| import Mathlib |
| import OAI.Probability.LogConcave.OraclePrograms.StandaloneCorrect |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace LogConcaveSampling |
| open Filter MeasureTheory ProbabilityTheory |
| open scoped Classical NNReal Topology |
|
|
| lemma LogPowerRate.envelope {q : ℕ → ℝ} {a : ℝ} (hq : LogPowerRate q a) : |
| ∃S : ℝ,0≤S ∧ ∃k : ℕ,∀ᶠ d : ℕ in atTop,|q d|≤S*dimensionLog d^k*(d:ℝ)^(-a) := by |
| obtain ⟨k,hk⟩ := hq |
| obtain ⟨C,hC⟩ := hk.bound |
| refine ⟨|C|,abs_nonneg _,k,?_⟩ |
| filter_upwards [hC] with d hd |
| have hn : 0≤dimensionLog d^k*(d:ℝ)^(-a) := by positivity [dimensionLog_nonneg d] |
| rw [Real.norm_eq_abs,Real.norm_of_nonneg hn] at hd |
| exact (hd.trans (mul_le_mul_of_nonneg_right (le_abs_self C) hn)).trans_eq (by ring) |
|
|
| lemma source_error_square (κ J : ℝ) {d : ℕ} (_ : 0<d) : |
| ((d:ℝ)^(-(κ*J)))^2=(d:ℝ)^(-(2*κ*J)) := by |
| have hd0 : (0:ℝ)≤d := Nat.cast_nonneg d |
| rw [←Real.rpow_natCast,←Real.rpow_mul hd0] |
| congr 1 |
| push_cast |
| ring |
|
|
| namespace OracleCompiler |
|
|
| theorem exists_source_numerical_ready {κ J t w : ℝ} |
| (hκ : 0<κ) (hJ : 1≤J) (ht : 0<t) (hts : t<1/100) |
| (hw : 0<w) (hwt : w<t) : ∃n : ℕ, |
| J+10<w*((n:ℝ)-3) ∧ J+10<t*((n:ℝ)-3) ∧ 4*(J+10)<(n:ℝ) ∧ |
| ∀q : ℕ → ℝ,LogPowerRate q (κ*(1/2)) → |
| ∀ᶠ d : ℕ in atTop,∀C : CircuitParameters, |
| SourceShape κ t w n (numericalLayerCount J) d C → |
| ∀{F : Point d → ℝ} {lam : ℝ≥0},∀hF : Primitive F lam,0<lam → (lam:ℝ)≤1 → |
| ∀hql : q d≤1/4, |
| C.MeanNumericalReady hF (q d) ((d:ℝ)^(-(κ*J))) hql ∧ |
| C.SampleNumericalReady hF (q d) ((d:ℝ)^(-(κ*J))) := by |
| obtain ⟨n,hnw,hnt,hnn,hmean⟩ := exists_calibrated_mean_circuit hκ hJ |
| (b:=1/2) le_rfl ht hts hw hwt |
| have hnw' : J+10<w*((n+1:ℕ)-3) := by push_cast; linarith |
| refine ⟨n,hnw,hnt,hnn,?_⟩ |
| intro q hq |
| obtain ⟨S,hS,k,henv⟩ := (hq.const_mul 2).envelope |
| have hm := hmean hS k |
| have hs := calibrated_sample_circuit_order hκ hJ (b:=1/2) le_rfl hw (n+1) |
| (Nat.succ_pos n) hnw' hS k |
| filter_upwards [hm,hs,henv,eventually_ge_atTop (1:ℕ)] with d hmd hsd hqd hd |
| intro C hC F lam hF hlam hlam1 hql |
| have hd0 : 0<d := hd |
| have hsquare := source_error_square κ J hd0 |
| have heψ : -(κ*(4*t))=-(4*(κ*t)) := by ring |
| obtain ⟨hT0,hT1,hh,hmc⟩ := hmd |
| obtain ⟨hh',hsc⟩ := hsd |
| have hbound {r σ : ℝ} (hz : MeanSize lam (q d) r σ) : |
| (lam:ℝ)*r^2≤S*dimensionLog d^k*(d:ℝ)^(-(κ*(1/2))) ∧ |
| (lam:ℝ)*r/(σ/2)≤S*dimensionLog d^k*(d:ℝ)^(-(κ*(1/2))) := by |
| have hq0 := hz.nonneg lam.coe_nonneg |
| have he : 2*q d≤S*dimensionLog d^k*(d:ℝ)^(-(κ*(1/2))) := (le_abs_self _).trans hqd |
| constructor |
| · linarith [hz.2.2.2.1] |
| · apply le_trans _ he |
| apply (div_le_iff₀ (by positivity [hz.2.2.1] : 0<σ/2)).mpr |
| nlinarith [hz.2.2.2.2] |
| constructor |
| · intro x r σ hz |
| have hL : (lam:ℝ)*r≤1 := (mul_le_of_le_one_left hz.1.le hlam1).trans hz.2.1 |
| obtain ⟨hl,hp⟩ := hmc hF x hz.1 hz.2.1 hlam hL |
| (show 0<σ/2 by positivity [hz.2.2.1]) (hbound hz).1 (hbound hz).2 |
| simpa only [CircuitParameters.MeanNumericalReady,hC.n_eq,hC.N_eq,hC.T_eq,hC.h_eq, |
| hC.ψ_eq,heψ,hsquare] using hp |
| · intro x r η hz hl |
| have hL : (lam:ℝ)*r≤1 := (mul_le_of_le_one_left hz.1.le hlam1).trans hz.2.1 |
| have hq0 := hz.nonneg lam.coe_nonneg |
| have hb : (lam:ℝ)*r^2≤S*dimensionLog d^k*(d:ℝ)^(-(κ*(1/2))) := by |
| have he := hz.2.2.2.2.trans (mul_le_of_le_one_right hq0 hz.2.2.2.1) |
| have hhq : 2*q d≤S*dimensionLog d^k*(d:ℝ)^(-(κ*(1/2))) := (le_abs_self _).trans hqd |
| linarith |
| obtain ⟨hl',hp⟩ := hsc hF x hz.1 hz.2.1 hlam hL hb |
| have hη := sampleCorrelation_properties hz.2.2.1 hz.2.2.2.1 |
| change ∀{T : ℝ},1/2≤T → T<1 → ∀e : ProbabilityNode T ((d:ℝ)^(-(κ*w))) (n+1), |
| Integrable (fun z => ‖sampleMeanCircuit F x r T ((d:ℝ)^(-(κ*w))) (n+1) (numericalLayerCount J) e z- |
| T⁻¹ • fullProbabilityFlow hF x hz.1.le hl' (probabilityNodeTime T ((d:ℝ)^(-(κ*w))) (n+1) e) z‖^2) |
| (stdGaussian (Point d)) ∧ |
| (∫z,‖sampleMeanCircuit F x r T ((d:ℝ)^(-(κ*w))) (n+1) (numericalLayerCount J) e z- |
| T⁻¹ • fullProbabilityFlow hF x hz.1.le hl' (probabilityNodeTime T ((d:ℝ)^(-(κ*w))) (n+1) e) z‖^2 |
| ∂stdGaussian (Point d))≤circuitD F x^2*(d:ℝ)^(-(2*κ*J)) at hp |
| rw [←hC.h_eq,←hC.n_eq,←hC.N_eq] at hp |
| have he := hp hη.1 hη.2.1 (C.sampleEndpoint η hz.2.2.1 hz.2.2.2.1) |
| simpa only [CircuitParameters.sampleEndpoint, |
| probabilityEndpoint_time _ _ _ _ (Nat.succ_pos C.n),hsquare] using he |
| end OracleCompiler |
| end LogConcaveSampling |
|
|
| end |
|
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| end |
|
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| end OAI |
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