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dd14c26 | 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 | import Mathlib
import OAI.Probability.LogConcave.Numerics.ScalarPicard
namespace OAI
section
noncomputable section
namespace LogConcaveSampling.OracleCompiler
open MeanTree
open scoped Classical
variable {d : ℕ}
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)
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
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
end
end
end OAI
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