LeanMachineLearning exposition

Bandits.ArrayModel.hist_add_one_eq_IicSuccProd๐Ÿ”—

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

hist_add_one_eq_IicSuccProd๐Ÿ”—

LemmaBandits.ArrayModel.hist_add_one_eq_IicSuccProd
Details

No docstring.

๐Ÿ”—theorem
Bandits.ArrayModel.hist_add_one_eq_IicSuccProd.{u_1, u_2} {๐“ : Type u_1}
  {R : Type u_2} {m๐“ : MeasurableSpace ๐“} {mR : MeasurableSpace R}
  [Nonempty ๐“] [StandardBorelSpace ๐“] [DecidableEq ๐“]
  (alg : Learning.Algorithm ๐“ R) (ฯ‰ : probSpace ๐“ R) (n : โ„•) :
  hist alg ฯ‰ (n + 1) =
    (MeasurableEquiv.symm
        (MeasurableEquiv.IicSuccProd (fun x => ๐“ ร— R) n))
      (hist alg ฯ‰ n, action alg (n + 1) ฯ‰, reward alg (n + 1) ฯ‰)
Bandits.ArrayModel.hist_add_one_eq_IicSuccProd.{u_1,
    u_2}
  {๐“ : Type u_1} {R : Type u_2}
  {m๐“ : MeasurableSpace ๐“}
  {mR : MeasurableSpace R} [Nonempty ๐“]
  [StandardBorelSpace ๐“] [DecidableEq ๐“]
  (alg : Learning.Algorithm ๐“ R)
  (ฯ‰ : probSpace ๐“ R) (n : โ„•) :
  hist alg ฯ‰ (n + 1) =
    (MeasurableEquiv.symm
        (MeasurableEquiv.IicSuccProd
          (fun x => ๐“ ร— R) n))
      (hist alg ฯ‰ n, action alg (n + 1) ฯ‰,
        reward alg (n + 1) ฯ‰)

Code

lemma hist_add_one_eq_IicSuccProd [DecidableEq ๐“] (alg : Algorithm ๐“ R) (ฯ‰ : probSpace ๐“ R)
    (n : โ„•) :
    hist alg ฯ‰ (n + 1) =
      (MeasurableEquiv.IicSuccProd (fun _ โ†ฆ ๐“ ร— R) n).symm
        (hist alg ฯ‰ n, (action alg (n + 1) ฯ‰, reward alg (n + 1) ฯ‰))
Type uses (6)
Body uses (5)
Used by (1)

Actions: Source ยท Open Issue

Proof
by
  rw [hist_add_one_eq_IicSuccProd', reward_add_one, action_add_one_eq]

Dependency graph

Type dependencies (6)

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)

Actions: Source ยท Open Issue

probSpace๐Ÿ”—

DefinitionBandits.ArrayModel.probSpace
Details

Probability space for the array model of stochastic bandits.

๐Ÿ”—def
Bandits.ArrayModel.probSpace.{u_1, u_2} (๐“ : Type u_1) (R : Type u_2) :
  Type (max u_1 u_2)
Bandits.ArrayModel.probSpace.{u_1, u_2}
  (๐“ : Type u_1) (R : Type u_2) :
  Type (max u_1 u_2)

Code

def probSpace : Type _ := (โ„• โ†’ I) ร— (โ„• โ†’ ๐“ โ†’ R)
Used by (64)

Actions: Source ยท Open Issue

hist๐Ÿ”—

DefinitionBandits.ArrayModel.hist
Details

History of actions and rewards up to time n in the array model.

๐Ÿ”—def
Bandits.ArrayModel.hist.{u_1, u_2} {๐“ : Type u_1} {R : Type u_2}
  {m๐“ : MeasurableSpace ๐“} {mR : MeasurableSpace R} [Nonempty ๐“]
  [StandardBorelSpace ๐“] [DecidableEq ๐“] (alg : Learning.Algorithm ๐“ R)
  (ฯ‰ : probSpace ๐“ R) (n : โ„•) : โ†ฅ(Finset.Iic n) โ†’ ๐“ ร— R
Bandits.ArrayModel.hist.{u_1, u_2}
  {๐“ : Type u_1} {R : Type u_2}
  {m๐“ : MeasurableSpace ๐“}
  {mR : MeasurableSpace R} [Nonempty ๐“]
  [StandardBorelSpace ๐“] [DecidableEq ๐“]
  (alg : Learning.Algorithm ๐“ R)
  (ฯ‰ : probSpace ๐“ R) (n : โ„•) :
  โ†ฅ(Finset.Iic n) โ†’ ๐“ ร— R

Code

noncomputable
def hist [DecidableEq ๐“] (alg : Algorithm ๐“ R) (ฯ‰ : probSpace ๐“ R) : (n : โ„•) โ†’ Iic n โ†’ ๐“ ร— R
| 0 => fun _ โ†ฆ (initAlgFunction alg (ฯ‰.1 0), ฯ‰.2 0 (initAlgFunction alg (ฯ‰.1 0)))
| n + 1 =>
  let hn : Iic n โ†’ ๐“ ร— R := hist alg ฯ‰ n
  let a : ๐“ := algFunction alg n hn (ฯ‰.1 (n + 1))
  fun i โ†ฆ if hin : i โ‰ค n then hn โŸจi, by simp [hin]โŸฉ else (a, ฯ‰.2 (pullCount' n hn a) a)
Type uses (2)
Body uses (3)
Used by (30)

Actions: Source ยท Open Issue

IicSuccProd๐Ÿ”—

DefinitionMeasurableEquiv.IicSuccProd
Details

Measurable equivalence between a product up to n + 1 and the pair of the product up to n and the space at n + 1.

๐Ÿ”—def
MeasurableEquiv.IicSuccProd.{u_3} (X : โ„• โ†’ Type u_3)
  [(n : โ„•) โ†’ MeasurableSpace (X n)] (n : โ„•) :
  ((i : โ†ฅ(Finset.Iic (n + 1))) โ†’ X โ†‘i) โ‰ƒแต
    ((i : โ†ฅ(Finset.Iic n)) โ†’ X โ†‘i) ร— X (n + 1)
MeasurableEquiv.IicSuccProd.{u_3}
  (X : โ„• โ†’ Type u_3)
  [(n : โ„•) โ†’ MeasurableSpace (X n)]
  (n : โ„•) :
  ((i : โ†ฅ(Finset.Iic (n + 1))) โ†’ X โ†‘i) โ‰ƒแต
    ((i : โ†ฅ(Finset.Iic n)) โ†’ X โ†‘i) ร—
      X (n + 1)

Code

def _root_.MeasurableEquiv.IicSuccProd (X : โ„• โ†’ Type*) [โˆ€ n, MeasurableSpace (X n)] (n : โ„•) :
    MeasurableEquiv (ฮ  i : Iic (n + 1), X i) ((ฮ  i : Iic n, X i) ร— X (n + 1)) :=
  (MeasurableEquiv.IicProdIoc (Nat.le_succ n)).symm.trans
    (MeasurableEquiv.prodCongr (MeasurableEquiv.refl _) (MeasurableEquiv.piSingleton n).symm)
Used by (11)

Actions: Source ยท Open Issue

action๐Ÿ”—

DefinitionBandits.ArrayModel.action
Details

Action taken at time n in the array model.

๐Ÿ”—def
Bandits.ArrayModel.action.{u_1, u_2} {๐“ : Type u_1} {R : Type u_2}
  {m๐“ : MeasurableSpace ๐“} {mR : MeasurableSpace R} [Nonempty ๐“]
  [StandardBorelSpace ๐“] [DecidableEq ๐“] (alg : Learning.Algorithm ๐“ R)
  (n : โ„•) (ฯ‰ : probSpace ๐“ R) : ๐“
Bandits.ArrayModel.action.{u_1, u_2}
  {๐“ : Type u_1} {R : Type u_2}
  {m๐“ : MeasurableSpace ๐“}
  {mR : MeasurableSpace R} [Nonempty ๐“]
  [StandardBorelSpace ๐“] [DecidableEq ๐“]
  (alg : Learning.Algorithm ๐“ R) (n : โ„•)
  (ฯ‰ : probSpace ๐“ R) : ๐“

Code

noncomputable
def action [DecidableEq ๐“] (alg : Algorithm ๐“ R) (n : โ„•) (ฯ‰ : probSpace ๐“ R) : ๐“ :=
  (hist alg ฯ‰ n โŸจn, by simpโŸฉ).1
Type uses (2)
Body uses (1)
Used by (43)

Actions: Source ยท Open Issue

reward๐Ÿ”—

DefinitionBandits.ArrayModel.reward
Details

Reward received at time n in the array model.

๐Ÿ”—def
Bandits.ArrayModel.reward.{u_1, u_2} {๐“ : Type u_1} {R : Type u_2}
  {m๐“ : MeasurableSpace ๐“} {mR : MeasurableSpace R} [Nonempty ๐“]
  [StandardBorelSpace ๐“] [DecidableEq ๐“] (alg : Learning.Algorithm ๐“ R)
  (n : โ„•) (ฯ‰ : probSpace ๐“ R) : R
Bandits.ArrayModel.reward.{u_1, u_2}
  {๐“ : Type u_1} {R : Type u_2}
  {m๐“ : MeasurableSpace ๐“}
  {mR : MeasurableSpace R} [Nonempty ๐“]
  [StandardBorelSpace ๐“] [DecidableEq ๐“]
  (alg : Learning.Algorithm ๐“ R) (n : โ„•)
  (ฯ‰ : probSpace ๐“ R) : R

Code

noncomputable
def reward [DecidableEq ๐“] (alg : Algorithm ๐“ R) (n : โ„•) (ฯ‰ : probSpace ๐“ R) : R :=
  (hist alg ฯ‰ n โŸจn, by simpโŸฉ).2
Type uses (2)
Body uses (1)
Used by (24)

Actions: Source ยท Open Issue

All dependencies, transitively (5)

instIsProbabilityMeasureP0๐Ÿ”—

InstanceLearning.instIsProbabilityMeasureP0
Details

No docstring.

๐Ÿ”—theorem
Learning.instIsProbabilityMeasureP0.{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) :
  MeasureTheory.IsProbabilityMeasure (Algorithm.p0 alg)
Learning.instIsProbabilityMeasureP0.{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) :
  MeasureTheory.IsProbabilityMeasure
    (Algorithm.p0 alg)

Code

instance (alg : Algorithm ๐“ ๐“จ) : IsProbabilityMeasure alg.p0
Type uses (1)
Used by (13)

Actions: Source ยท Open Issue

Proof
alg.hp0

initAlgFunction๐Ÿ”—

DefinitionBandits.ArrayModel.initAlgFunction
Details

The initial action is the image of a uniform random variable by this function.

๐Ÿ”—def
Bandits.ArrayModel.initAlgFunction.{u_1, u_2} {๐“ : Type u_1}
  {R : Type u_2} {m๐“ : MeasurableSpace ๐“} {mR : MeasurableSpace R}
  [Nonempty ๐“] [StandardBorelSpace ๐“] (alg : Learning.Algorithm ๐“ R) :
  โ†‘unitInterval โ†’ ๐“
Bandits.ArrayModel.initAlgFunction.{u_1,
    u_2}
  {๐“ : Type u_1} {R : Type u_2}
  {m๐“ : MeasurableSpace ๐“}
  {mR : MeasurableSpace R} [Nonempty ๐“]
  [StandardBorelSpace ๐“]
  (alg : Learning.Algorithm ๐“ R) :
  โ†‘unitInterval โ†’ ๐“

Code

noncomputable
def initAlgFunction (alg : Algorithm ๐“ R) : I โ†’ ๐“ :=
  (Measure.exists_measurable_map_eq alg.p0).choose
Type uses (1)
Body uses (1)
Used by (12)

Actions: Source ยท Open Issue

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)

Actions: Source ยท Open Issue

Proof
alg.h_policy n

algFunction๐Ÿ”—

DefinitionBandits.ArrayModel.algFunction
Details

The next action is the image of the history and a uniform random variable by this function.

๐Ÿ”—def
Bandits.ArrayModel.algFunction.{u_1, u_2} {๐“ : Type u_1} {R : Type u_2}
  {m๐“ : MeasurableSpace ๐“} {mR : MeasurableSpace R} [Nonempty ๐“]
  [StandardBorelSpace ๐“] (alg : Learning.Algorithm ๐“ R) (n : โ„•) :
  (โ†ฅ(Finset.Iic n) โ†’ ๐“ ร— R) โ†’ โ†‘unitInterval โ†’ ๐“
Bandits.ArrayModel.algFunction.{u_1, u_2}
  {๐“ : Type u_1} {R : Type u_2}
  {m๐“ : MeasurableSpace ๐“}
  {mR : MeasurableSpace R} [Nonempty ๐“]
  [StandardBorelSpace ๐“]
  (alg : Learning.Algorithm ๐“ R) (n : โ„•) :
  (โ†ฅ(Finset.Iic n) โ†’ ๐“ ร— R) โ†’
    โ†‘unitInterval โ†’ ๐“

Code

noncomputable
def algFunction (alg : Algorithm ๐“ R) (n : โ„•) :
    (Iic n โ†’ ๐“ ร— R) โ†’ I โ†’ ๐“ :=
  (Kernel.exists_measurable_map_eq_unitInterval (alg.policy n)).choose
Type uses (1)
Body uses (1)
Used by (17)

Actions: Source ยท Open Issue

pullCount'๐Ÿ”—

DefinitionLearning.pullCount'
Details

Number of pulls of arm a up to (and including) time n. This is the number of entries in h in which the arm is a.

๐Ÿ”—def
Learning.pullCount'.{u_1, u_2} {๐“ : Type u_1} {R : Type u_2}
  [DecidableEq ๐“] (n : โ„•) (h : โ†ฅ(Finset.Iic n) โ†’ ๐“ ร— R) (a : ๐“) : โ„•
Learning.pullCount'.{u_1, u_2}
  {๐“ : Type u_1} {R : Type u_2}
  [DecidableEq ๐“] (n : โ„•)
  (h : โ†ฅ(Finset.Iic n) โ†’ ๐“ ร— R) (a : ๐“) :
  โ„•

Code

noncomputable
def pullCount' (n : โ„•) (h : Iic n โ†’ ๐“ ร— R) (a : ๐“) := #{s | (h s).1 = a}
Used by (29)

Actions: Source ยท Open Issue