Bandits.TS.policy

`policy`🔗

DefinitionBandits.TS.policy

Details

The Thompson sampling policy samples an action according to its probability of being optimal under the posterior over environments given the history so far. The posterior under a uniform algorithm is used to avoid a circular definition.

def

Bandits.TS.policy.{u_1} {K : ℕ} {𝓔 : Type u_1} [MeasurableSpace 𝓔]
  [StandardBorelSpace 𝓔] [Nonempty 𝓔] (hK : 0 < K)
  (Q : MeasureTheory.Measure 𝓔) [MeasureTheory.IsProbabilityMeasure Q]
  (κ : ProbabilityTheory.Kernel (𝓔 × Fin K) ℝ)
  [ProbabilityTheory.IsMarkovKernel κ] (n : ℕ) :
  ProbabilityTheory.Kernel (↥(Finset.Iic n) → Fin K × ℝ) (Fin K)
Bandits.TS.policy.{u_1} {K : ℕ}
  {𝓔 : Type u_1} [MeasurableSpace 𝓔]
  [StandardBorelSpace 𝓔] [Nonempty 𝓔]
  (hK : 0 < K)
  (Q : MeasureTheory.Measure 𝓔)
  [MeasureTheory.IsProbabilityMeasure Q]
  (κ :
    ProbabilityTheory.Kernel (𝓔 × Fin K)
      ℝ)
  [ProbabilityTheory.IsMarkovKernel κ]
  (n : ℕ) :
  ProbabilityTheory.Kernel
    (↥(Finset.Iic n) → Fin K × ℝ) (Fin K)

Code

noncomputable
def TS.policy (hK : 0 < K) (Q : Measure 𝓔) [IsProbabilityMeasure Q] (κ : Kernel (𝓔 × Fin K) ℝ)
    [IsMarkovKernel κ] (n : ℕ) : Kernel (Iic n → (Fin K) × ℝ) (Fin K) :=
  have : Nonempty (Fin K) := Fin.pos_iff_nonempty.mp hK
  (IT.bayesTrajMeasurePosterior Q κ uniformAlgorithm n).map (bestAction κ id)

Body uses (3)

Used by (2)

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`Algorithm`🔗

StructureLearning.Algorithm

Details

A stochastic, sequential algorithm.

structure

Learning.Algorithm.{u_4, u_5} (𝓐 : Type u_4) (𝓨 : Type u_5)
  [MeasurableSpace 𝓐] [MeasurableSpace 𝓨] : Type (max u_4 u_5)
Learning.Algorithm.{u_4, u_5}
  (𝓐 : Type u_4) (𝓨 : Type u_5)
  [MeasurableSpace 𝓐]
  [MeasurableSpace 𝓨] : Type (max u_4 u_5)

Code

structure Algorithm (𝓐 𝓨 : Type*) [MeasurableSpace 𝓐] [MeasurableSpace 𝓨] where
  /-- Policy or sampling rule: distribution of the next action. -/
  policy : (n : ℕ) → Kernel (Iic n → 𝓐 × 𝓨) 𝓐
  /-- The policy is a Markov kernel. -/
  [h_policy : ∀ n, IsMarkovKernel (policy n)]
  /-- Distribution of the first action. -/
  p0 : Measure 𝓐
  /-- The first action distribution is a probability measure. -/
  [hp0 : IsProbabilityMeasure p0]

Used by (216)

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`history`🔗

DefinitionLearning.history

Details

History of the algorithm-environment sequence up to time n.

def

Learning.history.{u_1, u_2, u_3} {𝓐 : Type u_1} {𝓨 : Type u_2}
  {Ω : Type u_3} (A : ℕ → Ω → 𝓐) (Y : ℕ → Ω → 𝓨) (n : ℕ) (ω : Ω) :
  ↥(Finset.Iic n) → 𝓐 × 𝓨
Learning.history.{u_1, u_2, u_3}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {Ω : Type u_3} (A : ℕ → Ω → 𝓐)
  (Y : ℕ → Ω → 𝓨) (n : ℕ) (ω : Ω) :
  ↥(Finset.Iic n) → 𝓐 × 𝓨

Code

def history (A : ℕ → Ω → 𝓐) (Y : ℕ → Ω → 𝓨) (n : ℕ) (ω : Ω) : Iic n → 𝓐 × 𝓨 :=
  fun i ↦ (A i ω, Y i ω)

Used by (72)

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`action`🔗

DefinitionLearning.IT.action

Details

action n is the action pulled at time n. This is a random variable on the measurable space ℕ → 𝓐 × 𝓨.

def

Learning.IT.action.{u_1, u_2} {𝓐 : Type u_1} {𝓨 : Type u_2} (n : ℕ)
  (h : ℕ → 𝓐 × 𝓨) : 𝓐
Learning.IT.action.{u_1, u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2} (n : ℕ)
  (h : ℕ → 𝓐 × 𝓨) : 𝓐

Code

def action (n : ℕ) (h : ℕ → 𝓐 × 𝓨) : 𝓐 := (h n).1

Used by (31)

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`Environment`🔗

StructureLearning.Environment

Details

A stochastic environment.

structure

Learning.Environment.{u_4, u_5} (𝓐 : Type u_4) (𝓨 : Type u_5)
  [MeasurableSpace 𝓐] [MeasurableSpace 𝓨] : Type (max u_4 u_5)
Learning.Environment.{u_4, u_5}
  (𝓐 : Type u_4) (𝓨 : Type u_5)
  [MeasurableSpace 𝓐]
  [MeasurableSpace 𝓨] : Type (max u_4 u_5)

Code

structure Environment (𝓐 𝓨 : Type*) [MeasurableSpace 𝓐] [MeasurableSpace 𝓨] where
  /-- Distribution of the next observation as function of the past history. -/
  feedback : (n : ℕ) → Kernel ((Iic n → 𝓐 × 𝓨) × 𝓐) 𝓨
  /-- The feedback kernels are Markov kernels. -/
  [h_feedback : ∀ n, IsMarkovKernel (feedback n)]
  /-- Distribution of the first observation given the first action. -/
  ν0 : Kernel 𝓐 𝓨
  /-- The initial observation kernel is a Markov kernel. -/
  [hp0 : IsMarkovKernel ν0]

Used by (128)

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`stepKernel`🔗

DefinitionLearning.stepKernel

Details

Kernel describing the distribution of the next action-feedback pair given the history up to n.

def

Learning.stepKernel.{u_1, u_2} {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  (alg : Algorithm 𝓐 𝓨) (env : Environment 𝓐 𝓨) (n : ℕ) :
  ProbabilityTheory.Kernel (↥(Finset.Iic n) → 𝓐 × 𝓨) (𝓐 × 𝓨)
Learning.stepKernel.{u_1, u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  (alg : Algorithm 𝓐 𝓨)
  (env : Environment 𝓐 𝓨) (n : ℕ) :
  ProbabilityTheory.Kernel
    (↥(Finset.Iic n) → 𝓐 × 𝓨) (𝓐 × 𝓨)

Code

noncomputable
def stepKernel (alg : Algorithm 𝓐 𝓨) (env : Environment 𝓐 𝓨) (n : ℕ) :
    Kernel (Iic n → 𝓐 × 𝓨) (𝓐 × 𝓨) :=
  alg.policy n ⊗ₖ env.feedback n
deriving IsMarkovKernel

Type uses (2)

Used by (17)

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`instIsMarkovKernelForallSubtypeNatMemFinsetIicProdPolicy`🔗

InstanceLearning.instIsMarkovKernelForallSubtypeNatMemFinsetIicProdPolicy

Details

No docstring.

theorem

Learning.instIsMarkovKernelForallSubtypeNatMemFinsetIicProdPolicy.{u_1,
    u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2} {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨} (alg : Algorithm 𝓐 𝓨) (n : ℕ) :
  ProbabilityTheory.IsMarkovKernel (Algorithm.policy alg n)
Learning.instIsMarkovKernelForallSubtypeNatMemFinsetIicProdPolicy.{u_1,
    u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  (alg : Algorithm 𝓐 𝓨) (n : ℕ) :
  ProbabilityTheory.IsMarkovKernel
    (Algorithm.policy alg n)

Code

instance (alg : Algorithm 𝓐 𝓨) (n : ℕ) : IsMarkovKernel (alg.policy n)

Type uses (1)

Algorithm

Used by (14)

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Proof

alg.h_policy n

`instIsMarkovKernelProdForallSubtypeNatMemFinsetIicFeedback`🔗

InstanceLearning.instIsMarkovKernelProdForallSubtypeNatMemFinsetIicFeedback

Details

No docstring.

theorem

Learning.instIsMarkovKernelProdForallSubtypeNatMemFinsetIicFeedback.{u_1,
    u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2} {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨} (env : Environment 𝓐 𝓨) (n : ℕ) :
  ProbabilityTheory.IsMarkovKernel (Environment.feedback env n)
Learning.instIsMarkovKernelProdForallSubtypeNatMemFinsetIicFeedback.{u_1,
    u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  (env : Environment 𝓐 𝓨) (n : ℕ) :
  ProbabilityTheory.IsMarkovKernel
    (Environment.feedback env n)

Code

instance (env : Environment 𝓐 𝓨) (n : ℕ) : IsMarkovKernel (env.feedback n)

Type uses (1)

Environment

Used by (5)

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Proof

env.h_feedback n

`instIsMarkovKernelForallSubtypeNatMemFinsetIicProdStepKernel`🔗

InstanceLearning.instIsMarkovKernelForallSubtypeNatMemFinsetIicProdStepKernel

Details

No docstring.

theorem

Learning.instIsMarkovKernelForallSubtypeNatMemFinsetIicProdStepKernel.{u_1,
    u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2} {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨} (alg : Algorithm 𝓐 𝓨) (env : Environment 𝓐 𝓨)
  (n : ℕ) : ProbabilityTheory.IsMarkovKernel (stepKernel alg env n)
Learning.instIsMarkovKernelForallSubtypeNatMemFinsetIicProdStepKernel.{u_1,
    u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  (alg : Algorithm 𝓐 𝓨)
  (env : Environment 𝓐 𝓨) (n : ℕ) :
  ProbabilityTheory.IsMarkovKernel
    (stepKernel alg env n)

Code

deriving IsMarkovKernel

Type uses (3)

Body uses (2)

Used by (10)

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Proof

deriving IsMarkovKernel

`trajMeasure`🔗

DefinitionLearning.trajMeasure

Details

Measure on the sequence of actions and observations generated by the algorithm/environment.

def

Learning.trajMeasure.{u_1, u_2} {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  (alg : Algorithm 𝓐 𝓨) (env : Environment 𝓐 𝓨) :
  MeasureTheory.Measure (ℕ → 𝓐 × 𝓨)
Learning.trajMeasure.{u_1, u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  (alg : Algorithm 𝓐 𝓨)
  (env : Environment 𝓐 𝓨) :
  MeasureTheory.Measure (ℕ → 𝓐 × 𝓨)

Code

noncomputable
def trajMeasure (alg : Algorithm 𝓐 𝓨) (env : Environment 𝓐 𝓨) :
    Measure (ℕ → 𝓐 × 𝓨) :=
  Kernel.trajMeasure (alg.p0 ⊗ₘ env.ν0) (stepKernel alg env)
deriving IsProbabilityMeasure

Type uses (2)

Body uses (2)

Used by (19)

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`instIsProbabilityMeasureP0`🔗

InstanceLearning.instIsProbabilityMeasureP0

Details

No docstring.

theorem

Learning.instIsProbabilityMeasureP0.{u_1, u_2} {𝓐 : Type u_1}
  {𝓨 : Type u_2} {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  (alg : Algorithm 𝓐 𝓨) :
  MeasureTheory.IsProbabilityMeasure (Algorithm.p0 alg)
Learning.instIsProbabilityMeasureP0.{u_1,
    u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  (alg : Algorithm 𝓐 𝓨) :
  MeasureTheory.IsProbabilityMeasure
    (Algorithm.p0 alg)

Code

instance (alg : Algorithm 𝓐 𝓨) : IsProbabilityMeasure alg.p0

Type uses (1)

Algorithm

Used by (13)

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Proof

alg.hp0

`prodLeft`🔗

DefinitionLearning.Algorithm.prodLeft

Details

An algorithm with observations in 𝓧 × 𝓨 obtained from an algorithm with observations in 𝓨 by ignoring the 𝓧 component of each observation.

def

Learning.Algorithm.prodLeft.{u_1, u_2, u_4} {𝓐 : Type u_1}
  {𝓨 : Type u_2} {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  (𝓧 : Type u_4) [MeasurableSpace 𝓧] (alg : Algorithm 𝓐 𝓨) :
  Algorithm 𝓐 (𝓧 × 𝓨)
Learning.Algorithm.prodLeft.{u_1, u_2,
    u_4}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨} (𝓧 : Type u_4)
  [MeasurableSpace 𝓧]
  (alg : Algorithm 𝓐 𝓨) :
  Algorithm 𝓐 (𝓧 × 𝓨)

Code

def Algorithm.prodLeft (𝓧 : Type*) [MeasurableSpace 𝓧] (alg : Algorithm 𝓐 𝓨) :
    Algorithm 𝓐 (𝓧 × 𝓨) where
  policy n := (alg.policy n).comap (fun h i ↦ ((h i).1, (h i).2.2)) (by fun_prop)
  p0 := alg.p0

Type uses (1)

Algorithm

Body uses (2)

Used by (6)

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`bayesStationaryEnv`🔗

DefinitionLearning.bayesStationaryEnv

Details

An environment with observations in 𝓔 × 𝓨. The first element e of an observation is sampled from Q once and remains constant. The second element of an observation is sampled from κ (e, a), where a is the corresponding action.

def

Learning.bayesStationaryEnv.{u_1, u_2, u_3} {𝓔 : Type u_1}
  {𝓐 : Type u_2} {𝓨 : Type u_3} [MeasurableSpace 𝓔] [MeasurableSpace 𝓐]
  [MeasurableSpace 𝓨] (Q : MeasureTheory.Measure 𝓔)
  [MeasureTheory.IsProbabilityMeasure Q]
  (κ : ProbabilityTheory.Kernel (𝓔 × 𝓐) 𝓨)
  [ProbabilityTheory.IsMarkovKernel κ] : Environment 𝓐 (𝓔 × 𝓨)
Learning.bayesStationaryEnv.{u_1, u_2,
    u_3}
  {𝓔 : Type u_1} {𝓐 : Type u_2}
  {𝓨 : Type u_3} [MeasurableSpace 𝓔]
  [MeasurableSpace 𝓐] [MeasurableSpace 𝓨]
  (Q : MeasureTheory.Measure 𝓔)
  [MeasureTheory.IsProbabilityMeasure Q]
  (κ : ProbabilityTheory.Kernel (𝓔 × 𝓐) 𝓨)
  [ProbabilityTheory.IsMarkovKernel κ] :
  Environment 𝓐 (𝓔 × 𝓨)

Code

noncomputable
def bayesStationaryEnv (Q : Measure 𝓔) [IsProbabilityMeasure Q] (κ : Kernel (𝓔 × 𝓐) 𝓨)
    [IsMarkovKernel κ] : Environment 𝓐 (𝓔 × 𝓨) where
  feedback n :=
    let g : (Iic n → 𝓐 × 𝓔 × 𝓨) × 𝓐 → 𝓔 × 𝓐 := fun (h, a) => ((h ⟨0, by simp⟩).2.1, a)
    (Kernel.deterministic (Prod.fst ∘ g) (by fun_prop)) ×ₖ (κ.comap g (by fun_prop))
  ν0 := (Kernel.const _ Q) ⊗ₖ κ.swapLeft

Type uses (1)

Environment

Used by (4)

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`bayesTrajMeasure`🔗

DefinitionLearning.IT.bayesTrajMeasure

Details

A measure P on a measurable space that carries random variables E, A, and Y such that IsBayesAlgEnvSeq Q κ alg E A Y P.

def

Learning.IT.bayesTrajMeasure.{u_1, u_2, u_3} {𝓔 : Type u_1}
  {𝓐 : Type u_2} {𝓨 : Type u_3} [MeasurableSpace 𝓔] [MeasurableSpace 𝓐]
  [MeasurableSpace 𝓨] (Q : MeasureTheory.Measure 𝓔)
  [MeasureTheory.IsProbabilityMeasure Q]
  (κ : ProbabilityTheory.Kernel (𝓔 × 𝓐) 𝓨)
  [ProbabilityTheory.IsMarkovKernel κ] (alg : Algorithm 𝓐 𝓨) :
  MeasureTheory.Measure (ℕ → 𝓐 × 𝓔 × 𝓨)
Learning.IT.bayesTrajMeasure.{u_1, u_2,
    u_3}
  {𝓔 : Type u_1} {𝓐 : Type u_2}
  {𝓨 : Type u_3} [MeasurableSpace 𝓔]
  [MeasurableSpace 𝓐] [MeasurableSpace 𝓨]
  (Q : MeasureTheory.Measure 𝓔)
  [MeasureTheory.IsProbabilityMeasure Q]
  (κ : ProbabilityTheory.Kernel (𝓔 × 𝓐) 𝓨)
  [ProbabilityTheory.IsMarkovKernel κ]
  (alg : Algorithm 𝓐 𝓨) :
  MeasureTheory.Measure (ℕ → 𝓐 × 𝓔 × 𝓨)

Code

noncomputable
def bayesTrajMeasure (Q : Measure 𝓔) [IsProbabilityMeasure Q] (κ : Kernel (𝓔 × 𝓐) 𝓨)
    [IsMarkovKernel κ] (alg : Algorithm 𝓐 𝓨) : Measure (ℕ → 𝓐 × 𝓔 × 𝓨) :=
  trajMeasure (alg.prodLeft 𝓔) (bayesStationaryEnv Q κ)
deriving IsProbabilityMeasure

Type uses (1)

Algorithm

Body uses (3)

Used by (5)

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`instIsMarkovKernelν0`🔗

InstanceLearning.instIsMarkovKernelν0

Details

No docstring.

theorem

Learning.instIsMarkovKernelν0.{u_1, u_2} {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  (env : Environment 𝓐 𝓨) :
  ProbabilityTheory.IsMarkovKernel (Environment.ν0 env)
Learning.instIsMarkovKernelν0.{u_1, u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  (env : Environment 𝓐 𝓨) :
  ProbabilityTheory.IsMarkovKernel
    (Environment.ν0 env)

Code

instance (env : Environment 𝓐 𝓨) : IsMarkovKernel env.ν0

Type uses (1)

Environment

Used by (8)

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Proof

env.hp0

`instIsProbabilityMeasureForallNatProdTrajMeasure`🔗

InstanceLearning.instIsProbabilityMeasureForallNatProdTrajMeasure

Details

No docstring.

theorem

Learning.instIsProbabilityMeasureForallNatProdTrajMeasure.{u_1, u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2} {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨} (alg : Algorithm 𝓐 𝓨)
  (env : Environment 𝓐 𝓨) :
  MeasureTheory.IsProbabilityMeasure (trajMeasure alg env)
Learning.instIsProbabilityMeasureForallNatProdTrajMeasure.{u_1,
    u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  (alg : Algorithm 𝓐 𝓨)
  (env : Environment 𝓐 𝓨) :
  MeasureTheory.IsProbabilityMeasure
    (trajMeasure alg env)

Code

deriving IsProbabilityMeasure

Type uses (3)

Body uses (4)

Used by (8)

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Proof

deriving IsProbabilityMeasure

`instIsProbabilityMeasureForallNatProdBayesTrajMeasure`🔗

InstanceLearning.IT.instIsProbabilityMeasureForallNatProdBayesTrajMeasure

Details

No docstring.

theorem

Learning.IT.instIsProbabilityMeasureForallNatProdBayesTrajMeasure.{u_1,
    u_2, u_3}
  {𝓔 : Type u_1} {𝓐 : Type u_2} {𝓨 : Type u_3} [MeasurableSpace 𝓔]
  [MeasurableSpace 𝓐] [MeasurableSpace 𝓨] (Q : MeasureTheory.Measure 𝓔)
  [MeasureTheory.IsProbabilityMeasure Q]
  (κ : ProbabilityTheory.Kernel (𝓔 × 𝓐) 𝓨)
  [ProbabilityTheory.IsMarkovKernel κ] (alg : Algorithm 𝓐 𝓨) :
  MeasureTheory.IsProbabilityMeasure (bayesTrajMeasure Q κ alg)
Learning.IT.instIsProbabilityMeasureForallNatProdBayesTrajMeasure.{u_1,
    u_2, u_3}
  {𝓔 : Type u_1} {𝓐 : Type u_2}
  {𝓨 : Type u_3} [MeasurableSpace 𝓔]
  [MeasurableSpace 𝓐] [MeasurableSpace 𝓨]
  (Q : MeasureTheory.Measure 𝓔)
  [MeasureTheory.IsProbabilityMeasure Q]
  (κ : ProbabilityTheory.Kernel (𝓔 × 𝓐) 𝓨)
  [ProbabilityTheory.IsMarkovKernel κ]
  (alg : Algorithm 𝓐 𝓨) :
  MeasureTheory.IsProbabilityMeasure
    (bayesTrajMeasure Q κ alg)

Code

deriving IsProbabilityMeasure

Type uses (2)

Body uses (3)

Used by (4)

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Proof

deriving IsProbabilityMeasure

`bayesTrajMeasurePosterior`🔗

DefinitionLearning.IT.bayesTrajMeasurePosterior

Details

A kernel that represents the posterior over E given the history up to time n.

def

Learning.IT.bayesTrajMeasurePosterior.{u_1, u_2, u_3} {𝓔 : Type u_1}
  {𝓐 : Type u_2} {𝓨 : Type u_3} [MeasurableSpace 𝓔] [MeasurableSpace 𝓐]
  [MeasurableSpace 𝓨] [StandardBorelSpace 𝓔] [Nonempty 𝓔]
  (Q : MeasureTheory.Measure 𝓔) [MeasureTheory.IsProbabilityMeasure Q]
  (κ : ProbabilityTheory.Kernel (𝓔 × 𝓐) 𝓨)
  [ProbabilityTheory.IsMarkovKernel κ] (alg : Algorithm 𝓐 𝓨) (n : ℕ) :
  ProbabilityTheory.Kernel (↥(Finset.Iic n) → 𝓐 × 𝓨) 𝓔
Learning.IT.bayesTrajMeasurePosterior.{u_1,
    u_2, u_3}
  {𝓔 : Type u_1} {𝓐 : Type u_2}
  {𝓨 : Type u_3} [MeasurableSpace 𝓔]
  [MeasurableSpace 𝓐] [MeasurableSpace 𝓨]
  [StandardBorelSpace 𝓔] [Nonempty 𝓔]
  (Q : MeasureTheory.Measure 𝓔)
  [MeasureTheory.IsProbabilityMeasure Q]
  (κ : ProbabilityTheory.Kernel (𝓔 × 𝓐) 𝓨)
  [ProbabilityTheory.IsMarkovKernel κ]
  (alg : Algorithm 𝓐 𝓨) (n : ℕ) :
  ProbabilityTheory.Kernel
    (↥(Finset.Iic n) → 𝓐 × 𝓨) 𝓔

Code

noncomputable
def bayesTrajMeasurePosterior [StandardBorelSpace 𝓔] [Nonempty 𝓔]
    (Q : Measure 𝓔) [IsProbabilityMeasure Q] (κ : Kernel (𝓔 × 𝓐) 𝓨) [IsMarkovKernel κ]
    (alg : Algorithm 𝓐 𝓨) (n : ℕ) : Kernel (Iic n → 𝓐 × 𝓨) 𝓔 :=
  condDistrib (fun ω ↦ (ω 0).2.1) (history action (fun n ω ↦ (ω n).2.2) n)
    (bayesTrajMeasure Q κ alg)
deriving IsMarkovKernel

Type uses (1)

Algorithm

Body uses (4)

Used by (4)

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`randomSampling`🔗

DefinitionLearning.randomSampling

Details

The Random Sampling algorithm, which samples from a fixed probability measure at each iteration.

def

Learning.randomSampling.{u_1, u_2} {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  (μ : MeasureTheory.Measure 𝓐) [MeasureTheory.IsProbabilityMeasure μ] :
  Algorithm 𝓐 𝓨
Learning.randomSampling.{u_1, u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  (μ : MeasureTheory.Measure 𝓐)
  [MeasureTheory.IsProbabilityMeasure μ] :
  Algorithm 𝓐 𝓨

Code

noncomputable def randomSampling (μ : Measure 𝓐) [IsProbabilityMeasure μ] : Algorithm 𝓐 𝓨 where
  policy _ := Kernel.const _ μ
  p0 := μ

Type uses (1)

Algorithm

Used by (5)

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`uniformAlgorithm`🔗

DefinitionLearning.uniformAlgorithm

Details

The Uniform algorithm: actions are chosen uniformly at random.

def

Learning.uniformAlgorithm.{u_1, u_2} {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨} [Finite 𝓐]
  [Nonempty 𝓐] : Algorithm 𝓐 𝓨
Learning.uniformAlgorithm.{u_1, u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨} [Finite 𝓐]
  [Nonempty 𝓐] : Algorithm 𝓐 𝓨

Code

noncomputable
def uniformAlgorithm [Finite 𝓐] [Nonempty 𝓐] : Algorithm 𝓐 𝓨 := randomSampling (uniformOn Set.univ)

Type uses (1)

Algorithm

Body uses (1)

randomSampling

Used by (4)

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`max`🔗

DefinitionFunction.max

Details

The maximum value of a tuple.

def

Function.max.{u_1, u_2} {ι : Type u_1} {α : Type u_2} [LinearOrder α]
  [Fintype ι] [Nonempty ι] (f : ι → α) : α
Function.max.{u_1, u_2} {ι : Type u_1}
  {α : Type u_2} [LinearOrder α]
  [Fintype ι] [Nonempty ι] (f : ι → α) : α

Code

abbrev max : α := univ.sup' univ_nonempty f

Used by (8)

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`exists_argmax`🔗

Lemmaexists_argmax

Details

No docstring.

theorem

exists_argmax.{u_1, u_2} {ι : Type u_1} {α : Type u_2} [LinearOrder α]
  [Fintype ι] [Nonempty ι] (f : ι → α) : ∃ i, f i = Function.max f
exists_argmax.{u_1, u_2} {ι : Type u_1}
  {α : Type u_2} [LinearOrder α]
  [Fintype ι] [Nonempty ι] (f : ι → α) :
  ∃ i, f i = Function.max f

Code

lemma exists_argmax : ∃ i, f i = f.max

Type uses (1)

max

Used by (3)

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Proof

by
  obtain ⟨i, -, hi⟩ := Finset.exists_mem_eq_sup' (by simp : Finset.univ.Nonempty) f
  exact ⟨i, hi.symm⟩

`argmax`🔗

Definitionargmax

Details

The index of the maximum value of a tuple.

def

argmax.{u_1, u_2} {ι : Type u_1} {α : Type u_2} [LinearOrder α]
  [Fintype ι] [Nonempty ι] (f : ι → α) : ι
argmax.{u_1, u_2} {ι : Type u_1}
  {α : Type u_2} [LinearOrder α]
  [Fintype ι] [Nonempty ι] (f : ι → α) : ι

Code

noncomputable def argmax := (exists_argmax f).choose

Body uses (2)

Used by (17)

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`actionMean`🔗

DefinitionLearning.IsBayesAlgEnvSeq.actionMean

Details

A random variable that gives the mean feedback of action a.

def

Learning.IsBayesAlgEnvSeq.actionMean.{u_1, u_2, u_4} {𝓔 : Type u_1}
  {𝓐 : Type u_2} {Ω : Type u_4} [MeasurableSpace 𝓔] [MeasurableSpace 𝓐]
  (κ : ProbabilityTheory.Kernel (𝓔 × 𝓐) ℝ) (E : Ω → 𝓔) (a : 𝓐) (ω : Ω) :
  ℝ
Learning.IsBayesAlgEnvSeq.actionMean.{u_1,
    u_2, u_4}
  {𝓔 : Type u_1} {𝓐 : Type u_2}
  {Ω : Type u_4} [MeasurableSpace 𝓔]
  [MeasurableSpace 𝓐]
  (κ : ProbabilityTheory.Kernel (𝓔 × 𝓐) ℝ)
  (E : Ω → 𝓔) (a : 𝓐) (ω : Ω) : ℝ

Code

noncomputable
def actionMean (κ : Kernel (𝓔 × 𝓐) ℝ) (E : Ω → 𝓔) (a : 𝓐) (ω : Ω) : ℝ := (κ (E ω, a))[id]

Used by (12)

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`bestAction`🔗

DefinitionLearning.IsBayesAlgEnvSeq.bestAction

Details

A random variable that gives the action with the highest mean feedback.

def

Learning.IsBayesAlgEnvSeq.bestAction.{u_1, u_2, u_4} {𝓔 : Type u_1}
  {𝓐 : Type u_2} {Ω : Type u_4} [MeasurableSpace 𝓔] [MeasurableSpace 𝓐]
  [Nonempty 𝓐] [Fintype 𝓐] (κ : ProbabilityTheory.Kernel (𝓔 × 𝓐) ℝ)
  (E : Ω → 𝓔) (ω : Ω) : 𝓐
Learning.IsBayesAlgEnvSeq.bestAction.{u_1,
    u_2, u_4}
  {𝓔 : Type u_1} {𝓐 : Type u_2}
  {Ω : Type u_4} [MeasurableSpace 𝓔]
  [MeasurableSpace 𝓐] [Nonempty 𝓐]
  [Fintype 𝓐]
  (κ : ProbabilityTheory.Kernel (𝓔 × 𝓐) ℝ)
  (E : Ω → 𝓔) (ω : Ω) : 𝓐

Code

noncomputable
def bestAction [Nonempty 𝓐] [Fintype 𝓐] (κ : Kernel (𝓔 × 𝓐) ℝ) (E : Ω → 𝓔) (ω : Ω) : 𝓐 :=
  argmax (fun a ↦ actionMean κ E a ω)

Body uses (2)

Used by (12)

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LeanMachineLearning exposition

`Bandits.TS.policy`🔗

`policy`🔗

`Algorithm`🔗

`history`🔗

`action`🔗

`Environment`🔗

`stepKernel`🔗

`instIsMarkovKernelForallSubtypeNatMemFinsetIicProdPolicy`🔗

`instIsMarkovKernelProdForallSubtypeNatMemFinsetIicFeedback`🔗

`instIsMarkovKernelForallSubtypeNatMemFinsetIicProdStepKernel`🔗

`trajMeasure`🔗

`instIsProbabilityMeasureP0`🔗

`prodLeft`🔗

`bayesStationaryEnv`🔗

`bayesTrajMeasure`🔗

`instIsMarkovKernelν0`🔗

`instIsProbabilityMeasureForallNatProdTrajMeasure`🔗

`instIsProbabilityMeasureForallNatProdBayesTrajMeasure`🔗

`bayesTrajMeasurePosterior`🔗

`randomSampling`🔗

`uniformAlgorithm`🔗

`max`🔗

`exists_argmax`🔗

`argmax`🔗

`actionMean`🔗

`bestAction`🔗

Bandits.TS.policy🔗

policy🔗

Algorithm🔗

history🔗

action🔗

Environment🔗

stepKernel🔗

instIsMarkovKernelForallSubtypeNatMemFinsetIicProdPolicy🔗

instIsMarkovKernelProdForallSubtypeNatMemFinsetIicFeedback🔗

instIsMarkovKernelForallSubtypeNatMemFinsetIicProdStepKernel🔗

trajMeasure🔗

instIsProbabilityMeasureP0🔗

prodLeft🔗

bayesStationaryEnv🔗

bayesTrajMeasure🔗

instIsMarkovKernelν0🔗

instIsProbabilityMeasureForallNatProdTrajMeasure🔗

instIsProbabilityMeasureForallNatProdBayesTrajMeasure🔗

bayesTrajMeasurePosterior🔗

randomSampling🔗

uniformAlgorithm🔗

max🔗

exists_argmax🔗

argmax🔗

actionMean🔗

bestAction🔗

`Bandits.TS.policy`🔗

`policy`🔗

`Algorithm`🔗

`history`🔗

`action`🔗

`Environment`🔗

`stepKernel`🔗

`instIsMarkovKernelForallSubtypeNatMemFinsetIicProdPolicy`🔗

`instIsMarkovKernelProdForallSubtypeNatMemFinsetIicFeedback`🔗

`instIsMarkovKernelForallSubtypeNatMemFinsetIicProdStepKernel`🔗

`trajMeasure`🔗

`instIsProbabilityMeasureP0`🔗

`prodLeft`🔗

`bayesStationaryEnv`🔗

`bayesTrajMeasure`🔗

`instIsMarkovKernelν0`🔗

`instIsProbabilityMeasureForallNatProdTrajMeasure`🔗

`instIsProbabilityMeasureForallNatProdBayesTrajMeasure`🔗

`bayesTrajMeasurePosterior`🔗

`randomSampling`🔗

`uniformAlgorithm`🔗

`max`🔗

`exists_argmax`🔗

`argmax`🔗

`actionMean`🔗

`bestAction`🔗