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2f887db | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 | import OAI.Computability.PerfectCompleteness.Decoding.FixedStoppedDecoderLaw
import OAI.Computability.PerfectCompleteness.Machines.LowerCutInstalledInput
import OAI.Computability.PerfectCompleteness.Reduction.FixedNativeHigh
import OAI.Computability.PerfectCompleteness.Reduction.FixedStoppedDirections
namespace OAI
section
namespace PerfectCompleteness.FixedStoppedHighBound
noncomputable section
open scoped Classical
open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays
open UniqueGamesTheorem.Foundations.Games
open UniqueGamesTheorem.Appendix.RankLevelFilter (linearMapFintype)
attribute [local instance] linearMapFintype
variable {δ : ℚ} {hδ : 0 < δ} (parameters : FixedParameters.Parameters δ hδ)
(i j : Fin parameters.plan.depth) (hij : i < j) (input : List Bool)
abbrev Prefixes := FixedStoppedDecoderContext.Prefixes parameters i j
def setup (pref : Prefixes parameters i j)
(external : FixedStoppedDecoderContext.OutsideQuestions parameters i j hij input pref)
(exterior : FixedStoppedDecoderContext.Exterior parameters i j hij input pref external) :=
FixedDecoderCleanRate.setup parameters
(StoppedCleanGeometry.geometry parameters input
(FixedStoppedDecoderContext.upperPath parameters i j pref)
(FixedStoppedDecoderContext.lowerPath parameters i j hij pref) hij external exterior) input
theorem setup_node (pref : Prefixes parameters i j)
(external : FixedStoppedDecoderContext.OutsideQuestions parameters i j hij input pref)
(exterior : FixedStoppedDecoderContext.Exterior parameters i j hij input pref external) :
WholeArrayInteriorExterior.upperNode
(CleanDecoderRate.path (setup parameters i j hij input pref external exterior)) =
FixedStoppedDirections.lowerNode parameters i j hij pref := by
change WholeArrayInteriorExterior.upperNode
(CleanRecordDecoderInput.path _ _
(StoppedCleanGeometry.geometry parameters input
(FixedStoppedDecoderContext.upperPath parameters i j pref)
(FixedStoppedDecoderContext.lowerPath parameters i j hij pref) hij external exterior).cut) = _
rw [StoppedCleanGeometry.geometry_path]
exact (StoppedCleanGeometry.lowerNode_descendant
(FixedStoppedDecoderContext.upperPath parameters i j pref)
(FixedStoppedDecoderContext.lowerPath parameters i j hij pref) hij).symm
def highProbability (strategy : FixedPreliminaryGame.Strategy parameters input)
(pref : Prefixes parameters i j)
(external : FixedStoppedDecoderContext.OutsideQuestions parameters i j hij input pref)
(A : FixedStoppedDecoderContext.Advice parameters i j pref)
(exterior : FixedStoppedDecoderContext.Exterior parameters i j hij input pref external)
(a : Block (FixedRows.rows parameters.plan)
(FixedStoppedDirections.lowerNode parameters i j hij pref)) : ℝ :=
(CleanDecoderPairLaw.pairLaw (setup parameters i j hij input pref external exterior)
(FixedStoppedDecoderLaw.labeling parameters input strategy)
(CleanPhysicalDecoderLaw.useful (setup parameters i j hij input pref external exterior)
(FixedStoppedDecoderLaw.markFamily parameters i j hij input strategy pref)
(InitialParameters.useful δ)) parameters.plan.order parameters.plan.density A
(LinearMap.ker A) a (FixedStoppedDecoderLaw.threshold parameters j)
(parameters.cubeSize i.val) (FixedDecoderCleanRate.cubePositive parameters i.val)).probability
(fun z => DecoderRankSplit.highAgrees
(setup parameters i j hij input pref external exterior)
(FixedDecoderCleanRate.branchPositive parameters)
(FixedRows.cutoff parameters.plan hδ) z.1 z.2)
def directionMean (strategy : FixedPreliminaryGame.Strategy parameters input)
(pref : Prefixes parameters i j)
(external : FixedStoppedDecoderContext.OutsideQuestions parameters i j hij input pref)
(A : FixedStoppedDecoderContext.Advice parameters i j pref)
(exterior : FixedStoppedDecoderContext.Exterior parameters i j hij input pref external) : ℝ :=
(FixedStoppedDirections.lowerLaw parameters i j hij pref).expectation
(fun a => highProbability parameters i j hij input strategy pref external A exterior a.val)
private theorem castDirection_eq {branch rows : Nat → Nat} {n : Nat}
{node node' : Nodes branch n} (h : node = node')
(a : BucketSampler.Direction (rows (Nodes.height node))) :
ProjectedFiberRawPair.castDirection h a =
LowerCutInstalledInput.directionCast rows h a := by
cases h
rfl
theorem directionMean_eq_native
(strategy : FixedPreliminaryGame.Strategy parameters input)
(pref : Prefixes parameters i j)
(external : FixedStoppedDecoderContext.OutsideQuestions parameters i j hij input pref)
(A : FixedStoppedDecoderContext.Advice parameters i j pref)
(exterior : FixedStoppedDecoderContext.Exterior parameters i j hij input pref external) :
directionMean parameters i j hij input strategy pref external A exterior =
FixedNativeHigh.probability parameters (setup parameters i j hij input pref external exterior)
(setup_node parameters i j hij input pref external exterior)
(FixedStoppedDecoderLaw.markFamily parameters i j hij input strategy pref)
(InitialParameters.useful δ) (FixedStoppedDecoderLaw.labeling parameters input strategy)
parameters.plan.order parameters.plan.density A
(FixedStoppedDecoderLaw.threshold parameters j) := by
let : Nonempty (BucketSampler.Direction (FixedRows.rows parameters.plan (i.val + 1))) :=
PreliminarySampler.directionNonempty _
(parameters.plan.rows_pos (parameters.plan.depth - (i.val + 1)))
let e := LowerCutInstalledInput.lowerDirectionEquiv (FixedRows.rows parameters.plan)
(CleanDecoderRate.path (setup parameters i j hij input pref external exterior))
(FixedStoppedDecoderContext.upper parameters i j pref) (i.val + 1)
(FixedStoppedDecoderContext.lower parameters i j hij pref)
(setup_node parameters i j hij input pref external exterior)
have htransport :
(FiniteDistribution.uniform
(BucketSampler.Direction (FixedRows.rows parameters.plan (i.val + 1)))).pushforward e =
FixedStoppedDirections.lowerLaw parameters i j hij pref := by
rw [← FiniteDistribution.transport_eq_pushforward]
exact UniformConditioning.uniform_transport e
unfold directionMean
rw [← htransport, FiniteDistribution.expectation_pushforward]
unfold FixedNativeHigh.probability
apply FiniteDistribution.expectation_congr
intro a
have hdirection : e a =
ProjectedFiberRawPair.direction
(CleanDecoderRate.path (setup parameters i j hij input pref external exterior))
(FixedStoppedDecoderContext.upper parameters i j pref) (i.val + 1)
(FixedStoppedDecoderContext.lower parameters i j hij pref)
(setup_node parameters i j hij input pref external exterior) a := by
unfold e LowerCutInstalledInput.lowerDirectionEquiv ProjectedFiberRawPair.direction
exact (castDirection_eq _ _).symm
rw [hdirection]
rfl
theorem directionMean_square_le
(strategy : FixedPreliminaryGame.Strategy parameters input)
(pref : Prefixes parameters i j)
(external : FixedStoppedDecoderContext.OutsideQuestions parameters i j hij input pref)
(A : FixedStoppedDecoderContext.Advice parameters i j pref)
(exterior : FixedStoppedDecoderContext.Exterior parameters i j hij input pref external) :
directionMean parameters i j hij input strategy pref external A exterior ^ 2 ≤
FixedRankContradiction.gamma parameters (j.val + 1) ^ 2 / 8 + 2 * parameters.accuracy := by
rw [directionMean_eq_native]
simpa only [WholeArrayInteriorExterior.upperNode_height] using
FixedNativeHigh.probability_square_le parameters
(setup parameters i j hij input pref external exterior)
(setup_node parameters i j hij input pref external exterior)
(FixedStoppedDecoderLaw.markFamily parameters i j hij input strategy pref)
(InitialParameters.useful δ) (FixedStoppedDecoderLaw.labeling parameters input strategy)
parameters.plan.order parameters.plan.density A
(FixedStoppedDecoderLaw.threshold parameters j) parameters.plan.density_pos
theorem probability_eq_mean (strategy : FixedPreliminaryGame.Strategy parameters input) :
(FixedStoppedDecoderLaw.law parameters i j hij input strategy).probability
(FixedStoppedDecoderLaw.highAgreement parameters i j hij input) =
(FixedStoppedDecoderContext.prefixLaw parameters i j hij).expectation (fun pref =>
(FiniteDistribution.uniform
(FixedStoppedDecoderContext.OutsideQuestions parameters i j hij input pref)).expectation
(fun external =>
(FiniteDistribution.uniform
(FixedStoppedDecoderContext.Advice parameters i j pref)).expectation (fun A =>
(FiniteDistribution.uniform
(FixedStoppedDecoderContext.Exterior parameters i j hij input pref external)).expectation
(directionMean parameters i j hij input strategy pref external A)))) := by
exact (DecoderFamilyLaw.probability_eq_expectation
(FixedStoppedDecoderLaw.setups parameters i j hij input)
(FixedStoppedDecoderLaw.labeling parameters input strategy)
(FixedStoppedDecoderLaw.useful parameters i j hij input strategy)
parameters.plan.order parameters.plan.density
(FixedStoppedDecoderContext.advice parameters i j hij input)
(FixedStoppedDecoderContext.direction parameters i j hij input)
(FixedStoppedDecoderLaw.threshold parameters j)
(FixedStoppedDecoderContext.law parameters i j hij input)
(parameters.cubeSize i.val) (FixedDecoderCleanRate.cubePositive parameters i.val)
(FixedStoppedDecoderLaw.highAgreement parameters i j hij input)).trans
(FixedStoppedDirections.expectation_direction parameters i j hij input
(highProbability parameters i j hij input strategy))
private theorem mean_square_le_const {Ω : Type*} [Fintype Ω]
(μ : FiniteDistribution Ω) (f : Ω → ℝ) (bound : ℝ)
(hbound : ∀ x, f x ^ 2 ≤ bound) : μ.expectation f ^ 2 ≤ bound := by
calc
_ ≤ μ.expectation (fun x => f x ^ 2) := CollisionAveraging.mean_square_le μ f
_ ≤ μ.expectation (fun _ => bound) := SmallBias.expectation_mono μ hbound
_ = bound := SmallBias.expectation_const μ bound
theorem probability_square_le (strategy : FixedPreliminaryGame.Strategy parameters input) :
(FixedStoppedDecoderLaw.law parameters i j hij input strategy).probability
(FixedStoppedDecoderLaw.highAgreement parameters i j hij input) ^ 2 ≤
FixedRankContradiction.gamma parameters (j.val + 1) ^ 2 / 8 + 2 * parameters.accuracy := by
rw [probability_eq_mean]
apply mean_square_le_const
intro pref
apply mean_square_le_const
intro external
apply mean_square_le_const
intro A
apply mean_square_le_const
intro exterior
exact directionMean_square_le parameters i j hij input strategy pref external A exterior
end
end PerfectCompleteness.FixedStoppedHighBound
end
end OAI
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