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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 | import OAI.Combinatorics.ContingencyTables.Sampling.ExactGatePublic
import OAI.Combinatorics.ContingencyTables.Sampling.ExactPublicLaw
import OAI.Combinatorics.ContingencyTables.Machines.QuadraticPrefixRuntime
namespace OAI
/-! # Summability of the actual exact sampler's public runtime -/
namespace ContingencyTables.ExactExpectedTime
open MatchingFPRAS MatchingFPRAS.PhysicalStream MatchingFPRAS.PhysicalPost Turing
open DensePrograms SmallGraphProfiles PhysicalRetryTail
open scoped BigOperators Classical
noncomputable section
variable {m n : ℕ} (r : Fin m → ℕ) (c : Fin n → ℕ)
local notation "dim" => CompletionCounts.dimensionAllowance (I := Fin m) (J := Fin n)
attribute [local irreducible] ExactPublicPrefix.program ExactTablePrefix.experiment
PublishedInputs.FiniteBoxPrekopaLeindler PublishedInputs.FiniteCheeger
/-- Expected runtime is bounded by the explicit costs of the ordinary
prefix and the probability-weighted correction branch. -/
theorem bound
(hPL : ∀ q,PublishedInputs.FiniteBoxPrekopaLeindler q)
(hCheeger : ∀ q k,PublishedInputs.FiniteCheeger
(GridBoundary.Grid (I := Fin q) (4*k^8)))
(htotal : (∑ i,r i)=∑ j,c j) (B L : ℕ)
(hB : ∀ X : Table r c,GreedyFeasibleTable.printMatrixRealizer.cost (matrixCode X.val)≤B)
(hL : ∀ X : Table r c,(ExactPublicPrefix.word r c X).length≤L) :
let S := ExactGatePublic.budget r c B L
let T := ExactPublicCost.budget r c 1 B L+1
let tail := PhysicalRetryTail.tail ExactPublicPrefix.program.table
(ExactPublicPrefix.program.initial (Tape.mk₁ (Algorithms.encodeMargins r c)))
Summable tail ∧ (∑' t,tail t)≤(S:ℝ)+11*(1/(2:ℝ)^correctionPrecision r c)*T := by
dsimp only
let S := ExactGatePublic.budget r c B L
let T := ExactPublicCost.budget r c 1 B L+1
let M := ExactPublicPrefix.program.table
let init := ExactPublicPrefix.program.initial (Tape.mk₁ (Algorithms.encodeMargins r c))
refine QuadraticPrefixRuntime.expected_time (PhysicalRetryTail.tail M init) S T
(Nat.zero_lt_succ _) (1/(2:ℝ)^correctionPrecision r c) (by positivity)
(tail_nonnegative M init) ?_ (tail_antitone M init) ?_ ?_
· intro t
have h := PhysicalEventBounds.mean_mono (experiment M t init) live (fun _ => 1) live_le_one
rw [PhysicalPartialSequence.mean_const] at h
unfold PhysicalRetryTail.tail
exact_mod_cast h
· obtain ⟨p,hp,hm,hc⟩ := ExactGatePublic.prefix_correct_cost r c htotal B L hB hL
have h := tail_le_failure M (ExactGatePublic.publicOutput r c) init p hp
(ExactGatePublic.publicOutput_halted r c) S hc
rw [hm,ExactGatePublic.failure r c htotal] at h
simpa only [Rat.cast_div,Rat.cast_one,Rat.cast_pow,Rat.cast_ofNat] using h
· intro N
obtain ⟨p,hp,hm,hc⟩ := ExactPublicCost.prefix_correct_cost r c htotal N B L hB hL
have hcost : p.cost≤S+T*(N+1)^2 := by
have hq := ExactPublicCost.quadratic_budget r c N B L
have ht := Nat.mul_le_mul_right ((N+1)^2)
(Nat.le_succ (ExactPublicCost.budget r c 1 B L))
exact (hc.trans (hq.trans ht)).trans (Nat.le_add_left _ _)
have h := tail_le_failure M (ExactPublicPrefix.output r c) init p hp
(ExactPublicPrefix.output_halted r c) _ hcost
rw [hm] at h
have hf := ExactTablePrefix.failure_bound r c (hPL _) (hCheeger _ dim) htotal N
have hfr := (Rat.cast_le (K := ℝ)).mpr hf
exact h.trans (by
simpa only [Rat.cast_mul,Rat.cast_div,Rat.cast_one,Rat.cast_pow,Rat.cast_ofNat] using hfr)
/-- The physical tail is summable for every equal-total margin pair. -/
theorem summable
(hPL : ∀ q,PublishedInputs.FiniteBoxPrekopaLeindler q)
(hCheeger : ∀ q k,PublishedInputs.FiniteCheeger
(GridBoundary.Grid (I := Fin q) (4*k^8)))
(htotal : (∑ i,r i)=∑ j,c j) :
Summable (fun t => 1-Algorithms.haltMass ExactPublicPrefix.machine (Algorithms.encodeMargins r c) t) := by
obtain ⟨p,hp⟩ := ExactPrintCost.polynomial_print_bound
let B := p.eval (ExactPrintCost.matrixSize (n := n) r)
have hB (X : Table r c) : GreedyFeasibleTable.printMatrixRealizer.cost (matrixCode X.val)≤B :=
(Nat.le_add_right _ _).trans (hp m n r c X)
have hL (X : Table r c) : (ExactPublicPrefix.word r c X).length≤B :=
(MatchingFPRAS.TreeTyped.list_length_le_weight _).trans
((Nat.le_add_left _ _).trans (hp m n r c X))
have hs := (bound r c hPL hCheeger htotal B B hB hL).1
have he : (fun t => 1-Algorithms.haltMass ExactPublicPrefix.machine (Algorithms.encodeMargins r c) t)=
PhysicalRetryTail.tail ExactPublicPrefix.program.table
(ExactPublicPrefix.program.initial (Tape.mk₁ (Algorithms.encodeMargins r c))) := by
funext t
exact PhysicalPublicLaw.tail_eq _ _ _
rw [he]
exact hs
end
end ContingencyTables.ExactExpectedTime
end OAI
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