| 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) |
|
|
| / |
| 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 |
|
|
| / |
| 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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|
