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d5b7741 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 | import OAI.Combinatorics.ContingencyTables.Sampling.ExactIntegerRetry
import OAI.Combinatorics.ContingencyTables.Machines.OnlineIntegerRetry
import OAI.Combinatorics.ContingencyTables.Machines.FiniteWordExecution
import OAI.Combinatorics.ContingencyTables.Machines.IntegerMatrixProgram
import OAI.Combinatorics.ContingencyTables.Dense.DenseWordProgram
namespace OAI
/-! # The concrete fair-bit integer trial used by exact correction -/
namespace ContingencyTables.ExactIntegerOnline
open MatchingFPRAS MatchingFPRAS.TreeTyped MatchingFPRAS.PhysicalStream
open ExactIntegerChoice ExactIntegerPrograms FirstSuccess ResidualMixture
open scoped BigOperators Classical
def trialBits (w : List ℕ) : ℕ := Nat.clog 2 w.sum
def trialBitsRealizer : Realizer trialBits :=
composition listSum DensePrograms.ceilLogTwoRealizer
lemma polynomial_trialBits : PolynomialTime trialBitsRealizer :=
polynomial_composition polynomial_listSum DensePrograms.polynomial_ceilLogTwo
def trial (w : List ℕ) : Experiment (Option ℕ) :=
FiniteWordExecution.experiment trialBits weightedIndex w
noncomputable def trialRealizer : OnlineTree.Realizer trial :=
FiniteWordExecution.realizer trialBitsRealizer weightedIndexRealizer
lemma trialBits_le_weight (w : List ℕ) : trialBits w ≤ weight w := by
rw [trialBits,DensePrograms.clog_two_eq_size_pred]
exact (Nat.size_le_size (Nat.sub_le _ _)).trans (IntegerMatrixPrograms.sum_size_le_weight w)
theorem polynomial_trial : OnlineTree.PolynomialTime trialRealizer := by
apply OnlineTree.work_remeasure
(FiniteWordExecution.work_realizer polynomial_trialBits polynomial_weightedIndex)
(2*Polynomial.X)
intro w
have hb:=trialBits_le_weight w
simp only [FiniteWordExecution.wordMeasure,Polynomial.eval_mul,Polynomial.eval_ofNat,
Polynomial.eval_X]
omega
private lemma map_expectation {X Y : Type*} [Fintype X] [Fintype Y] [DecidableEq Y]
(p : RationalLaw X) (f : X → Y) (g : Y → ℚ) :
(∑ y,(mapLaw p f).mass y*g y)=∑ x,p.mass x*g (f x) := by
simp only [mapLaw,Finset.sum_mul]
rw [Finset.sum_comm]
apply Finset.sum_congr rfl
intro x _
simp [ite_mul]
private lemma average_expectation {Y : Type*} [Fintype Y] [DecidableEq Y]
(k : ℕ) (f : (Fin k → Bool) → Y) (g : Y → ℚ) :
average (fun bits => g (f bits))=
∑ y,(mapLaw (uniformLaw (α := Fin k → Bool)) f).mass y*g y := by
rw [map_expectation]
simp only [average,uniformLaw,uniformMass,Fintype.card_fun,Fintype.card_bool,
Fintype.card_fin,Nat.cast_pow,Nat.cast_ofNat,one_div,inv_mul_eq_div,Finset.sum_div]
/-- The compiled variable-length trial has precisely the integer-grid
probabilities, with the unused suffix of the binary range returning none. -/
theorem trial_mean {r : ℕ} (w : Fin r → ℕ) (h : 0<totalWeight w) (g : Option ℕ → ℚ) :
(trial (List.ofFn w)).mean g=
∑ y : Option (Fin r),
(trialLaw w h (Nat.le_pow_clog (by decide : 1<2) (totalWeight w))).mass y*
g (y.map Fin.val) := by
let k:=Nat.clog 2 (totalWeight w)
let hQ:=Nat.le_pow_clog (by decide : 1<2) (totalWeight w)
have hL : trialBits (List.ofFn w)=k := by
rw [trialBits,List.sum_ofFn]
rfl
have he : trial (List.ofFn w)=(OnlineWord.prepend k []).map
(fun bs => weightedIndex (List.ofFn w,bs)) := by
unfold trial FiniteWordExecution.experiment FiniteWordExecution.source
rw [hL]
rw [he,Experiment.mean_map,OnlineWord.mean_prepend_nil]
simp_rw [weightedIndex_ofFn w h hQ]
rw [average_expectation k (fun bits => weightedTrial w h hQ (FairBits.binaryWordEquiv k bits))
(fun y => g (y.map Fin.val))]
have hlaw : mapLaw (uniformLaw (α := Fin k → Bool))
(fun bits => weightedTrial w h hQ (FairBits.binaryWordEquiv k bits))=trialLaw w h hQ := by
rw [←FairBits.map_equiv (FairBits.binaryWordEquiv k),FairBits.uniform_equiv]
rw [hlaw]
private lemma retry_mass_succ {B : Type*} [Fintype B] [DecidableEq B]
(p : RationalLaw (Option B)) (N : ℕ) (y : Option B) :
(retryLaw p (fun b => b.map some) none (N+1)).mass y=
(y.elim 0 (fun b => p.mass (some b)))+
p.mass none*(retryLaw p (fun b => b.map some) none N).mass y := by
rw [retryLaw,retryStep_mass]
cases y <;> simp [successMass,failureMass]
/-- Mean preservation for every finite prefix allows the online loop to
reuse the already proved exact rational retry law. -/
lemma prefix_mean {A B C : Type} [Fintype B] [DecidableEq B]
(f : A → Experiment (Option C)) (a : A) (p : RationalLaw (Option B)) (e : B → C)
(hf : ∀ g : Option C → ℚ,(f a).mean g=∑ y,p.mass y*g (y.map e))
(N : ℕ) (g : Option C → ℚ) :
(OnlineIntegerRetry.prefixExperiment (f := f) a N).mean g=
∑ y,(retryLaw p (fun b => b.map some) none N).mass y*g (y.map e) := by
induction N with
| zero => simp [OnlineIntegerRetry.prefixExperiment,Experiment.mean,retryLaw,pointLaw]
| succ N ih =>
rw [OnlineIntegerRetry.prefixExperiment,Experiment.mean_bind,hf]
simp only [Fintype.sum_option,Option.map_none,Option.map_some,Option.elim_none,
Option.elim_some,Experiment.mean,ih]
simp_rw [retry_mass_succ]
simp only [Option.elim_none,Option.elim_some,zero_add]
simp only [add_mul,Finset.sum_add_distrib,mul_assoc]
rw [←Finset.mul_sum]
ring
theorem integer_prefix_mean {r : ℕ} (w : Fin r → ℕ) (h : 0<totalWeight w)
(N : ℕ) (g : Option ℕ → ℚ) :
(OnlineIntegerRetry.prefixExperiment (f := trial) (List.ofFn w) N).mean g=
∑ y,(retryIntegerLaw w h
(Nat.le_pow_clog (by decide : 1<2) (totalWeight w)) N).mass y*g (y.map Fin.val) :=
prefix_mean trial (List.ofFn w) _ Fin.val (trial_mean w h) N g
theorem integer_prefix_failure {r : ℕ} (w : Fin r → ℕ) (h : 0<totalWeight w) (N : ℕ) :
(OnlineIntegerRetry.prefixExperiment (f := trial) (List.ofFn w) N).mean
(fun y => if y=none then 1 else 0)=
(failure w (Q := 2^Nat.clog 2 (totalWeight w)))^N := by
rw [integer_prefix_mean w h]
simp only [Option.map_eq_none_iff,mul_ite,mul_one,mul_zero,Finset.sum_ite_eq',
Finset.mem_univ,ite_true,retryIntegerLaw_none]
end ContingencyTables.ExactIntegerOnline
end OAI
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