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import OAI.Computability.PerfectCompleteness.Construction.HierarchicalAdviceFromBuckets
import OAI.Computability.PerfectCompleteness.Decoding.HierarchicalDecoderMeetingLemmas
import OAI.Computability.PerfectCompleteness.Foundations.HierarchicalProjectedSliceLemmas
import OAI.Computability.PerfectCompleteness.Reduction.FixedCallBudgetLemmas
namespace OAI
namespace PerfectCompleteness.FixedComparisonErrors
open FixedParameters ChildBlockCardinality
noncomputable section
variable {δ : ℚ} {hδ : 0 < δ} (p : Parameters δ hδ)
theorem comparison_errors {retained : Nat} (hcalls : retained ≤ calls p.plan)
(height : Nat) :
let L : ℝ := bound (branch p) height (FixedRows.sourceLength p.plan hδ)
retained (FixedRows.rows p.plan)
let m := p.cubeSize height
Real.sqrt (L ^ 2 / (m : ℝ) ^ 3) / 2 < p.accuracy ∧
Real.sqrt ((1 + L ^ 2 / (m : ℝ) ^ 4) ^ (m ^ 3) - 1) / 2 < p.accuracy ∧
2 * ((1 / (m : ℝ) ^ 2) * L / (1 - 1 / (m : ℝ) ^ 2)) < p.accuracy := by
let L : ℝ := bound (branch p) height (FixedRows.sourceLength p.plan hδ)
retained (FixedRows.rows p.plan)
let B : ℝ := bound (branch p) height (FixedRows.sourceLength p.plan hδ)
(calls p.plan) (FixedRows.rows p.plan)
let m := p.cubeSize height
have hL : 0 ≤ L := Nat.cast_nonneg _
have hB : 0 ≤ B := Nat.cast_nonneg _
have hLB : L ≤ B := by
dsimp only [L, B]
exact_mod_cast FixedCallBudget.child_bound_mono hcalls (branch p) height
(FixedRows.sourceLength p.plan hδ) (FixedRows.rows p.plan)
have hsq : L ^ 2 ≤ B ^ 2 := (sq_le_sq₀ hL hB).mpr hLB
have hm : (2 : ℝ) ≤ m := by exact_mod_cast (p.cube_errors height).1
have hm2 : (0 : ℝ) < (m : ℝ) ^ 2 := by positivity
have hprob : (1 : ℝ) / (m : ℝ) ^ 2 < 1 := by
apply (div_lt_one hm2).mpr
nlinarith
have hfixed := (p.cube_errors height).2
change Real.sqrt (L ^ 2 / (m : ℝ) ^ 3) / 2 < p.accuracy ∧
Real.sqrt ((1 + L ^ 2 / (m : ℝ) ^ 4) ^ (m ^ 3) - 1) / 2 < p.accuracy ∧
2 * ((1 / (m : ℝ) ^ 2) * L / (1 - 1 / (m : ℝ) ^ 2)) < p.accuracy
refine ⟨?_, ?_, ?_⟩
· apply lt_of_le_of_lt _ hfixed.1
apply div_le_div_of_nonneg_right _ (by norm_num : (0 : ℝ) ≤ 2)
apply Real.sqrt_le_sqrt
exact div_le_div_of_nonneg_right hsq (by positivity)
· apply lt_of_le_of_lt _ hfixed.2.1
apply div_le_div_of_nonneg_right _ (by norm_num : (0 : ℝ) ≤ 2)
apply Real.sqrt_le_sqrt
apply sub_le_sub_right
apply pow_le_pow_left₀ (by positivity)
exact add_le_add_right
(div_le_div_of_nonneg_right hsq
(by positivity : (0 : ℝ) ≤ (m : ℝ) ^ 4)) 1
· apply lt_of_le_of_lt _ hfixed.2.2.1
apply mul_le_mul_of_nonneg_left _ (by norm_num : (0 : ℝ) ≤ 2)
apply div_le_div_of_nonneg_right _ (by linarith)
exact mul_le_mul_of_nonneg_left hLB (by positivity)
end
end PerfectCompleteness.FixedComparisonErrors
namespace PerfectCompleteness.HierarchicalFixedAdvice
noncomputable section
open scoped Classical
open TreeSourceSpaces HierarchicalArrays
open InitialParameters UpperParameterScalars
open UniqueGamesTheorem.Foundations.Games
open UniqueGamesTheorem.Appendix.RankLevelFilter (linearMapFintype)
attribute [local instance] linearMapFintype
variable {δ : ℚ} {hδ : 0 < δ} (p : FixedParameters.Parameters δ hδ)
{branch : Nat → Nat} {t : Nat} {Ω P : Type*} [Fintype Ω] [Fintype P]
(S : HierarchicalAdviceExperiment.Experiment branch (FixedRows.rows p.plan) p.plan.depth t Ω)
local instance backgroundFintype : Fintype (HierarchicalAdviceExperiment.Background S) :=
Fintype.ofFinite _
local instance rowSpaceFintype : Fintype (NodeEmbedding.RowSpace S.slots S.upper) :=
Fintype.ofFinite _
local instance inputFintype :
(b : HierarchicalAdviceExperiment.Background S) → Fintype (HierarchicalAdviceExperiment.Input S b) :=
HierarchicalAdviceExperiment.inputFintype S
local instance inputFiniteDimensional :
(b : HierarchicalAdviceExperiment.Background S) →
FiniteDimensional F2 (HierarchicalAdviceExperiment.Input S b) :=
HierarchicalAdviceExperiment.inputFiniteDimensional S
theorem rows_positive : 0 < FixedRows.rows p.plan (Nodes.height S.upper) := by
exact p.plan.rows_pos (p.plan.depth - Nodes.height S.upper)
theorem adviceFactor_eq :
HierarchicalAdviceFromBuckets.adviceFactor S p.plan.order (inverse δ) =
q (inverse δ) p.plan.order (FixedRows.rows p.plan (Nodes.height S.upper)) := by
simp only [HierarchicalAdviceFromBuckets.adviceFactor, q, g, h,
div_eq_mul_inv, pow_two]
ring
theorem paired_error_small : 10 * p.accuracy ≤ simultaneous δ ^ 2 / 16 := by
have hs := p.accuracy_small 0 (Nat.zero_le _)
simp only [CommonAccuracy.tolerance, lt_min_iff] at hs
linarith [hs.2.1]
theorem coarse_error_small :
10 * p.accuracy ≤ p.plan.density *
HierarchicalAdviceFromBuckets.adviceFactor S p.plan.order (inverse δ) / 16 := by
rw [adviceFactor_eq p S]
have hs := p.accuracy_small (Nodes.height S.upper) (Nodes.height_le S.upper)
simp only [CommonAccuracy.tolerance, lt_min_iff] at hs
linarith [hs.2.2.1]
variable (externalLaw : FiniteDistribution P)
(background : P → HierarchicalAdviceExperiment.Background S)
(scalarLaw : P → FiniteDistribution (NodeEmbedding.RowSpace S.slots S.upper))
theorem prediction_difference
(hmass : simultaneous δ / 2 ≤ HierarchicalAdviceFromBuckets.usefulProbability
S (useful δ) externalLaw background scalarLaw (rows_positive p S))
(hpaired : (HierarchicalAdviceFromBuckets.pairedLaw S externalLaw background
scalarLaw (rows_positive p S)).totalVariation
(HierarchicalAgreementMean.referenceLaw S.slots S.upper
(HierarchicalAdviceFromBuckets.backgroundLaw S externalLaw background)
(rows_positive p S)) ≤ 10 * p.accuracy)
(hcoarse : ((HierarchicalAdviceExperiment.originalLaw S p.plan.order).pushforward
(HierarchicalAdviceExperiment.observe S p.plan.order)).totalVariation
((HierarchicalAdviceExperiment.referenceLaw S
(HierarchicalAdviceFromBuckets.backgroundLaw S externalLaw background) p.plan.order).pushforward (HierarchicalAdviceExperiment.referenceObserve S p.plan.order)) ≤
10 * p.accuracy) :
b (useful δ) p.plan.density (inverse δ) p.plan.order
(FixedRows.rows p.plan (Nodes.height S.upper)) ≤
(HierarchicalAdviceExperiment.originalLaw S p.plan.order).probability
(HierarchicalAdviceExperiment.prediction S (useful δ) p.plan.order p.plan.density) -
rhoPred (useful δ) p.plan.density *
(HierarchicalAdviceExperiment.originalLaw S p.plan.order).probability
(fun x => HierarchicalAdviceExperiment.J S (useful δ) p.plan.order p.plan.density
(HierarchicalAdviceExperiment.observe S p.plan.order x)) := by
have hbound := HierarchicalAdviceFromBuckets.prediction_difference S (useful δ)
externalLaw background scalarLaw (rows_positive p S) (useful_pos hδ).le
(simultaneous δ) (simultaneous_pos hδ).le hmass
(hpaired.trans (paired_error_small p))
p.plan.order p.plan.density (inverse δ) (10 * p.accuracy)
(inverse_pos hδ) (by unfold inverse; linarith)
p.plan.density_pos p.plan.density_lt p.plan.spectral
(by have := p.accuracy_pos; positivity) (coarse_error_small p S) hcoarse
rw [adviceFactor_eq p S] at hbound
have hfactor : useful δ * p.plan.density *
q (inverse δ) p.plan.order (FixedRows.rows p.plan (Nodes.height S.upper)) / 4 =
b (useful δ) p.plan.density (inverse δ) p.plan.order
(FixedRows.rows p.plan (Nodes.height S.upper)) := by
unfold b rhoPred
ring
rw [hfactor] at hbound
exact hbound
theorem good_record_mass_ge {R : Type*} [Fintype R]
(hmass : simultaneous δ / 2 ≤ HierarchicalAdviceFromBuckets.usefulProbability
S (useful δ) externalLaw background scalarLaw (rows_positive p S))
(hpaired : (HierarchicalAdviceFromBuckets.pairedLaw S externalLaw background
scalarLaw (rows_positive p S)).totalVariation
(HierarchicalAgreementMean.referenceLaw S.slots S.upper
(HierarchicalAdviceFromBuckets.backgroundLaw S externalLaw background)
(rows_positive p S)) ≤ 10 * p.accuracy)
(hcoarse : ((HierarchicalAdviceExperiment.originalLaw S p.plan.order).pushforward
(HierarchicalAdviceExperiment.observe S p.plan.order)).totalVariation
((HierarchicalAdviceExperiment.referenceLaw S
(HierarchicalAdviceFromBuckets.backgroundLaw S externalLaw background) p.plan.order).pushforward (HierarchicalAdviceExperiment.referenceObserve S p.plan.order)) ≤
10 * p.accuracy)
(record : Ω × HierarchicalAdviceExperiment.Advice S p.plan.order → R) :
b (useful δ) p.plan.density (inverse δ) p.plan.order
(FixedRows.rows p.plan (Nodes.height S.upper)) ≤
(HierarchicalAdviceExperiment.originalLaw S p.plan.order).probability
(fun x => HierarchicalAdviceExperiment.J S (useful δ) p.plan.order p.plan.density
(HierarchicalAdviceExperiment.observe S p.plan.order x) &&
PositiveFiberMass.goodRecord (HierarchicalAdviceExperiment.originalLaw S p.plan.order) record
(fun y => HierarchicalAdviceExperiment.J S (useful δ) p.plan.order p.plan.density
(HierarchicalAdviceExperiment.observe S p.plan.order y))
(HierarchicalAdviceExperiment.prediction S (useful δ) p.plan.order p.plan.density)
(rhoPred (useful δ) p.plan.density) (record x)) :=
(prediction_difference p S externalLaw background scalarLaw hmass hpaired hcoarse).trans
(HierarchicalAdviceExperiment.prediction_difference_le_good_mass S (useful δ)
(useful_pos hδ).le p.plan.order p.plan.density p.plan.density_pos.le record)
end
end PerfectCompleteness.HierarchicalFixedAdvice
namespace PerfectCompleteness.HierarchicalFixedProjectedMeeting
noncomputable section
open scoped Classical TensorProduct
open TreeSourceSpaces HierarchicalArrays HierarchicalProjectedSlice
open UpperParameterScalars
open UniqueGamesTheorem.Foundations.Games
open UniqueGamesTheorem.Appendix.RankLevelFilter (linearMapFintype)
attribute [local instance] linearMapFintype
attribute [local instance] RightDecoder.characterFintype RightDecoder.scalarFintype
variable {δ : ℚ} (plan : FixedRows.Plan δ) (hδ : 0 < δ)
{branch : Nat → Nat} {t : Nat}
(slots projected : RecursiveSpaces.Slots branch plan.depth → Fin t → MixedSupport.Slot)
(projection : ∀ s j, MixedSupport.Projection (slots s j) (projected s j))
(upper : Nodes branch plan.depth) (lowerLevel : Nat)
(background : HierarchicalMatrixTable.Background
(rows := FixedRows.rows plan) slots upper)
local instance rowSpaceFintype : Fintype (NodeEmbedding.RowSpace slots upper) :=
Fintype.ofFinite _
local instance valueFintype : Fintype
(Block (FixedRows.rows plan) upper ×
HierarchicalMatrixTable.SideOutput (rows := FixedRows.rows plan) upper) :=
Fintype.ofFinite _
local instance upperDualFintype : Fintype (Module.Dual F2 (NodeEmbedding.RowSpace slots upper)) :=
LeftDecoder.dualFintype (V := NodeEmbedding.RowSpace slots upper)
variable {Ω : Type*} [Fintype Ω]
(original : FiniteDistribution Ω) (arrays : Ω → Arrays slots (FixedRows.rows plan))
(lowerEvent : Ω → Bool)
(σ : KeyStrategy.Strategy (TreeCanonical.locationCount branch plan.depth t))
(A : ManyGoodRows.RowMap (Block (FixedRows.rows plan) upper) plan.order)
(known : HiddenBucketBias.VisibleDirection (LinearMap.ker A) → NodeEmbedding.NodeH projected upper)
(lower : Nodes branch plan.depth) (a : Block (FixedRows.rows plan) lower)
(cut : OwnInputReference.Cut upper lower)
(input : OwnInputReference.Input projected (FixedRows.rows plan) upper lower (LinearMap.ker A) a)
include hδ
theorem meeting_probability_lower
(hgood : VisibleGood slots projected projection upper lowerLevel background
original arrays lowerEvent (InitialParameters.useful δ) σ plan.density A known)
(hne : ∃ T, J slots projected projection upper lowerLevel background
original arrays lowerEvent (InitialParameters.useful δ) σ plan.density A known T = true)
(hmatch : rhoPred (InitialParameters.useful δ) plan.density ≤
RightDecoderWitness.conditionalMatch projected (FixedRows.rows plan) upper lower
(LinearMap.ker A) a (FixedRows.repeats plan) cut σ input
(columnSpace slots projected projection upper lowerLevel background
original arrays lowerEvent (InitialParameters.useful δ) σ plan.density A known hgood)
(target slots projected projection upper lowerLevel background
original arrays lowerEvent (InitialParameters.useful δ) σ plan.density A known hgood hne)
(nativeCorrection slots projected projection upper lowerLevel background
original arrays lowerEvent (InitialParameters.useful δ) σ plan.density A known hgood)
(visibleOffset slots projected projection upper lowerLevel background
original arrays lowerEvent (InitialParameters.useful δ) σ plan.density A known hgood)) :
(1 / (2 : ℝ) ^ plan.order) *
(tau (InitialParameters.useful δ) plan.density plan.order
(FixedRows.rows plan (Nodes.height upper)) ^ 2 /
(2 * (2 : ℝ) ^ FixedRows.rows plan (Nodes.height upper))) ≤
((HierarchicalProjectedMeeting.leftLaw slots projected projection upper lowerLevel
background original arrays lowerEvent (InitialParameters.useful δ) σ plan.density A known).product
(RightDecoder.law projected (FixedRows.rows plan) upper lower (LinearMap.ker A) a
(FixedRows.repeats plan) cut σ input
(tau (InitialParameters.useful δ) plan.density plan.order
(FixedRows.rows plan (Nodes.height upper))))).probability
(fun pair => decide ((ProjectedNodeEmbedding.intoOriginal projection upper).dualMap
pair.1 = pair.2)) := by
simpa only [UpperParameterScalars.tau] using
(HierarchicalProjectedMeeting.meeting_probability_lower slots projected projection upper lowerLevel
background original arrays lowerEvent (InitialParameters.useful δ) σ plan.density A known
lower a (FixedRows.repeats plan) cut input hgood hne
(rhoPred (InitialParameters.useful δ) plan.density)
(h plan.order (FixedRows.rows plan (Nodes.height upper)))
(rhoPred_pos (InitialParameters.useful_pos hδ) plan.density_pos)
(h_pos plan.order (FixedRows.rows plan (Nodes.height upper)))
(HierarchicalFixedDecoderMeeting.repetitionsBalanced plan)
(HierarchicalFixedDecoderMeeting.slice_error plan (Nodes.height upper))
(HierarchicalFixedDecoderMeeting.list_error plan (Nodes.height upper))
(HierarchicalFixedDecoderMeeting.kernel_excluded plan (Nodes.height upper) A)
le_rfl hmatch)
end
end PerfectCompleteness.HierarchicalFixedProjectedMeeting
end OAI
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