Online.Bandit.Regret

`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)

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`gap_nonneg`🔗

LemmaBandits.gap_nonneg

Details

No docstring.

theorem

Bandits.gap_nonneg.{u_1} {𝓐 : Type u_1} {m𝓐 : MeasurableSpace 𝓐}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ} {a : 𝓐} [Finite 𝓐] : 0 ≤ gap ν a
Bandits.gap_nonneg.{u_1} {𝓐 : Type u_1}
  {m𝓐 : MeasurableSpace 𝓐}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ}
  {a : 𝓐} [Finite 𝓐] : 0 ≤ gap ν a

Code

lemma gap_nonneg [Finite 𝓐] : 0 ≤ gap ν a

Type uses (1)

gap

Used by (5)

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Proof

by
  rw [gap, sub_nonneg]
  exact le_ciSup (f := fun i ↦ (ν i)[id]) (by simp) a

`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)

gap

Used by (1)

gap_nonneg_of_le

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Proof

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

`gap_le_of_mem_Icc`🔗

LemmaBandits.gap_le_of_mem_Icc

Details

No docstring.

theorem

Bandits.gap_le_of_mem_Icc.{u_1} {𝓐 : Type u_1} {m𝓐 : MeasurableSpace 𝓐}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ} {a : 𝓐} [Nonempty 𝓐] {l u : ℝ}
  (h : ∀ (a : 𝓐), ∫ (x : ℝ), id x ∂ν a ∈ Set.Icc l u) : gap ν a ≤ u - l
Bandits.gap_le_of_mem_Icc.{u_1}
  {𝓐 : Type u_1} {m𝓐 : MeasurableSpace 𝓐}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ}
  {a : 𝓐} [Nonempty 𝓐] {l u : ℝ}
  (h :
    ∀ (a : 𝓐),
      ∫ (x : ℝ), id x ∂ν a ∈
        Set.Icc l u) :
  gap ν a ≤ u - l

Code

lemma gap_le_of_mem_Icc [Nonempty 𝓐] {l u : ℝ} (h : ∀ a, (ν a)[id] ∈ Set.Icc l u) :
    gap ν a ≤ u - l

Type uses (1)

gap

Used by (1)

gap_le_of_mem_Icc

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Proof

by
  grind [gap, ciSup_le (fun i ↦ (h i).2)]

`regret`🔗

DefinitionBandits.regret

Details

Regret of a sequence of pulls k : ℕ → 𝓐 at time t for the reward kernel ν ; Kernel 𝓐 ℝ.

def

Bandits.regret.{u_1, u_2} {𝓐 : Type u_1} {Ω : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} (ν : ProbabilityTheory.Kernel 𝓐 ℝ)
  (A : ℕ → Ω → 𝓐) (t : ℕ) (ω : Ω) : ℝ
Bandits.regret.{u_1, u_2} {𝓐 : Type u_1}
  {Ω : Type u_2} {m𝓐 : MeasurableSpace 𝓐}
  (ν : ProbabilityTheory.Kernel 𝓐 ℝ)
  (A : ℕ → Ω → 𝓐) (t : ℕ) (ω : Ω) : ℝ

Code

noncomputable
def regret (ν : Kernel 𝓐 ℝ) (A : ℕ → Ω → 𝓐) (t : ℕ) (ω : Ω) : ℝ :=
  t * (⨆ a, (ν a)[id]) - ∑ s ∈ range t, (ν (A s ω))[id]

Used by (11)

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`regret_eq_sum_gap`🔗

LemmaBandits.regret_eq_sum_gap

Details

No docstring.

theorem

Bandits.regret_eq_sum_gap.{u_1, u_2} {𝓐 : Type u_1} {Ω : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {ν : ProbabilityTheory.Kernel 𝓐 ℝ}
  {A : ℕ → Ω → 𝓐} {ω : Ω} {t : ℕ} :
  regret ν A t ω = ∑ s ∈ Finset.range t, gap ν (A s ω)
Bandits.regret_eq_sum_gap.{u_1, u_2}
  {𝓐 : Type u_1} {Ω : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ}
  {A : ℕ → Ω → 𝓐} {ω : Ω} {t : ℕ} :
  regret ν A t ω =
    ∑ s ∈ Finset.range t, gap ν (A s ω)

Code

lemma regret_eq_sum_gap : regret ν A t ω = ∑ s ∈ range t, gap ν (A s ω)

Type uses (2)

Used by (3)

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Proof

by
  simp [regret, gap]

`regret_nonneg`🔗

LemmaBandits.regret_nonneg

Details

No docstring.

theorem

Bandits.regret_nonneg.{u_1, u_2} {𝓐 : Type u_1} {Ω : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {ν : ProbabilityTheory.Kernel 𝓐 ℝ}
  {A : ℕ → Ω → 𝓐} {ω : Ω} {t : ℕ} [Finite 𝓐] : 0 ≤ regret ν A t ω
Bandits.regret_nonneg.{u_1, u_2}
  {𝓐 : Type u_1} {Ω : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ}
  {A : ℕ → Ω → 𝓐} {ω : Ω} {t : ℕ}
  [Finite 𝓐] : 0 ≤ regret ν A t ω

Code

lemma regret_nonneg [Finite 𝓐] : 0 ≤ regret ν A t ω

Type uses (1)

regret

Body uses (3)

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Proof

by
  rw [regret_eq_sum_gap]
  exact sum_nonneg (fun _ _ ↦ gap_nonneg)

`gap_eq_zero_of_regret_eq_zero`🔗

LemmaBandits.gap_eq_zero_of_regret_eq_zero

Details

No docstring.

theorem

Bandits.gap_eq_zero_of_regret_eq_zero.{u_1, u_2} {𝓐 : Type u_1}
  {Ω : Type u_2} {m𝓐 : MeasurableSpace 𝓐}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ} {A : ℕ → Ω → 𝓐} {ω : Ω} {t : ℕ}
  [Finite 𝓐] (hr : regret ν A t ω = 0) {s : ℕ} (hs : s < t) :
  gap ν (A s ω) = 0
Bandits.gap_eq_zero_of_regret_eq_zero.{u_1,
    u_2}
  {𝓐 : Type u_1} {Ω : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ}
  {A : ℕ → Ω → 𝓐} {ω : Ω} {t : ℕ}
  [Finite 𝓐] (hr : regret ν A t ω = 0)
  {s : ℕ} (hs : s < t) : gap ν (A s ω) = 0

Code

lemma gap_eq_zero_of_regret_eq_zero [Finite 𝓐] (hr : regret ν A t ω = 0) {s : ℕ} (hs : s < t) :
    gap ν (A s ω) = 0

Type uses (2)

Body uses (2)

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Proof

by
  rw [regret_eq_sum_gap] at hr
  exact (sum_eq_zero_iff_of_nonneg fun _ _ ↦ gap_nonneg).1 hr s (mem_range.2 hs)

`regret_eq_sum_pullCount_mul_gap`🔗

LemmaBandits.regret_eq_sum_pullCount_mul_gap

Details

No docstring.

theorem

Bandits.regret_eq_sum_pullCount_mul_gap.{u_1, u_2} {𝓐 : Type u_1}
  {Ω : Type u_2} [DecidableEq 𝓐] {m𝓐 : MeasurableSpace 𝓐}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ} {A : ℕ → Ω → 𝓐} {ω : Ω} {t : ℕ}
  [Fintype 𝓐] :
  regret ν A t ω = ∑ a, ↑(Learning.pullCount A a t ω) * gap ν a
Bandits.regret_eq_sum_pullCount_mul_gap.{u_1,
    u_2}
  {𝓐 : Type u_1} {Ω : Type u_2}
  [DecidableEq 𝓐] {m𝓐 : MeasurableSpace 𝓐}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ}
  {A : ℕ → Ω → 𝓐} {ω : Ω} {t : ℕ}
  [Fintype 𝓐] :
  regret ν A t ω =
    ∑ a,
      ↑(Learning.pullCount A a t ω) *
        gap ν a

Code

lemma regret_eq_sum_pullCount_mul_gap [Fintype 𝓐] :
    regret ν A t ω = ∑ a, pullCount A a t ω * gap ν a

Type uses (3)

Body uses (2)

Used by (2)

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Proof

by
  simp_rw [regret_eq_sum_gap, sum_pullCount_mul]

`integral_regret_eq_sum_gap_mul_integral_pullCount`🔗

LemmaBandits.integral_regret_eq_sum_gap_mul_integral_pullCount

Details

No docstring.

theorem

Bandits.integral_regret_eq_sum_gap_mul_integral_pullCount.{u_1, u_2}
  {𝓐 : Type u_1} {Ω : Type u_2} [DecidableEq 𝓐] {m𝓐 : MeasurableSpace 𝓐}
  {mΩ : MeasurableSpace Ω} {ν : ProbabilityTheory.Kernel 𝓐 ℝ}
  {A : ℕ → Ω → 𝓐} {n : ℕ} [StandardBorelSpace 𝓐] [Fintype 𝓐]
  {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P]
  (hA : ∀ (n : ℕ), Measurable (A n)) :
  ∫ (x : Ω), regret ν A n x ∂P =
    ∑ a,
      gap ν a * ∫ (x : Ω), (fun ω => ↑(Learning.pullCount A a n ω)) x ∂P
Bandits.integral_regret_eq_sum_gap_mul_integral_pullCount.{u_1,
    u_2}
  {𝓐 : Type u_1} {Ω : Type u_2}
  [DecidableEq 𝓐] {m𝓐 : MeasurableSpace 𝓐}
  {mΩ : MeasurableSpace Ω}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ}
  {A : ℕ → Ω → 𝓐} {n : ℕ}
  [StandardBorelSpace 𝓐] [Fintype 𝓐]
  {P : MeasureTheory.Measure Ω}
  [MeasureTheory.IsProbabilityMeasure P]
  (hA : ∀ (n : ℕ), Measurable (A n)) :
  ∫ (x : Ω), regret ν A n x ∂P =
    ∑ a,
      gap ν a *
        ∫ (x : Ω),
          (fun ω =>
              ↑(Learning.pullCount A a n
                  ω))
            x ∂P

Code

lemma integral_regret_eq_sum_gap_mul_integral_pullCount
    [StandardBorelSpace 𝓐] [Fintype 𝓐] {P : Measure Ω} [IsProbabilityMeasure P]
    (hA : ∀ n, Measurable (A n)) :
    P[regret ν A n] = ∑ a, gap ν a * P[fun ω ↦ (pullCount A a n ω : ℝ)]

Type uses (3)

Body uses (2)

Used by (1)

integral_regret_le_of_forall_integral_pullCount_le

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Proof

by
  simp_rw [regret_eq_sum_pullCount_mul_gap]
  rw [integral_finsetSum]
  swap; · exact fun i _ ↦ (integrable_pullCount hA i n).mul_const _
  congr with a
  rw [integral_mul_const, mul_comm]

`integral_regret_le_of_forall_integral_pullCount_le`🔗

LemmaBandits.integral_regret_le_of_forall_integral_pullCount_le

Details

To bound the expected regret, it suffices to bound the expected number of pulls for each action with positive gap.

theorem

Bandits.integral_regret_le_of_forall_integral_pullCount_le.{u_1, u_2}
  {𝓐 : Type u_1} {Ω : Type u_2} [DecidableEq 𝓐] {m𝓐 : MeasurableSpace 𝓐}
  {mΩ : MeasurableSpace Ω} {ν : ProbabilityTheory.Kernel 𝓐 ℝ}
  {A : ℕ → Ω → 𝓐} {R : ℕ → Ω → ℝ} {n : ℕ} [StandardBorelSpace 𝓐]
  [Fintype 𝓐] {P : MeasureTheory.Measure Ω}
  [MeasureTheory.IsProbabilityMeasure P] {alg : Learning.Algorithm 𝓐 ℝ}
  {env : Learning.Environment 𝓐 ℝ} {B : 𝓐 → ℝ}
  (h : Learning.IsAlgEnvSeq A R alg env P)
  (h_le :
    ∀ (a : 𝓐),
      gap ν a ≠ 0 → ∫ (ω : Ω), ↑(Learning.pullCount A a n ω) ∂P ≤ B a) :
  ∫ (x : Ω), regret ν A n x ∂P ≤ ∑ a, gap ν a * B a
Bandits.integral_regret_le_of_forall_integral_pullCount_le.{u_1,
    u_2}
  {𝓐 : Type u_1} {Ω : Type u_2}
  [DecidableEq 𝓐] {m𝓐 : MeasurableSpace 𝓐}
  {mΩ : MeasurableSpace Ω}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ}
  {A : ℕ → Ω → 𝓐} {R : ℕ → Ω → ℝ} {n : ℕ}
  [StandardBorelSpace 𝓐] [Fintype 𝓐]
  {P : MeasureTheory.Measure Ω}
  [MeasureTheory.IsProbabilityMeasure P]
  {alg : Learning.Algorithm 𝓐 ℝ}
  {env : Learning.Environment 𝓐 ℝ}
  {B : 𝓐 → ℝ}
  (h : Learning.IsAlgEnvSeq A R alg env P)
  (h_le :
    ∀ (a : 𝓐),
      gap ν a ≠ 0 →
        ∫ (ω : Ω),
            ↑(Learning.pullCount A a n
                ω) ∂P ≤
          B a) :
  ∫ (x : Ω), regret ν A n x ∂P ≤
    ∑ a, gap ν a * B a

Code

lemma integral_regret_le_of_forall_integral_pullCount_le
    [StandardBorelSpace 𝓐] [Fintype 𝓐] {P : Measure Ω} [IsProbabilityMeasure P]
    {alg : Algorithm 𝓐 ℝ} {env : Environment 𝓐 ℝ} {B : 𝓐 → ℝ}
    (h : IsAlgEnvSeq A R alg env P)
    (h_le : ∀ a, gap ν a ≠ 0 → ∫ ω, (pullCount A a n ω : ℝ) ∂P ≤ B a) :
    P[regret ν A n] ≤ ∑ a, gap ν a * B a

Type uses (6)

Body uses (2)

Used by (2)

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Proof

by
  have hA := h.measurable_action
  rw [integral_regret_eq_sum_gap_mul_integral_pullCount hA]
  gcongr 1 with a
  by_cases h_gap : gap ν a = 0
  · simp [h_gap]
  gcongr
  · exact gap_nonneg
  · exact h_le a h_gap

`bestArm`🔗

DefinitionBandits.bestArm

Details

action with the highest mean.

def

Bandits.bestArm.{u_1} {𝓐 : Type u_1} {m𝓐 : MeasurableSpace 𝓐}
  [Fintype 𝓐] [Nonempty 𝓐] (ν : ProbabilityTheory.Kernel 𝓐 ℝ) : 𝓐
Bandits.bestArm.{u_1} {𝓐 : Type u_1}
  {m𝓐 : MeasurableSpace 𝓐} [Fintype 𝓐]
  [Nonempty 𝓐]
  (ν : ProbabilityTheory.Kernel 𝓐 ℝ) : 𝓐

Code

noncomputable def bestArm (ν : Kernel 𝓐 ℝ) : 𝓐 :=
  (exists_max_image univ (fun a ↦ (ν a)[id]) (univ_nonempty_iff.mpr inferInstance)).choose

Used by (18)

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`le_bestArm`🔗

LemmaBandits.le_bestArm

Details

No docstring.

theorem

Bandits.le_bestArm.{u_1} {𝓐 : Type u_1} {m𝓐 : MeasurableSpace 𝓐}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ} [Fintype 𝓐] [Nonempty 𝓐] (a : 𝓐) :
  ∫ (x : ℝ), id x ∂ν a ≤ ∫ (x : ℝ), id x ∂ν (bestArm ν)
Bandits.le_bestArm.{u_1} {𝓐 : Type u_1}
  {m𝓐 : MeasurableSpace 𝓐}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ}
  [Fintype 𝓐] [Nonempty 𝓐] (a : 𝓐) :
  ∫ (x : ℝ), id x ∂ν a ≤
    ∫ (x : ℝ), id x ∂ν (bestArm ν)

Code

lemma le_bestArm (a : 𝓐) : (ν a)[id] ≤ (ν (bestArm ν))[id]

Type uses (1)

bestArm

Used by (2)

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Proof

(exists_max_image univ (fun a ↦ (ν a)[id])
    (univ_nonempty_iff.mpr inferInstance)).choose_spec.2 _ (mem_univ a)

`gap_eq_bestArm_sub`🔗

LemmaBandits.gap_eq_bestArm_sub

Details

No docstring.

theorem

Bandits.gap_eq_bestArm_sub.{u_1} {𝓐 : Type u_1} {m𝓐 : MeasurableSpace 𝓐}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ} {a : 𝓐} [Fintype 𝓐] [Nonempty 𝓐] :
  gap ν a = ∫ (x : ℝ), id x ∂ν (bestArm ν) - ∫ (x : ℝ), id x ∂ν a
Bandits.gap_eq_bestArm_sub.{u_1}
  {𝓐 : Type u_1} {m𝓐 : MeasurableSpace 𝓐}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ}
  {a : 𝓐} [Fintype 𝓐] [Nonempty 𝓐] :
  gap ν a =
    ∫ (x : ℝ), id x ∂ν (bestArm ν) -
      ∫ (x : ℝ), id x ∂ν a

Code

lemma gap_eq_bestArm_sub : gap ν a = (ν (bestArm ν))[id] - (ν a)[id]

Type uses (2)

Body uses (1)

le_bestArm

Used by (4)

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Proof

by
  rw [gap]
  congr
  refine le_antisymm ?_ (le_ciSup (f := fun a ↦ (ν a)[id]) (by simp) (bestArm ν))
  exact ciSup_le le_bestArm

`gap_bestArm`🔗

LemmaBandits.gap_bestArm

Details

No docstring.

theorem

Bandits.gap_bestArm.{u_1} {𝓐 : Type u_1} {m𝓐 : MeasurableSpace 𝓐}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ} [Fintype 𝓐] [Nonempty 𝓐] :
  gap ν (bestArm ν) = 0
Bandits.gap_bestArm.{u_1} {𝓐 : Type u_1}
  {m𝓐 : MeasurableSpace 𝓐}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ}
  [Fintype 𝓐] [Nonempty 𝓐] :
  gap ν (bestArm ν) = 0

Code

lemma gap_bestArm : gap ν (bestArm ν) = 0

Type uses (2)

Body uses (1)

gap_eq_bestArm_sub

Used by (1)

probReal_sum_le_sum_streamMeasure

Actions: Source · Open Issue

Proof

by
  rw [gap_eq_bestArm_sub, sub_self]

`integral_eq_of_gap_eq_zero`🔗

LemmaBandits.integral_eq_of_gap_eq_zero

Details

No docstring.

theorem

Bandits.integral_eq_of_gap_eq_zero.{u_1} {𝓐 : Type u_1}
  {m𝓐 : MeasurableSpace 𝓐} {ν : ProbabilityTheory.Kernel 𝓐 ℝ} {a : 𝓐}
  [Fintype 𝓐] [Nonempty 𝓐] (hg : gap ν a = 0) :
  ∫ (x : ℝ), id x ∂ν (bestArm ν) = ∫ (x : ℝ), id x ∂ν a
Bandits.integral_eq_of_gap_eq_zero.{u_1}
  {𝓐 : Type u_1} {m𝓐 : MeasurableSpace 𝓐}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ}
  {a : 𝓐} [Fintype 𝓐] [Nonempty 𝓐]
  (hg : gap ν a = 0) :
  ∫ (x : ℝ), id x ∂ν (bestArm ν) =
    ∫ (x : ℝ), id x ∂ν a

Code

lemma integral_eq_of_gap_eq_zero (hg : gap ν a = 0) : (ν (bestArm ν))[id] = (ν a)[id]

Type uses (2)

Body uses (1)

gap_eq_bestArm_sub

Actions: Source · Open Issue

Proof

by
  rwa [← sub_eq_zero, ← gap_eq_bestArm_sub]

`avg_mean_reward_tendsto_of_sublinear_regret`🔗

LemmaBandits.avg_mean_reward_tendsto_of_sublinear_regret

Details

If the regret is sublinear, the average mean reward tends to the highest mean of the arms.

theorem

Bandits.avg_mean_reward_tendsto_of_sublinear_regret.{u_1, u_2}
  {𝓐 : Type u_1} {Ω : Type u_2} {m𝓐 : MeasurableSpace 𝓐}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ} {A : ℕ → Ω → 𝓐} {ω : Ω}
  (hr : (fun x => regret ν A x ω) =o[Filter.atTop] fun t => ↑t) :
  Filter.Tendsto
    (fun t => (∑ s ∈ Finset.range t, ∫ (x : ℝ), id x ∂ν (A s ω)) / ↑t)
    Filter.atTop (nhds (⨆ a, ∫ (x : ℝ), id x ∂ν a))
Bandits.avg_mean_reward_tendsto_of_sublinear_regret.{u_1,
    u_2}
  {𝓐 : Type u_1} {Ω : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ}
  {A : ℕ → Ω → 𝓐} {ω : Ω}
  (hr :
    (fun x =>
        regret ν A x ω) =o[Filter.atTop]
      fun t => ↑t) :
  Filter.Tendsto
    (fun t =>
      (∑ s ∈ Finset.range t,
          ∫ (x : ℝ), id x ∂ν (A s ω)) /
        ↑t)
    Filter.atTop
    (nhds (⨆ a, ∫ (x : ℝ), id x ∂ν a))

Code

lemma avg_mean_reward_tendsto_of_sublinear_regret
    (hr : (regret ν A · ω) =o[atTop] fun t ↦ (t : ℝ)) :
    Tendsto (fun t ↦ (∑ s ∈ range t, (ν (A s ω))[id]) / (t : ℝ))
      atTop (nhds (⨆ a, (ν a)[id]))

Type uses (1)

regret

Actions: Source · Open Issue

Proof

by
  have ht : Tendsto (fun t ↦ (⨆ a, (ν a)[id]) - regret ν A t ω / t)
      atTop (nhds (⨆ a, (ν a)[id])) := by
    simpa using tendsto_const_nhds.sub hr.tendsto_div_nhds_zero
  apply ht.congr'
  filter_upwards [eventually_ne_atTop 0] with t ht
  rw [regret]
  field_simp
  ring

`pullCount_rate_tendsto_of_sublinear_regret`🔗

LemmaBandits.pullCount_rate_tendsto_of_sublinear_regret

Details

If the regret is sublinear, the rate of suboptimal arm pulls tends to zero.

theorem

Bandits.pullCount_rate_tendsto_of_sublinear_regret.{u_1, u_2}
  {𝓐 : Type u_1} {Ω : Type u_2} [DecidableEq 𝓐] {m𝓐 : MeasurableSpace 𝓐}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ} {A : ℕ → Ω → 𝓐} {ω : Ω} {a : 𝓐}
  [Finite 𝓐]
  (hr : (fun x => regret ν A x ω) =o[Filter.atTop] fun t => ↑t)
  (hg : 0 < gap ν a) :
  Filter.Tendsto (fun t => ↑(Learning.pullCount A a t ω) / ↑t)
    Filter.atTop (nhds 0)
Bandits.pullCount_rate_tendsto_of_sublinear_regret.{u_1,
    u_2}
  {𝓐 : Type u_1} {Ω : Type u_2}
  [DecidableEq 𝓐] {m𝓐 : MeasurableSpace 𝓐}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ}
  {A : ℕ → Ω → 𝓐} {ω : Ω} {a : 𝓐}
  [Finite 𝓐]
  (hr :
    (fun x =>
        regret ν A x ω) =o[Filter.atTop]
      fun t => ↑t)
  (hg : 0 < gap ν a) :
  Filter.Tendsto
    (fun t =>
      ↑(Learning.pullCount A a t ω) / ↑t)
    Filter.atTop (nhds 0)

Code

lemma pullCount_rate_tendsto_of_sublinear_regret [Finite 𝓐]
    (hr : (regret ν A · ω) =o[atTop] fun t ↦ (t : ℝ)) (hg : 0 < gap ν a) :
    Tendsto (fun t ↦ (pullCount A a t ω : ℝ) / t) atTop (nhds 0)

Type uses (3)

Body uses (3)

Actions: Source · Open Issue

Proof

by
  have := Fintype.ofFinite 𝓐
  have hb (t : ℕ) : (pullCount A a t ω : ℝ) * gap ν a ≤ regret ν A t ω := by
    rw [regret_eq_sum_pullCount_mul_gap]
    exact single_le_sum (f := fun a ↦ pullCount A a t ω * gap ν a)
      (fun _ _ ↦ mul_nonneg (Nat.cast_nonneg _) gap_nonneg) (mem_univ a)
  have hb' (t : ℕ) : (pullCount A a t ω : ℝ) / t ≤ regret ν A t ω / t / gap ν a := by
    obtain ht | ht := eq_or_ne t 0
    · simp [ht]
    · calc (pullCount A a t ω : ℝ) / t
          = pullCount A a t ω * gap ν a / gap ν a / t := by field_simp
        _ ≤ regret ν A t ω / gap ν a / t := by gcongr; exact hb t
        _ = regret ν A t ω / t / gap ν a := by ring
  apply squeeze_zero' (Eventually.of_forall fun _ ↦ by positivity) (Eventually.of_forall hb')
  simpa using hr.tendsto_div_nhds_zero.div_const (gap ν a)

LeanMachineLearning exposition

2.5. Online.Bandit.Regret🔗

`gap`🔗

`gap_nonneg`🔗

`gap_nonneg_of_le`🔗

`gap_le_of_mem_Icc`🔗

`regret`🔗

`regret_eq_sum_gap`🔗

`regret_nonneg`🔗

`gap_eq_zero_of_regret_eq_zero`🔗

`regret_eq_sum_pullCount_mul_gap`🔗

`integral_regret_eq_sum_gap_mul_integral_pullCount`🔗

`integral_regret_le_of_forall_integral_pullCount_le`🔗

`bestArm`🔗

`le_bestArm`🔗

`gap_eq_bestArm_sub`🔗

`gap_bestArm`🔗

`integral_eq_of_gap_eq_zero`🔗

`avg_mean_reward_tendsto_of_sublinear_regret`🔗

`pullCount_rate_tendsto_of_sublinear_regret`🔗

2.5. Online.Bandit.Regret🔗

gap🔗

gap_nonneg🔗

gap_nonneg_of_le🔗

gap_le_of_mem_Icc🔗

regret🔗

regret_eq_sum_gap🔗

regret_nonneg🔗

gap_eq_zero_of_regret_eq_zero🔗

regret_eq_sum_pullCount_mul_gap🔗

integral_regret_eq_sum_gap_mul_integral_pullCount🔗

integral_regret_le_of_forall_integral_pullCount_le🔗

bestArm🔗

le_bestArm🔗

gap_eq_bestArm_sub🔗

gap_bestArm🔗

integral_eq_of_gap_eq_zero🔗

avg_mean_reward_tendsto_of_sublinear_regret🔗

pullCount_rate_tendsto_of_sublinear_regret🔗

`gap`🔗

`gap_nonneg`🔗

`gap_nonneg_of_le`🔗

`gap_le_of_mem_Icc`🔗

`regret`🔗

`regret_eq_sum_gap`🔗

`regret_nonneg`🔗

`gap_eq_zero_of_regret_eq_zero`🔗

`regret_eq_sum_pullCount_mul_gap`🔗

`integral_regret_eq_sum_gap_mul_integral_pullCount`🔗

`integral_regret_le_of_forall_integral_pullCount_le`🔗

`bestArm`🔗

`le_bestArm`🔗

`gap_eq_bestArm_sub`🔗

`gap_bestArm`🔗

`integral_eq_of_gap_eq_zero`🔗

`avg_mean_reward_tendsto_of_sublinear_regret`🔗

`pullCount_rate_tendsto_of_sublinear_regret`🔗