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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 | import OAI.Combinatorics.ContingencyTables.Sampling.ExactPhysicalProgram
import OAI.Combinatorics.ContingencyTables.Sampling.ExactTypedCost
namespace OAI
/-! # A physical cost bound for every finite prefix of the exact sampler -/
namespace ContingencyTables.ExactPhysicalCost
open MatchingFPRAS MatchingFPRAS.PhysicalStream MatchingFPRAS.PhysicalPost
open MatchingFPRAS.TreeTyped MatchingFPRAS.TreePost Turing
open DensePrograms ProfilePrograms ExactPhysicalProgram
attribute [local irreducible] ExactTypedProgram.program
noncomputable def typedBudget (as : List Symbol) (N B : ℕ) : ℕ :=
exactParsedMarginsRealizer.cost as+ExactCoreCost.budget (exactParsedMargins as) N+B
noncomputable def budget (as : List Symbol) (N B : ℕ) : ℕ :=
typedBudget as N B*OnlineStack.Executes.factor ExactTypedProgram.program.code
(weight as) (typedBudget as N B)
/-- The compiler's quadratic overhead is charged to the actual finite
prefix cost. The executable program is still the same unbounded sampler. -/
theorem prefix_correct_cost (as : List Symbol) (N B : ℕ)
(hB : (ExactCoreProgram.prefixExperiment (exactParsedMargins as) N).All
(fun o => o.elim 0 GreedyFeasibleTable.printMatrixRealizer.cost≤B)) :
∃ q,Implements program.table (output as)
(program.initial (Tape.mk₁ (packed as))) q ∧
(∀ g : Option MatrixCode → ℚ,
q.mean g=(ExactCoreProgram.prefixExperiment (exactParsedMargins as) N).mean g) ∧
q.cost≤budget as N B := by
obtain ⟨p,hp,hpm,hpc⟩ := ExactTypedCost.prefix_correct_cost as .nil N B hB
obtain ⟨q,hq,hqm,hqc⟩ := hp.compile (weight as) (inputStacks_bound as)
have hu := hq.up
have hi := OnlineTree.cfg_packed ExactTypedProgram.program as (some default)
change upCfg (OnlineStack.cfg (OnlineTree.initial ExactTypedProgram.program (encode as) .nil))=
program.initial (Tape.mk₁ (packed as)) at hi
rw [hi] at hu
refine ⟨q,hu,(fun g => (hqm g).trans (hpm g)),?_⟩
exact hqc.trans (Nat.mul_le_mul hpc (OnlineStack.Executes.factor_mono
(weight as) (weight as) p.cost (typedBudget as N B) le_rfl hpc))
lemma typedBudget_le_scaled (as : List Symbol) (N B : ℕ) :
typedBudget as N B≤typedBudget as 1 B*(N+1) := by
have hc := Nat.mul_le_mul_left (typedBudget as 0 B) (show 1≤N+1 by omega)
unfold typedBudget ExactCoreCost.budget at hc ⊢
simp only [Nat.one_mul,Nat.zero_mul] at hc ⊢
nlinarith
lemma factor_le_scaled (H K V : ℕ) (hV : 1≤V) :
OnlineStack.Executes.factor ExactTypedProgram.program.code H (K*V)≤
OnlineStack.Executes.factor ExactTypedProgram.program.code H K*V := by
let A := OnlineStack.accesses ExactTypedProgram.program.code
let O := OnlineStack.overhead ExactTypedProgram.program.code
have hH : H+1≤(H+1)*V := by
simpa only [Nat.mul_one] using Nat.mul_le_mul_left (H+1) hV
have hinner : H+K*V*A+1≤(H+K*A+1)*V := by nlinarith
have hA := Nat.mul_le_mul_left (2*A) hinner
have hO : 1+2*A*(H+K*V*A+1)≤(1+2*A*(H+K*A+1))*V := by nlinarith
have hfinal := Nat.mul_le_mul_left O hO
change O*(1+2*A*(H+K*V*A+1))+1≤(O*(1+2*A*(H+K*A+1))+1)*V
nlinarith
/-- One fixed coefficient bounds the compiler overhead for all prefix lengths. -/
lemma quadratic_budget (as : List Symbol) (N B : ℕ) :
budget as N B≤budget as 1 B*(N+1)^2 := by
have ht := typedBudget_le_scaled as N B
have hf := OnlineStack.Executes.factor_mono (M := ExactTypedProgram.program.code)
(weight as) (weight as) (typedBudget as N B) (typedBudget as 1 B*(N+1)) le_rfl ht
have hs := factor_le_scaled (weight as) (typedBudget as 1 B) (N+1) (by omega)
calc
budget as N B≤(typedBudget as 1 B*(N+1))*
(OnlineStack.Executes.factor ExactTypedProgram.program.code (weight as)
(typedBudget as 1 B)*(N+1)) := Nat.mul_le_mul ht (hf.trans hs)
_ = budget as 1 B*(N+1)^2 := by unfold budget; ring
end ContingencyTables.ExactPhysicalCost
end OAI
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