| import Mathlib |
| import OAI.Probability.LogConcave.OraclePrograms.CircuitParameters |
| import OAI.Probability.LogConcave.Numerics.CompileDeclaredRootNormalized |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace LogConcaveSampling.OracleCompiler |
| open MeanTree MeasureTheory Quadrature |
| open scoped Classical NNReal |
|
|
| variable {d : ℕ} |
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| def MeanSize (lam q r σ : ℝ) : Prop := |
| 0 < r ∧ r ≤ 1 ∧ 0 < σ ∧ lam*r^2 ≤ q ∧ lam*r ≤ q*σ |
|
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| def SampleSize (lam q r η : ℝ) : Prop := |
| 0 < r ∧ r ≤ 1 ∧ 0 < η ∧ η ≤ 1 ∧ lam*r^2 ≤ q*η |
|
|
| lemma MeanSize.nonneg {lam q r σ : ℝ} (hlam : 0 ≤ lam) (h : MeanSize lam q r σ) : 0 ≤ q := |
| (mul_nonneg hlam (sq_nonneg r)).trans h.2.2.2.1 |
| lemma SampleSize.nonneg {lam q r η : ℝ} (hlam : 0 ≤ lam) (h : SampleSize lam q r η) : 0 ≤ q := by |
| have hp := (mul_nonneg hlam (sq_nonneg r)).trans h.2.2.2.2 |
| exact nonneg_of_mul_nonneg_left hp h.2.2.1 |
|
|
| lemma internal_meanSize {lam Q r R D A r' n : ℝ} |
| (hlam : 0 ≤ lam) (hQ : 0 ≤ Q) (hr : 0 < r) (hr1 : r ≤ 1) |
| (hR : 0 < R) (hD : 0 < D) (hA : 0 < A) |
| (hl : lam*r^2 ≤ Q) (hlo : r*R ≤ r') (hhi : r' ≤ r) |
| (hn : r*R/(2*D) ≤ n) : |
| MeanSize lam (Q+4*D*lam*A/R) r' (declaredNoise n A) := by |
| have hr' : 0 < r' := (mul_pos hr hR).trans_le hlo |
| have hn0 : 0 < n := (div_pos (mul_pos hr hR) (by positivity)).trans_le hn |
| have hτ : 0 < declaredNoise n A := by unfold declaredNoise; positivity |
| have hloτ : r*R/(4*D*A) ≤ declaredNoise n A := by |
| have hh := div_le_div_of_nonneg_right hn (show 0 ≤ 2*A by positivity) |
| convert! hh using 1 |
| field_simp |
| ring |
| refine ⟨hr',hhi.trans hr1,hτ,?_,?_⟩ |
| · have he := (mul_le_mul_of_nonneg_left (pow_le_pow_left₀ hr'.le hhi 2) hlam).trans hl |
| exact he.trans (le_add_of_nonneg_right (by positivity)) |
| · have hh := mul_le_mul_of_nonneg_left hloτ (show 0 ≤ 4*D*lam*A/R by positivity) |
| have hid : (4*D*lam*A/R)*(r*R/(4*D*A))=lam*r := by field_simp |
| rw [hid] at hh |
| have hrh := mul_le_mul_of_nonneg_left hhi hlam |
| have hQt := mul_nonneg hQ hτ.le |
| nlinarith |
|
|
| namespace CircuitParameters |
| variable (C : CircuitParameters) |
| lemma meanCenterBudget_small {lam q r σ : ℝ} (h : MeanSize lam q r σ) (hA : 0 ≤ C.A) : |
| lam*C.meanCenterBudget r σ ≤ 2*C.A*q := by |
| have hs : lam*r/σ ≤ q := (div_le_iff₀ h.2.2.1).2 h.2.2.2.2 |
| have hh := mul_le_mul_of_nonneg_left (add_le_add hs h.2.2.2.1) hA |
| unfold meanCenterBudget |
| convert! hh using 1 <;> ring |
|
|
| lemma sampleCenterBudget_small {lam q r η : ℝ} (hlam : 0 ≤ lam) |
| (h : SampleSize lam q r η) (hA : 0 ≤ C.A) : |
| lam*C.sampleCenterBudget r ≤ C.A*q := by |
| have hq := h.nonneg hlam |
| have he := h.2.2.2.2.trans (mul_le_of_le_one_right hq h.2.2.2.1) |
| have hh := mul_le_mul_of_nonneg_left he hA |
| unfold sampleCenterBudget |
| convert! hh using 1 |
| ring |
|
|
| end CircuitParameters |
| end LogConcaveSampling.OracleCompiler |
|
|
| end |
|
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| end |
|
|
| section |
|
|
| noncomputable section |
| namespace LogConcaveSampling.OracleCompiler |
| open MeanTree Quadrature FinitePicard |
| open scoped Classical BigOperators NNReal |
|
|
| def fixedProbabilityBudget (n : ℕ) : ℝ := |
| (probabilityWeight_budget (n+1) (Nat.succ_pos n)).choose |
|
|
| def fixedHarmonicBudget (n : ℕ) : ℝ≥0 := |
| (harmonicWeight_budget (n+1) (Nat.succ_pos n)).choose |
|
|
| def fixedCenteringBudget (n : ℕ) : ℝ := |
| 1+(∑i : Fin (n+1),∑j,|centeringWeight n i j|)+ |
| ∑j,|Quadrature.weight (probabilityNodes n) j 0 1| |
|
|
| lemma fixedProbabilityBudget_spec (n : ℕ) : 0≤fixedProbabilityBudget n ∧ |
| ∀{T h : ℝ},0<T → T<1 → 0<h → |
| ∀i : ProbabilityNode T h (n+1),∑j,|probabilityWeight T h (n+1) i j|≤fixedProbabilityBudget n := |
| (probabilityWeight_budget (n+1) (Nat.succ_pos n)).choose_spec |
|
|
| lemma fixedCenteringBudget_spec (n : ℕ) : 0≤fixedCenteringBudget n ∧ |
| (∀i,∑j,|centeringWeight n i j|≤fixedCenteringBudget n) ∧ |
| (∑j,|Quadrature.weight (probabilityNodes n) j 0 1|≤fixedCenteringBudget n) := by |
| have h₁ : 0≤∑i : Fin (n+1),∑j,|centeringWeight n i j| := by positivity |
| have h₂ : 0≤∑j : Fin (n+1),|Quadrature.weight (probabilityNodes n) j 0 1| := by positivity |
| refine ⟨by unfold fixedCenteringBudget; positivity,?_,?_⟩ |
| · intro i |
| have he := Finset.single_le_sum (fun j (_ : j∈Finset.univ) => |
| show 0≤∑k,|centeringWeight n j k| by positivity) (Finset.mem_univ i) |
| unfold fixedCenteringBudget |
| linarith |
| · unfold fixedCenteringBudget |
| linarith |
|
|
| namespace CircuitParameters |
| variable (C : CircuitParameters) |
|
|
| def quadratureBudget : ℝ := ∑j,|terminalQuadratureWeight C.T C.h C.n j| |
| def derivativeBudget : ℝ := ∑j : Fin (C.m+1),|derivativeWeight (angleNodes C.m) j/C.ψ| |
| def actualA : ℝ := |
| 1+2*fixedCenteringBudget C.nc*(C.quadratureBudget*C.derivativeBudget* |
| (2*fixedProbabilityBudget C.n)+2)+4*fixedProbabilityBudget C.n+fixedHarmonicBudget C.n |
| def actualAf : ℝ := 1+fixedCenteringBudget C.nc+2*fixedProbabilityBudget C.n |
|
|
| lemma actualA_pos : 0<C.actualA := by |
| have hP := (fixedProbabilityBudget_spec C.n).1 |
| have hB := (fixedCenteringBudget_spec C.nc).1 |
| have hQ : 0≤C.quadratureBudget := by unfold quadratureBudget; positivity |
| have hS : 0≤C.derivativeBudget := by unfold derivativeBudget; positivity |
| unfold actualA |
| positivity |
| lemma actualAf_pos : 0<C.actualAf := by |
| have hP := (fixedProbabilityBudget_spec C.n).1 |
| have hB := (fixedCenteringBudget_spec C.nc).1 |
| unfold actualAf |
| positivity |
|
|
| lemma angle_mem (hψ : 0≤C.ψ) (j : Fin (C.m+1)) : |
| C.ψ*angleNodes C.m j∈Set.Icc (0:ℝ) 1 := by |
| have hj : 0≤angleNodes C.m j := by simp [angleNodes] |
| have hv : C.v≤1 := by |
| have hR : Real.sqrt (1-C.T^2)≤1 := by |
| apply (Real.sqrt_le_iff).mpr |
| exact ⟨by norm_num,by nlinarith [sq_nonneg C.T]⟩ |
| nlinarith [C.angle_small] |
| exact ⟨mul_nonneg hψ hj,(le_abs_self _).trans ((C.angle_bound j).trans hv)⟩ |
|
|
| lemma meanTree_actual_weights {d : ℕ} {r σ : ℝ} (hr : 0<r) (hσ : 0<σ) |
| (hψ : 0≤C.ψ) : weight (C.meanTree (d:=d) r σ)≤C.actualAf ∧ |
| centersBound (C.actualA*(r/σ+r^2)) (C.meanTree (d:=d) r σ) := by |
| have hT0 : 0<C.T := by linarith [C.T_lower] |
| have hP := fixedProbabilityBudget_spec C.n |
| have hB := fixedCenteringBudget_spec C.nc |
| have hP0 := hP.1 |
| have hB0 := hB.1 |
| have hQ : 0≤C.quadratureBudget := by unfold quadratureBudget; positivity |
| have hS : 0≤C.derivativeBudget := by unfold derivativeBudget; positivity |
| have hh := (harmonicWeight_budget (C.n+1) (Nat.succ_pos C.n)).choose_spec |
| apply literalMean_weights hr hσ (by rw [abs_of_pos hT0]; exact C.T_upper.le) |
| (by positivity : 0≤2*fixedProbabilityBudget C.n) (fixedHarmonicBudget C.n).coe_nonneg |
| hQ hS hB.1 C.n C.m C.N C.nc C.Nc C.meanEndpoint |
| · intro j |
| have ht := probabilityNodeTime_mem hT0 C.T_upper C.h_pos (Nat.succ_pos C.n) j |
| rw [abs_of_nonneg ht.1] |
| exact ht.2.trans C.T_upper.le |
| · intro a i |
| exact reanchorWeight_budget _ (hP.2 hT0 C.T_upper C.h_pos) a i |
| · intro k i |
| exact hh _ (C.angle_mem hψ k).1 (C.angle_mem hψ k).2 i |
| · exact le_rfl |
| · exact le_rfl |
| · exact hB.2.1 |
| · exact hB.2.2 |
| · unfold actualA |
| have h := (fixedHarmonicBudget C.n).coe_nonneg |
| linarith [hP.1] |
| · unfold actualA |
| have he : 0≤2*fixedCenteringBudget C.nc* |
| (C.quadratureBudget*C.derivativeBudget*(2*fixedProbabilityBudget C.n)+2) := by positivity |
| linarith |
| · unfold actualAf |
| linarith [hP.1] |
|
|
| lemma sampleTree_actual_weights {d : ℕ} {r η : ℝ} (hr : 0<r) (hη : 0<η) (hη1 : η≤1) : |
| weight (C.sampleTree (d:=d) r η hη hη1)≤C.actualAf*r ∧ |
| centersBound (C.actualA*r^2) (C.sampleTree (d:=d) r η hη hη1) := by |
| have hp := sampleCorrelation_properties hη hη1 |
| have hp0 : 0<sampleCorrelation η := by linarith [hp.1] |
| have hP := fixedProbabilityBudget_spec C.n |
| have hB := fixedCenteringBudget_spec C.nc |
| have hP0 := hP.1 |
| have hB0 := hB.1 |
| have hQ : 0≤C.quadratureBudget := by unfold quadratureBudget; positivity |
| have hS : 0≤C.derivativeBudget := by unfold derivativeBudget; positivity |
| apply literalSample_weights hr hP.1 hp.1 (C.n+1) C.N (C.sampleEndpoint η hη hη1) |
| · intro j |
| have ht := probabilityNodeTime_mem hp0 hp.2.1 C.h_pos (Nat.succ_pos C.n) j |
| rw [abs_of_nonneg ht.1] |
| exact ht.2.trans hp.2.1.le |
| · exact hP.2 hp0 hp.2.1 C.h_pos |
| · unfold actualA |
| have he : 0≤2*fixedCenteringBudget C.nc* |
| (C.quadratureBudget*C.derivativeBudget*(2*fixedProbabilityBudget C.n)+2) := by positivity |
| have hh := (fixedHarmonicBudget C.n).coe_nonneg |
| linarith [hP.1] |
| · unfold actualAf |
| linarith [hB.1] |
|
|
| end CircuitParameters |
| end LogConcaveSampling.OracleCompiler |
|
|
| end |
|
|
| end |
|
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| end OAI |
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