File size: 5,722 Bytes
97c3405 | 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 | import OAI.Combinatorics.ContingencyTables.Counting.CompanionEstimatorConfidence
import OAI.Combinatorics.ContingencyTables.Counting.CompanionEstimatorRange
namespace OAI
/-! # A single Boolean word for the literal companion weight estimator -/
namespace ContingencyTables.CompanionOracleDensity
open CompanionFactorialApplication ForestAssignmentSplit GridBoundary ResidualMixture FirstSuccess
open CommonBasesFPRAS FiniteIndependentAverages FairBits
open scoped BigOperators Classical
noncomputable section
variable {I J : Type*} [Fintype I] [Fintype J]
/-- The complete input length, with one independent walk for each ratio
sample and one walk plus integer-bin draw for each correction sample. -/
def estimatorWordBits (A : Instance I J) (C K j : ℕ) (delta failure : ℚ) : ℕ :=
j*(sampleCount j delta failure *
(oracleTime A C K 1 delta * oracleStepBits A C)) +
sampleCount j delta failure *
oracleIntegerBits A C (oracleTime A C K 1 (correctionTheta delta)) (offsetAccuracy A delta)
lemma estimatorWordBits_eq (A : Instance I J) (C K j : ℕ) (delta failure : ℚ) :
estimatorWordBits A C K j delta failure = sampleCount j delta failure *
(j*oracleTime A C K 1 delta*oracleStepBits A C +
oracleTime A C K 1 (correctionTheta delta)*oracleStepBits A C +
binWordBits A (offsetAccuracy A delta)) := by
unfold estimatorWordBits oracleIntegerBits
ring
def ratioBlockWordEquiv (A : Instance I J) (C K j : ℕ) (delta failure : ℚ) :
(Fin (sampleCount j delta failure *
(oracleTime A C K 1 delta * oracleStepBits A C)) → Bool) ≃
RatioBlock A C K j delta failure :=
(oracleWalkWordEquiv (sampleCount j delta failure)
(oracleTime A C K 1 delta * oracleStepBits A C)).trans
(Equiv.piCongrRight (fun _ =>
oracleWalkWordEquiv (oracleTime A C K 1 delta) (oracleStepBits A C)))
/-- This equivalence partitions, rather than reuses, the random bits of
all the individual experiments in the estimator. -/
def estimatorWordEquiv (A : Instance I J) (C K j : ℕ) (delta failure : ℚ) :
(Fin (estimatorWordBits A C K j delta failure) → Bool) ≃
EstimatorInput A C K j delta failure :=
(splitWordEquiv _ _).trans (Equiv.prodCongr
((oracleWalkWordEquiv j (sampleCount j delta failure *
(oracleTime A C K 1 delta * oracleStepBits A C))).trans
(Equiv.piCongrRight (fun _ => ratioBlockWordEquiv A C K j delta failure)))
(oracleWalkWordEquiv (sampleCount j delta failure)
(oracleIntegerBits A C (oracleTime A C K 1 (correctionTheta delta)) (offsetAccuracy A delta))))
def wordEstimate (A : Instance I J) (C : ℕ) (z : View A → ℕ)
(K : ℕ) (hK : 1 ≤ K) (hW : 100*A.d*K ≤ A.W) (j : ℕ) (epsilon failure : ℚ)
(v : Grid (I := Fin (dimension A)) K)
(bits : Fin (estimatorWordBits A C K j (innerTolerance epsilon j) failure) → Bool) : ℚ :=
estimateValue A C z K hK hW j (innerTolerance epsilon j) failure v
(estimatorWordEquiv A C K j (innerTolerance epsilon j) failure bits)
/-- The single fixed-length word has exactly the independent-product law. -/
theorem wordEstimate_failure (A : Instance I J) (C : ℕ) (hC : 0 < C) (z : View A → ℕ)
(hz : BalancedBox.InBox (bounds A) z) (K : ℕ) (hK : 1 ≤ K) (hW : 100*A.d*K ≤ A.W)
(hB : (K:ℝ)=binWidth A C) (j : ℕ) (hj : j ≤ rationalHeight A C)
(epsilon failure : ℚ) (he0 : 0 < epsilon) (he1 : epsilon < 1) (hf : 0 < failure)
(hPL : PublishedInputs.FiniteBoxPrekopaLeindler (dimension A))
(hCheeger : PublishedInputs.FiniteCheeger (Grid (I := Fin (dimension A)) K))
(v : Grid (I := Fin (dimension A)) K) :
uniformAverage (fun bits : Fin (estimatorWordBits A C K j (innerTolerance epsilon j) failure) → Bool =>
if (wordEstimate A C z K hK hW j epsilon failure v bits:ℝ) <
(1-(epsilon:ℝ))*h A ((j:ℝ)*C/height A C) z ∨
(1+(epsilon:ℝ))*h A ((j:ℝ)*C/height A C) z <
(wordEstimate A C z K hK hW j epsilon failure v bits:ℝ)
then (1:ℝ) else 0) ≤ (failure:ℝ)/4 := by
let test := fun draws : EstimatorInput A C K j (innerTolerance epsilon j) failure =>
if (estimateValue A C z K hK hW j (innerTolerance epsilon j) failure v draws:ℝ) <
(1-(epsilon:ℝ))*h A ((j:ℝ)*C/height A C) z ∨
(1+(epsilon:ℝ))*h A ((j:ℝ)*C/height A C) z <
(estimateValue A C z K hK hW j (innerTolerance epsilon j) failure v draws:ℝ)
then (1:ℝ) else 0
have he := uniformAverage_equiv
(estimatorWordEquiv A C K j (innerTolerance epsilon j) failure) test
change uniformAverage (fun bits => test
(estimatorWordEquiv A C K j (innerTolerance epsilon j) failure bits)) ≤ _
exact he.trans_le
(estimateValue_failure A C hC z hz K hK hW hB j hj epsilon failure he0 he1 hf hPL hCheeger v)
theorem wordEstimate_range (A : Instance I J) (C : ℕ) (hC : 0 < C) (z : View A → ℕ)
(hz : BalancedBox.InBox (bounds A) z) (K : ℕ) (hK : 1 ≤ K) (hW : 100*A.d*K ≤ A.W)
(hB : (K:ℝ)=binWidth A C) (j : ℕ) (hj : j ≤ rationalHeight A C)
(epsilon failure : ℚ) (he0 : 0 < epsilon) (he1 : epsilon < 1) (hf : 0 < failure)
(v : Grid (I := Fin (dimension A)) K)
(bits : Fin (estimatorWordBits A C K j (innerTolerance epsilon j) failure) → Bool) :
(assignmentCount A:ℝ)*(1/2:ℝ)^j/16 ≤
(wordEstimate A C z K hK hW j epsilon failure v bits:ℝ) ∧
(wordEstimate A C z K hK hW j epsilon failure v bits:ℝ) ≤ 8*assignmentCount A :=
estimateValue_range A C hC z hz K hK hW hB j hj (innerTolerance epsilon j) failure
(innerTolerance_pos epsilon j he0) (innerTolerance_lt_one epsilon j he1) hf v _
end
end ContingencyTables.CompanionOracleDensity
end OAI
|