤ u) :
  0 ≤ gap ν a

Code

lemma gap_nonneg_of_le {u : ℝ} (h : ∀ a, (ν a)[id] ≤ u) : 0 ≤ gap ν a
Type uses (1)
Used by (1)

Actions: Source · Open Issue

Proof
by
  rw [gap, sub_nonneg]
  exact le_ciSup ⟨u, Set.forall_mem_range.2 h⟩ a

Dependency graph

Type dependencies (1)

gap🔗

DefinitionBandits.gap
Details

Gap of an action a: difference between the highest mean of the actions and the mean of a.

🔗def
Bandits.gap.{u_1} {𝓐 : Type u_1} {m𝓐 : MeasurableSpace 𝓐}
  (ν : ProbabilityTheory.Kernel 𝓐 ℝ) (a : 𝓐) : ℝ
Bandits.gap.{u_1} {𝓐 : Type u_1}
  {m𝓐 : MeasurableSpace 𝓐}
  (ν : ProbabilityTheory.Kernel 𝓐 ℝ)
  (a : 𝓐) : ℝ

Code

noncomputable
def gap (ν : Kernel 𝓐 ℝ) (a : 𝓐) : ℝ := (⨆ i, (ν i)[id]) - (ν a)[id]
Used by (27)

Actions: Source · Open Issue