| import Mathlib |
| import OAI.Probability.LogConcave.Numerics.ScalarPicard |
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| namespace OAI |
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| section |
| noncomputable section |
| namespace LogConcaveSampling.OracleCompiler |
| open MeanTree |
| open scoped Classical |
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| variable {d : ℕ} |
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| def RoutineBudget (B : ℕ) (η₀ η₁ : ℝ) : {k : RoutineKind} → RoutineValue d k → Prop |
| | .mean,M => ∀r σ,MeanBudget B r (M r σ) |
| | .sample,S => ∀r η,η=η₀ ∨ η=η₁ → SampleBudget B r η (S r η) |
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| lemma RoutineBudget.mono {B C : ℕ} {η₀ η₁ : ℝ} {k : RoutineKind} {R : RoutineValue d k} |
| (h : RoutineBudget B η₀ η₁ R) (hBC : B≤C) : RoutineBudget C η₀ η₁ R := by |
| cases k with |
| | mean => exact fun r σ => (h r σ).mono hBC |
| | sample => exact fun r η hη => (h r η hη).mono hBC |
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| theorem constructRoutine_budget (C : CircuitParameters) {t : ℝ} (ht : 0<t) (hs : t≤1/44) |
| (c : ℕ) (η₁ : ℝ) (hD : 0<C.D) |
| (hcm : ∀r σ,cost (C.meanTree (d:=d) r σ)≤c) |
| (hcs : ∀r η,η=Real.sqrt (1-C.T^2)/C.T ∨ η=η₁ → |
| ∀hη : 0<η, ∀hη1 : η≤1,cost (C.sampleTree (d:=d) r η hη hη1)≤c) : |
| ∀L,((2*(40*c+40)^L.rank t : ℕ):ℝ)≤C.D^2 → |
| RoutineBudget (2*(40*c+40)^L.rank t) (Real.sqrt (1-C.T^2)/C.T) η₁ |
| (constructRoutine d C ht hs L) := by |
| let K := 40*c+40 |
| have hK : 0<K := by dsimp [K]; omega |
| have hp (n : ℕ) : 2≤2*K^n := by have := Nat.one_le_pow n _ hK; omega |
| apply routine_induction ht hs |
| · intro L hL hcap |
| rw [constructRoutine,dite_eq_left hL] |
| cases L with | mk k b p => |
| cases k with |
| | mean => exact fun r σ => (terminalMeanProgram_budget r σ).mono (hp _) |
| | sample => exact fun r η _ => (terminalReservedProgram_budget r η).mono (hp _) |
| · intro L hL ih hcap |
| let n := L.rank t-1 |
| let B := 2*K^n |
| have hi : (RoutineLabel.mk .mean (L.b-10*t) (L.p-1/4)).rank t<L.rank t := |
| RoutineStep.rank_lt ht hs hL (by cases L; exact RoutineStep.internal _ .mean _ _) |
| have hpos : 0<L.rank t := by omega |
| have hn : n+1=L.rank t := by dsimp [n]; omega |
| have hB : 1≤B := (by have := hp n; exact le_trans (by omega) this) |
| have hBparent : B≤2*K^L.rank t := Nat.mul_le_mul_left 2 (Nat.pow_le_pow_right hK (by dsimp [n]; omega)) |
| have hBD : (B:ℝ)≤C.D^2 := le_trans (by exact_mod_cast hBparent) hcap |
| have hbfinal : K*B=2*K^L.rank t := by rw [←hn]; dsimp [B]; rw [pow_succ]; ring |
| have child (L' : RoutineLabel) (hh : RoutineStep t L L') : |
| RoutineBudget B (Real.sqrt (1-C.T^2)/C.T) η₁ (constructRoutine d C ht hs L') := by |
| have hr := hh.rank_lt ht hs hL |
| have hb : 2*K^L'.rank t≤B := Nat.mul_le_mul_left 2 (Nat.pow_le_pow_right hK (by dsimp [n]; omega)) |
| exact (ih L' hh (le_trans (by exact_mod_cast hb.trans hBparent) hcap)).mono hb |
| rw [constructRoutine,dite_eq_right hL] |
| cases L with | mk k b p => |
| cases k with |
| | mean => |
| have hf := child ⟨.mean,b+t,p⟩ (RoutineStep.meanFinal b p) |
| have hi := child ⟨.mean,b-10*t,p-1/4⟩ (RoutineStep.internal .mean .mean b p) |
| have hs' := child ⟨.sample,b-10*t,p-1/4⟩ (RoutineStep.internal .mean .sample b p) |
| intro r σ |
| have hh := C.meanFrom_budget hB hD hBD |
| (constructRoutine d C ht hs ⟨.mean,b+t,p⟩) |
| (constructRoutine d C ht hs ⟨.mean,b-10*t,p-1/4⟩) |
| (constructRoutine d C ht hs ⟨.sample,b-10*t,p-1/4⟩) |
| (fun s r σ _ => by cases s; exact hi r σ; exact hf r σ) |
| (fun r _ => hs' r _ (Or.inl rfl)) r σ (hcm r σ) |
| rw [show (40*c+40)*B=2*(40*c+40)^({kind:=RoutineKind.mean,b:=b,p:=p}:RoutineLabel).rank t from hbfinal] at hh |
| exact hh |
| | sample => |
| have hf := child ⟨.mean,b,p⟩ (RoutineStep.sampleFinal b p) |
| have hi := child ⟨.mean,b-10*t,p-1/4⟩ (RoutineStep.internal .sample .mean b p) |
| intro r η hη |
| have hh := C.sampleFrom_budget hB hD hBD |
| (constructRoutine d C ht hs ⟨.mean,b,p⟩) |
| (constructRoutine d C ht hs ⟨.mean,b-10*t,p-1/4⟩) |
| (fun s r σ _ => by cases s; exact hi r σ; exact hf r σ) r η |
| (fun _ hηpos hη1 => hcs r η hη hηpos hη1) |
| rw [show (40*c+40)*B=2*(40*c+40)^({kind:=RoutineKind.sample,b:=b,p:=p}:RoutineLabel).rank t from hbfinal] at hh |
| exact hh |
| end LogConcaveSampling.OracleCompiler |
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| end |
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| end |
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| end OAI |
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