LeanMachineLearning exposition

Learning.IT.hasCondDistrib_feedback_zero๐Ÿ”—

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Minimal Lean file

hasCondDistrib_feedback_zero๐Ÿ”—

LemmaLearning.IT.hasCondDistrib_feedback_zero
Details

No docstring.

๐Ÿ”—theorem
Learning.IT.hasCondDistrib_feedback_zero.{u_1, u_2} {๐“ : Type u_1}
  {๐“จ : Type u_2} {m๐“ : MeasurableSpace ๐“} {m๐“จ : MeasurableSpace ๐“จ}
  (alg : Algorithm ๐“ ๐“จ) (env : Environment ๐“ ๐“จ) :
  ProbabilityTheory.HasCondDistrib (feedback 0) (action 0)
    (Environment.ฮฝ0 env) (trajMeasure alg env)
Learning.IT.hasCondDistrib_feedback_zero.{u_1,
    u_2}
  {๐“ : Type u_1} {๐“จ : Type u_2}
  {m๐“ : MeasurableSpace ๐“}
  {m๐“จ : MeasurableSpace ๐“จ}
  (alg : Algorithm ๐“ ๐“จ)
  (env : Environment ๐“ ๐“จ) :
  ProbabilityTheory.HasCondDistrib
    (feedback 0) (action 0)
    (Environment.ฮฝ0 env)
    (trajMeasure alg env)

Code

lemma hasCondDistrib_feedback_zero (alg : Algorithm ๐“ ๐“จ) (env : Environment ๐“ ๐“จ) :
    HasCondDistrib (feedback 0) (action 0) env.ฮฝ0 (trajMeasure alg env)
Type uses (5)
Body uses (5)
Used by (2)

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Proof
by
  have h_step := (hasLaw_step_zero alg env).map_eq
  have h_action := (hasLaw_action_zero alg env).map_eq
  exact โŸจby fun_prop, by rwa [h_action]โŸฉ

Dependency graph

Type dependencies (5)

Algorithm๐Ÿ”—

StructureLearning.Algorithm
Details

A stochastic, sequential algorithm.

๐Ÿ”—structure
Learning.Algorithm.{u_4, u_5} (๐“ : Type u_4) (๐“จ : Type u_5)
  [MeasurableSpace ๐“] [MeasurableSpace ๐“จ] : Type (max u_4 u_5)
Learning.Algorithm.{u_4, u_5}
  (๐“ : Type u_4) (๐“จ : Type u_5)
  [MeasurableSpace ๐“]
  [MeasurableSpace ๐“จ] : Type (max u_4 u_5)

Code

structure Algorithm (๐“ ๐“จ : Type*) [MeasurableSpace ๐“] [MeasurableSpace ๐“จ] where
  /-- Policy or sampling rule: distribution of the next action. -/
  policy : (n : โ„•) โ†’ Kernel (Iic n โ†’ ๐“ ร— ๐“จ) ๐“
  /-- The policy is a Markov kernel. -/
  [h_policy : โˆ€ n, IsMarkovKernel (policy n)]
  /-- Distribution of the first action. -/
  p0 : Measure ๐“
  /-- The first action distribution is a probability measure. -/
  [hp0 : IsProbabilityMeasure p0]
Used by (216)

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Environment๐Ÿ”—

StructureLearning.Environment
Details

A stochastic environment.

๐Ÿ”—structure
Learning.Environment.{u_4, u_5} (๐“ : Type u_4) (๐“จ : Type u_5)
  [MeasurableSpace ๐“] [MeasurableSpace ๐“จ] : Type (max u_4 u_5)
Learning.Environment.{u_4, u_5}
  (๐“ : Type u_4) (๐“จ : Type u_5)
  [MeasurableSpace ๐“]
  [MeasurableSpace ๐“จ] : Type (max u_4 u_5)

Code

structure Environment (๐“ ๐“จ : Type*) [MeasurableSpace ๐“] [MeasurableSpace ๐“จ] where
  /-- Distribution of the next observation as function of the past history. -/
  feedback : (n : โ„•) โ†’ Kernel ((Iic n โ†’ ๐“ ร— ๐“จ) ร— ๐“) ๐“จ
  /-- The feedback kernels are Markov kernels. -/
  [h_feedback : โˆ€ n, IsMarkovKernel (feedback n)]
  /-- Distribution of the first observation given the first action. -/
  ฮฝ0 : Kernel ๐“ ๐“จ
  /-- The initial observation kernel is a Markov kernel. -/
  [hp0 : IsMarkovKernel ฮฝ0]
Used by (128)

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feedback๐Ÿ”—

DefinitionLearning.IT.feedback
Details

feedback n is the feedback at time n. This is a random variable on the measurable space โ„• โ†’ ๐“ ร— ๐“จ.

๐Ÿ”—def
Learning.IT.feedback.{u_1, u_2} {๐“ : Type u_1} {๐“จ : Type u_2} (n : โ„•)
  (h : โ„• โ†’ ๐“ ร— ๐“จ) : ๐“จ
Learning.IT.feedback.{u_1, u_2}
  {๐“ : Type u_1} {๐“จ : Type u_2} (n : โ„•)
  (h : โ„• โ†’ ๐“ ร— ๐“จ) : ๐“จ

Code

def feedback (n : โ„•) (h : โ„• โ†’ ๐“ ร— ๐“จ) : ๐“จ := (h n).2
Used by (16)

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action๐Ÿ”—

DefinitionLearning.IT.action
Details

action n is the action pulled at time n. This is a random variable on the measurable space โ„• โ†’ ๐“ ร— ๐“จ.

๐Ÿ”—def
Learning.IT.action.{u_1, u_2} {๐“ : Type u_1} {๐“จ : Type u_2} (n : โ„•)
  (h : โ„• โ†’ ๐“ ร— ๐“จ) : ๐“
Learning.IT.action.{u_1, u_2}
  {๐“ : Type u_1} {๐“จ : Type u_2} (n : โ„•)
  (h : โ„• โ†’ ๐“ ร— ๐“จ) : ๐“

Code

def action (n : โ„•) (h : โ„• โ†’ ๐“ ร— ๐“จ) : ๐“ := (h n).1
Used by (31)

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trajMeasure๐Ÿ”—

DefinitionLearning.trajMeasure
Details

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

๐Ÿ”—def
Learning.trajMeasure.{u_1, u_2} {๐“ : Type u_1} {๐“จ : Type u_2}
  {m๐“ : MeasurableSpace ๐“} {m๐“จ : MeasurableSpace ๐“จ}
  (alg : Algorithm ๐“ ๐“จ) (env : Environment ๐“ ๐“จ) :
  MeasureTheory.Measure (โ„• โ†’ ๐“ ร— ๐“จ)
Learning.trajMeasure.{u_1, u_2}
  {๐“ : Type u_1} {๐“จ : Type u_2}
  {m๐“ : MeasurableSpace ๐“}
  {m๐“จ : MeasurableSpace ๐“จ}
  (alg : Algorithm ๐“ ๐“จ)
  (env : Environment ๐“ ๐“จ) :
  MeasureTheory.Measure (โ„• โ†’ ๐“ ร— ๐“จ)

Code

noncomputable
def trajMeasure (alg : Algorithm ๐“ ๐“จ) (env : Environment ๐“ ๐“จ) :
    Measure (โ„• โ†’ ๐“ ร— ๐“จ) :=
  Kernel.trajMeasure (alg.p0 โŠ—โ‚˜ env.ฮฝ0) (stepKernel alg env)
deriving IsProbabilityMeasure
Type uses (2)
Body uses (2)
Used by (19)

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All dependencies, transitively (4)

stepKernel๐Ÿ”—

DefinitionLearning.stepKernel
Details

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

๐Ÿ”—def
Learning.stepKernel.{u_1, u_2} {๐“ : Type u_1} {๐“จ : Type u_2}
  {m๐“ : MeasurableSpace ๐“} {m๐“จ : MeasurableSpace ๐“จ}
  (alg : Algorithm ๐“ ๐“จ) (env : Environment ๐“ ๐“จ) (n : โ„•) :
  ProbabilityTheory.Kernel (โ†ฅ(Finset.Iic n) โ†’ ๐“ ร— ๐“จ) (๐“ ร— ๐“จ)
Learning.stepKernel.{u_1, u_2}
  {๐“ : Type u_1} {๐“จ : Type u_2}
  {m๐“ : MeasurableSpace ๐“}
  {m๐“จ : MeasurableSpace ๐“จ}
  (alg : Algorithm ๐“ ๐“จ)
  (env : Environment ๐“ ๐“จ) (n : โ„•) :
  ProbabilityTheory.Kernel
    (โ†ฅ(Finset.Iic n) โ†’ ๐“ ร— ๐“จ) (๐“ ร— ๐“จ)

Code

noncomputable
def stepKernel (alg : Algorithm ๐“ ๐“จ) (env : Environment ๐“ ๐“จ) (n : โ„•) :
    Kernel (Iic n โ†’ ๐“ ร— ๐“จ) (๐“ ร— ๐“จ) :=
  alg.policy n โŠ—โ‚– env.feedback n
deriving IsMarkovKernel
Type uses (2)
Used by (17)

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instIsMarkovKernelForallSubtypeNatMemFinsetIicProdPolicy๐Ÿ”—

InstanceLearning.instIsMarkovKernelForallSubtypeNatMemFinsetIicProdPolicy
Details

No docstring.

๐Ÿ”—theorem
Learning.instIsMarkovKernelForallSubtypeNatMemFinsetIicProdPolicy.{u_1,
    u_2}
  {๐“ : Type u_1} {๐“จ : Type u_2} {m๐“ : MeasurableSpace ๐“}
  {m๐“จ : MeasurableSpace ๐“จ} (alg : Algorithm ๐“ ๐“จ) (n : โ„•) :
  ProbabilityTheory.IsMarkovKernel (Algorithm.policy alg n)
Learning.instIsMarkovKernelForallSubtypeNatMemFinsetIicProdPolicy.{u_1,
    u_2}
  {๐“ : Type u_1} {๐“จ : Type u_2}
  {m๐“ : MeasurableSpace ๐“}
  {m๐“จ : MeasurableSpace ๐“จ}
  (alg : Algorithm ๐“ ๐“จ) (n : โ„•) :
  ProbabilityTheory.IsMarkovKernel
    (Algorithm.policy alg n)

Code

instance (alg : Algorithm ๐“ ๐“จ) (n : โ„•) : IsMarkovKernel (alg.policy n)
Type uses (1)
Used by (14)

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Proof
alg.h_policy n

instIsMarkovKernelProdForallSubtypeNatMemFinsetIicFeedback๐Ÿ”—

InstanceLearning.instIsMarkovKernelProdForallSubtypeNatMemFinsetIicFeedback
Details

No docstring.

๐Ÿ”—theorem
Learning.instIsMarkovKernelProdForallSubtypeNatMemFinsetIicFeedback.{u_1,
    u_2}
  {๐“ : Type u_1} {๐“จ : Type u_2} {m๐“ : MeasurableSpace ๐“}
  {m๐“จ : MeasurableSpace ๐“จ} (env : Environment ๐“ ๐“จ) (n : โ„•) :
  ProbabilityTheory.IsMarkovKernel (Environment.feedback env n)
Learning.instIsMarkovKernelProdForallSubtypeNatMemFinsetIicFeedback.{u_1,
    u_2}
  {๐“ : Type u_1} {๐“จ : Type u_2}
  {m๐“ : MeasurableSpace ๐“}
  {m๐“จ : MeasurableSpace ๐“จ}
  (env : Environment ๐“ ๐“จ) (n : โ„•) :
  ProbabilityTheory.IsMarkovKernel
    (Environment.feedback env n)

Code

instance (env : Environment ๐“ ๐“จ) (n : โ„•) : IsMarkovKernel (env.feedback n)
Type uses (1)
Used by (5)

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Proof
env.h_feedback n

instIsMarkovKernelForallSubtypeNatMemFinsetIicProdStepKernel๐Ÿ”—

InstanceLearning.instIsMarkovKernelForallSubtypeNatMemFinsetIicProdStepKernel
Details

No docstring.

๐Ÿ”—theorem
Learning.instIsMarkovKernelForallSubtypeNatMemFinsetIicProdStepKernel.{u_1,
    u_2}
  {๐“ : Type u_1} {๐“จ : Type u_2} {m๐“ : MeasurableSpace ๐“}
  {m๐“จ : MeasurableSpace ๐“จ} (alg : Algorithm ๐“ ๐“จ) (env : Environment ๐“ ๐“จ)
  (n : โ„•) : ProbabilityTheory.IsMarkovKernel (stepKernel alg env n)
Learning.instIsMarkovKernelForallSubtypeNatMemFinsetIicProdStepKernel.{u_1,
    u_2}
  {๐“ : Type u_1} {๐“จ : Type u_2}
  {m๐“ : MeasurableSpace ๐“}
  {m๐“จ : MeasurableSpace ๐“จ}
  (alg : Algorithm ๐“ ๐“จ)
  (env : Environment ๐“ ๐“จ) (n : โ„•) :
  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