| import Mathlib |
| import OAI.Combinatorics.MatchingCount.Graphs.RowPeak |
|
|
| namespace OAI |
|
|
| namespace MatchingFPRAS.Concentration |
|
|
| open MeasureTheory ProbabilityTheory Finset |
|
|
| variable {Ω I J : Type*} [MeasurableSpace Ω] [Fintype I] |
|
|
| variable {μ : Measure Ω} [IsProbabilityMeasure μ] |
|
|
| theorem hoeffding_mean (X : I → Ω → ℝ) (hi : iIndepFun X μ) |
| (hm : ∀ i,AEMeasurable (X i) μ) (hb : ∀ i,∀ᵐ ω ∂μ,X i ω∈Set.Icc (0:ℝ) 1) |
| (m : ℝ) (hmean : ∀ i,(∫ ω,X i ω ∂μ)=m) |
| (hI : 0<Fintype.card I) (t : ℝ) (ht : 0≤t) : |
| μ.real {ω | t < |(∑ i,X i ω)/(Fintype.card I : ℝ)-m|} ≤ |
| 2*Real.exp (-2*(Fintype.card I : ℝ)*t^2) := by |
| classical |
| have hN : (0:ℝ)<Fintype.card I := Nat.cast_pos.mpr hI |
| have hc (i : I) : HasSubgaussianMGF (fun ω => X i ω-m) (1/4) μ := by |
| have h := hasSubgaussianMGF_of_mem_Icc (hm i) (hb i) |
| rw [hmean i] at h |
| norm_num at h ⊢ |
| exact h |
| have hind : iIndepFun (fun i ω => X i ω-m) μ := by |
| exact hi.comp (fun _ x => x-m) (fun _ => by fun_prop) |
| have hs : HasSubgaussianMGF (fun ω => ∑ i,(X i ω-m)) |
| ((Fintype.card I : NNReal)/4) μ := by |
| simpa only [Finset.sum_const,nsmul_eq_mul,mul_one_div,Finset.card_univ] using HasSubgaussianMGF.sum_of_iIndepFun hind (s:=univ) (fun i _ => hc i) |
| have hh := hs.measure_ge_le (mul_nonneg hN.le ht) |
| have hl := hs.neg.measure_ge_le (mul_nonneg hN.le ht) |
| have he : -((Fintype.card I : ℝ)*t)^2 / |
| (2*((Fintype.card I : NNReal)/4 : NNReal)) = -2*(Fintype.card I : ℝ)*t^2 := by |
| push_cast |
| field_simp [ne_of_gt hN] |
| ring |
| rw [he] at hh hl |
| calc |
| _ ≤ μ.real ({ω | (Fintype.card I : ℝ)*t ≤ ∑ i,(X i ω-m)} ∪ |
| {ω | (Fintype.card I : ℝ)*t ≤ -(∑ i,(X i ω-m))}) := by |
| refine measureReal_mono ?_ (by finiteness) |
| intro ω hω |
| simp only [Set.mem_ofPred_eq,Set.mem_union] at hω ⊢ |
| by_cases h : (Fintype.card I : ℝ)*t ≤ ∑ i,(X i ω-m) |
| · exact Or.inl h |
| · right |
| simp only [sum_sub_distrib,sum_const,nsmul_eq_mul,Finset.card_univ] at h ⊢ |
| rcases lt_abs.mp hω with hv | hv |
| · have hv' := (lt_div_iff₀ hN).mp (by linarith : m+t < (∑ i,X i ω)/(Fintype.card I : ℝ)) |
| nlinarith |
| · have hv' := (div_lt_iff₀ hN).mp (by linarith : (∑ i,X i ω)/(Fintype.card I : ℝ) < m-t) |
| nlinarith |
| _ ≤ μ.real {ω | (Fintype.card I : ℝ)*t ≤ ∑ i,(X i ω-m)} + |
| μ.real {ω | (Fintype.card I : ℝ)*t ≤ -(∑ i,(X i ω-m))} := measureReal_union_le _ _ |
| _ ≤ _ := by simpa only [Pi.neg_apply, two_mul] using add_le_add hh hl |
|
|
| theorem hoeffding_observables [Fintype J] (X : J → I → Ω → ℝ) |
| (hi : ∀ j,iIndepFun (X j) μ) (hm : ∀ j i,AEMeasurable (X j i) μ) |
| (hb : ∀ j i,∀ᵐ ω ∂μ,X j i ω∈Set.Icc (0:ℝ) 1) |
| (m : J → ℝ) (hmean : ∀ j i,(∫ ω,X j i ω ∂μ)=m j) |
| (hI : 0<Fintype.card I) (t : ℝ) (ht : 0≤t) : |
| μ.real {ω | ∃ j,t < |(∑ i,X j i ω)/(Fintype.card I : ℝ)-m j|} ≤ |
| 2*(Fintype.card J : ℝ)*Real.exp (-2*(Fintype.card I : ℝ)*t^2) := by |
| classical |
| have he : {ω | ∃ j,t < |(∑ i,X j i ω)/(Fintype.card I : ℝ)-m j|} = |
| ⋃ j,{ω | t < |(∑ i,X j i ω)/(Fintype.card I : ℝ)-m j|} := by ext; simp |
| rw [he] |
| apply (measureReal_iUnion_fintype_le _).trans |
| calc |
| _ ≤ ∑ _ : J,2*Real.exp (-2*(Fintype.card I : ℝ)*t^2) := by |
| apply sum_le_sum |
| intro j hj |
| exact hoeffding_mean (X j) (hi j) (hm j) (hb j) (m j) (hmean j) hI t ht |
| _ = _ := by simp; ring |
|
|
|
|
|
|
| open MeasureTheory ProbabilityTheory Finset |
| open scoped Classical |
|
|
| variable {Ω I J : Type*} [Fintype Ω] [Fintype I] [Fintype J] |
|
|
| noncomputable def finitePMF (p : Ω → ℝ) (hp : ∀ x,0 ≤ p x) (hs : ∑ x,p x=1) : PMF Ω := |
| PMF.ofFintype (fun x => ENNReal.ofReal (p x)) (by |
| rw [←ENNReal.ofReal_sum_of_nonneg (fun x _ => hp x),hs] |
| simp) |
|
|
| lemma finitePMF_apply (p : Ω → ℝ) (hp : ∀ x,0 ≤ p x) (hs : ∑ x,p x=1) (x : Ω) : |
| (finitePMF p hp hs x).toReal=p x := by simp [finitePMF,hp x] |
|
|
| theorem finite_hoeffding [DecidableEq I] (p : Ω → ℝ) (hp : ∀ x,0 ≤ p x) (hs : ∑ x,p x=1) |
| (f : J → Ω → ℝ) (hf : ∀ j x,f j x∈Set.Icc (0:ℝ) 1) |
| (hI : 0<Fintype.card I) (t : ℝ) (ht : 0≤t) : |
| ∑ v : I → Ω,(if ∃ j,t < |(∑ i,f j (v i))/(Fintype.card I : ℝ)-∑ x,p x*f j x| |
| then ∏ i,p (v i) else 0) ≤ |
| 2*(Fintype.card J : ℝ)*Real.exp (-2*(Fintype.card I : ℝ)*t^2) := by |
| classical |
| let : MeasurableSpace Ω := ⊤ |
| let q := finitePMF p hp hs |
| let μ : Measure (I → Ω) := Measure.pi (fun _ : I => q.toMeasure) |
| have hi (j : J) : iIndepFun (fun i (v : I → Ω) => f j (v i)) μ := by |
| exact iIndepFun_pi (fun _ : I => (measurable_of_finite (f j)).aemeasurable) |
| have hm (j : J) (i : I) : AEMeasurable (fun v : I → Ω => f j (v i)) μ := |
| (measurable_of_finite _).aemeasurable |
| have hb (j : J) (i : I) : ∀ᵐ v ∂μ,f j (v i)∈Set.Icc (0:ℝ) 1 := |
| Filter.Eventually.of_forall (fun v => hf j (v i)) |
| have he (j : J) (i : I) : (∫ v,f j (v i) ∂μ)=∑ x,p x*f j x := by |
| have hev := (measurePreserving_eval (fun _ : I => q.toMeasure) i).map_eq |
| have hmap := integral_map ((measurable_of_finite (Function.eval i)).aemeasurable (μ:=μ)) |
| ((measurable_of_finite (f j)).aestronglyMeasurable (μ:=Measure.map (Function.eval i) μ)) |
| rw [hev] at hmap |
| rw [←hmap, PMF.integral_eq_sum] |
| apply sum_congr rfl |
| intro x hx |
| simp [q,finitePMF_apply,smul_eq_mul] |
| have h := hoeffding_observables (μ:=μ) (fun j i v => f j (v i)) hi hm hb |
| (fun j => ∑ x,p x*f j x) he hI t ht |
| have hsingle (v : I → Ω) : μ.real {v}=∏ i,p (v i) := by |
| simp only [Measure.real,μ,Measure.pi_singleton,ENNReal.toReal_prod] |
| apply prod_congr rfl |
| intro i hi |
| simp [q,finitePMF,hp (v i)] |
| let B : Finset (I → Ω) := univ.filter (fun v => ∃ j,t < |(∑ i,f j (v i))/(Fintype.card I : ℝ)-∑ x,p x*f j x|) |
| have hB : (B : Set (I → Ω))={v | ∃ j,t < |(∑ i,f j (v i))/(Fintype.card I : ℝ)-∑ x,p x*f j x|} := by |
| ext v; simp [B] |
| rw [←hB,←sum_measureReal_singleton] at h |
| simpa only [hsingle, B, sum_filter] using h |
|
|
|
|
|
|
| lemma observables_exp_tail (n K : ℕ) (hK : 0<K) : |
| 2*((n:ℝ)^2+2)*Real.exp (-20*((n:ℝ)+K+1)) < 1/(100*K) := by |
| have hK0 : (0:ℝ)<K := Nat.cast_pos.mpr hK |
| have hn0 : (0:ℝ)≤n := Nat.cast_nonneg n |
| have h1 : (2:ℝ)≤Real.exp 1 := by linarith [Real.add_one_le_exp (1:ℝ)] |
| have hc : (200:ℝ)<Real.exp 8 := by |
| calc |
| (200:ℝ) < 2^8 := by norm_num |
| _ ≤ (Real.exp 1)^8 := pow_le_pow_left₀ (by norm_num) h1 _ |
| _ = Real.exp 8 := by rw [← Real.exp_nat_mul]; norm_num |
| have hn : (n:ℝ)^2+2≤Real.exp (2*((n:ℝ)+1)) := by |
| have h := Real.add_one_le_exp ((n:ℝ)+1) |
| have hh : 0≤(n:ℝ)+2 := by positivity |
| calc |
| _ ≤ ((n:ℝ)+2)^2 := by nlinarith |
| _ ≤ (Real.exp ((n:ℝ)+1))^2 := pow_le_pow_left₀ hh (by linarith) _ |
| _ = _ := by rw [← Real.exp_nat_mul]; norm_num |
| have hk : (K:ℝ)≤Real.exp (K:ℝ) := by linarith [Real.add_one_le_exp (K:ℝ)] |
| have hprod : 200*(K:ℝ)*((n:ℝ)^2+2)<Real.exp (20*((n:ℝ)+K+1)) := by |
| calc |
| _ ≤ 200*Real.exp (K:ℝ)*Real.exp (2*((n:ℝ)+1)) := by gcongr |
| _ < Real.exp 8*Real.exp (K:ℝ)*Real.exp (2*((n:ℝ)+1)) := by gcongr |
| _ = Real.exp (8+(K:ℝ)+2*((n:ℝ)+1)) := by rw [Real.exp_add,Real.exp_add] |
| _ < _ := Real.exp_lt_exp.mpr (by nlinarith) |
| have he : 0<Real.exp (20*((n:ℝ)+K+1)) := Real.exp_pos _ |
| rw [show -20*((n:ℝ)+K+1)= -(20*((n:ℝ)+K+1)) by ring,Real.exp_neg] |
| apply (lt_div_iff₀ (by positivity : (0:ℝ)<100*K)).mpr |
| rw [← div_eq_mul_inv, (div_mul_eq_mul_div), div_lt_one he] |
| nlinarith [hprod] |
|
|
| lemma stage_hoeffding_budget (n K S : ℕ) (hK : 0<K) (sigma : ℝ) |
| (hS : 10*((n:ℝ)+K+1)≤(S:ℝ)*sigma^2) : |
| 2*((n:ℝ)^2+2)*Real.exp (-2*(S:ℝ)*sigma^2)<1/(100*K) := by |
| apply lt_of_le_of_lt _ (observables_exp_tail n K hK) |
| apply mul_le_mul_of_nonneg_left _ (by positivity) |
| apply Real.exp_le_exp.mpr |
| nlinarith |
|
|
| def trials (ell : ℕ) : ℕ := 10*(ell+1)+1 |
|
|
| lemma median_tail (ell : ℕ) : |
| Real.exp (-2*(4/10:ℝ)^2*(trials ell)) < (2:ℝ)^(-(ell:ℤ)) := by |
| have hlog : Real.log 2<1 := by |
| rw [Real.log_lt_iff_lt_exp (by norm_num)] |
| have hx := Real.add_one_lt_exp (by norm_num : (1:ℝ)≠0) |
| norm_num at hx |
| exact hx |
| have he : Real.exp (-((ell:ℝ)*Real.log 2))=(2:ℝ)^(-(ell:ℤ)) := by |
| rw [Real.exp_neg,Real.exp_nat_mul,Real.exp_log (by norm_num : (0:ℝ)<2)] |
| simp |
| rw [← he] |
| apply Real.exp_lt_exp.mpr |
| have hh := mul_le_mul_of_nonneg_left hlog.le (Nat.cast_nonneg ell : (0:ℝ)≤ell) |
| dsimp [trials] |
| push_cast |
| norm_num at * |
| nlinarith [(Nat.cast_nonneg ell : (0:ℝ)≤ell)] |
|
|
| end MatchingFPRAS.Concentration |
|
|
| namespace MatchingFPRAS.Enlargement |
|
|
| open MatchingFPRAS.Switching MatchingFPRAS.Subdivision MatchingFPRAS.RealGeometry |
| open MatchingFPRAS.Hierarchy MatchingFPRAS.Transport MatchingFPRAS.StageAnalysis Finset |
| open scoped Classical |
|
|
| variable {V : Type*} [Fintype V] [LinearOrder V] [Nontrivial V] |
|
|
| noncomputable local instance (w : V → V → ℝ) (K : ℕ) : DecidableEq (SourceLabel w K) := Classical.decEq _ |
|
|
| noncomputable local instance (w : V → V → ℝ) (K : ℕ) : Fintype (BagEdge (sourceBags w K)) := Fintype.ofFinite _ |
|
|
| omit [Fintype V] [Nontrivial V] in |
| lemma logicalActivity_le_off (w w₁ : V → V → ℝ) |
| (h : ∀ i j,i≠j → w₁ i j ≤ w i j) (e : LogicalEdge V) : |
| logicalActivity w₁ e ≤ logicalActivity w e := by |
| unfold logicalActivity edgeActivity |
| split_ifs |
| · exact h _ _ e.property.ne |
| · exact h _ _ e.property.ne.symm |
|
|
| lemma bagReweight_range_off (w w₁ : V → V → ℝ) (hw : ∀ x y,0 < w x y) |
| (hw₁ : ∀ x y,0 ≤ w₁ x y) (hle : ∀ x y,x≠y → w₁ x y ≤ w x y) |
| (K : ℕ) (e : BagEdge (sourceBags w K)) : |
| 0 ≤ bagReweight w w₁ K e ∧ bagReweight w w₁ K e ≤ 1 := by |
| unfold bagReweight |
| split_ifs with he |
| · unfold rawReweight |
| split_ifs |
| · have h₀ : 0 < logicalActivity w he.choose.1 := by |
| unfold logicalActivity edgeActivity; split_ifs <;> exact hw _ _ |
| have h₁ : 0 ≤ logicalActivity w₁ he.choose.1 := by |
| unfold logicalActivity edgeActivity; split_ifs <;> exact hw₁ _ _ |
| exact ⟨div_nonneg h₁ h₀.le,(div_le_one h₀).mpr (logicalActivity_le_off w w₁ hle _)⟩ |
| · norm_num |
| · norm_num |
|
|
| lemma observableD_range_off (w w₁ : V → V → ℝ) (hw : ∀ x y,0 < w x y) |
| (hw₁ : ∀ x y,0 ≤ w₁ x y) (hle : ∀ x y,x≠y → w₁ x y ≤ w x y) |
| (K : ℕ) (a : SourceMarked w K) : |
| 0 ≤ observableD w w₁ K a ∧ observableD w w₁ K a ≤ 1 := by |
| unfold observableD |
| split_ifs |
| · exact ⟨Finset.prod_nonneg (fun e _ => (bagReweight_range_off w w₁ hw hw₁ hle K e).1), |
| Finset.prod_le_one₀ (fun e _ => (bagReweight_range_off w w₁ hw hw₁ hle K e).1) |
| (fun e _ => (bagReweight_range_off w w₁ hw hw₁ hle K e).2)⟩ |
| · norm_num |
|
|
| lemma observableProbe_range_off (w w₁ : V → V → ℝ) (hw : ∀ x y,0 < w x y) |
| (hw₁ : ∀ x y,0 ≤ w₁ x y) (hle : ∀ x y,x≠y → w₁ x y ≤ w x y) |
| (K : ℕ) (e₀ : LogicalEdge V) (a : SourceMarked w K) : |
| 0 ≤ observableProbe w w₁ K e₀ a ∧ observableProbe w w₁ K e₀ a ≤ 1 := by |
| unfold observableProbe |
| split_ifs |
| · exact ⟨Finset.prod_nonneg (fun e _ => (bagReweight_range_off w w₁ hw hw₁ hle K e).1), |
| Finset.prod_le_one₀ (fun e _ => (bagReweight_range_off w w₁ hw hw₁ hle K e).1) |
| (fun e _ => (bagReweight_range_off w w₁ hw hw₁ hle K e).2)⟩ |
| · norm_num |
|
|
| abbrev ObservableIndex (V : Type*) [LinearOrder V] := Fin 2 ⊕ LogicalEdge V |
|
|
| noncomputable def sourceObservable (w w₁ : V → V → ℝ) (K : ℕ) : |
| ObservableIndex V → SourceMarked w K → ℝ |
| | Sum.inl i => if i=0 then observableA w K else observableD w w₁ K |
| | Sum.inr e => observableProbe w w₁ K e |
|
|
| lemma sourceObservable_range (w w₁ : V → V → ℝ) (hw : ∀ x y,0 < w x y) |
| (hw₁ : ∀ x y,0 ≤ w₁ x y) (hle : ∀ x y,x≠y → w₁ x y ≤ w x y) |
| (K : ℕ) (j : ObservableIndex V) (a : SourceMarked w K) : |
| sourceObservable w w₁ K j a∈Set.Icc (0:ℝ) 1 := by |
| cases j with |
| | inl i => |
| dsimp [sourceObservable] |
| split_ifs |
| · exact observableA_range w K a |
| · exact observableD_range_off w w₁ hw hw₁ hle K a |
| | inr e => exact observableProbe_range_off w w₁ hw hw₁ hle K e a |
|
|
| omit [Nontrivial V] in |
| lemma observableIndex_card : Fintype.card (ObservableIndex V) ≤ (Fintype.card V)^2+2 := by |
| have h := Fintype.card_subtype_le (fun e : V×V => e.1<e.2) |
| change Fintype.card (LogicalEdge V) ≤ Fintype.card (V×V) at h |
| simp only [Fintype.card_prod] at h |
| simpa only [ObservableIndex,Fintype.card_sum,Fintype.card_fin,pow_two,Nat.add_comm] using Nat.add_le_add_left h 2 |
|
|
| noncomputable def sampleAverage {Ω : Type*} {S : ℕ} (f : Ω → ℝ) (x : Fin S → Ω) : ℝ := |
| (∑ i,f (x i))/(S:ℝ) |
|
|
| noncomputable def sourceMean (w w₁ : V → V → ℝ) (K : ℕ) (j : ObservableIndex V) : ℝ := |
| ∑ x : SourceMarked w K,sourceLaw w K (sourceD (V:=V) K) x*sourceObservable w w₁ K j x |
|
|
| def StageGood {S : ℕ} (w w₁ : V → V → ℝ) (K : ℕ) (σ : ℝ) (x : Fin S → SourceMarked w K) : Prop := |
| ∀ j : ObservableIndex V, |sampleAverage (sourceObservable w w₁ K j) x-sourceMean w w₁ K j| ≤ σ |
| |
| theorem source_ideal_concentration (w w₁ : V → V → ℝ) (hw : ∀ x y,0 < w x y) |
| (hw₁ : ∀ x y,0 ≤ w₁ x y) (hle : ∀ x y,x≠y → w₁ x y ≤ w x y) |
| (hz : 0 < partition w ∅) (K S : ℕ) (hS : 0<S) (σ : ℝ) (hσ : 0≤σ) : |
| ∑ x : Fin S → SourceMarked w K,(if ¬StageGood w w₁ K σ x |
| then ∏ i,sourceLaw w K (sourceD (V:=V) K) (x i) else 0) ≤ |
| 2*((Fintype.card V:ℝ)^2+2)*Real.exp (-2*(S:ℝ)*σ^2) := by |
| have hp := sourceBagZ_pos w hw hz K (sourceD (V:=V) K) |
| have h := MatchingFPRAS.Concentration.finite_hoeffding (I:=Fin S) |
| (sourceLaw w K (sourceD (V:=V) K)) (sourceLaw_nonneg w hw K _ hp.le) |
| (sourceLaw_sum w K _ hp.ne') (sourceObservable w w₁ K) (sourceObservable_range w w₁ hw hw₁ hle K) |
| (by simpa using hS) σ hσ |
| have he : ∀ x : Fin S → SourceMarked w K, |
| (¬StageGood w w₁ K σ x) ↔ |
| (∃ j,σ < |(∑ i,sourceObservable w w₁ K j (x i))/(Fintype.card (Fin S):ℝ)- |
| ∑ a,sourceLaw w K (sourceD (V:=V) K) a*sourceObservable w w₁ K j a|) := by |
| intro x |
| simp only [StageGood,not_forall,not_le,sampleAverage,sourceMean,Fintype.card_fin] |
| simp_rw [he] at ⊢ |
| apply h.trans |
| have hc : (Fintype.card (ObservableIndex V):ℝ) ≤ (Fintype.card V:ℝ)^2+2 := by |
| exact_mod_cast (observableIndex_card (V:=V)) |
| simpa only [Fintype.card_fin] using mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left hc (by norm_num)) (Real.exp_pos _).le |
|
|
| variable {n K T S : ℕ} [Nontrivial (Fin n)] |
|
|
| noncomputable def empiricalData (c : StageData n K T) (x : Fin S → SourceMarked c.w K) |
| (h : StageGood c.w c.next K (RealStage.sigma (sourceD (V:=Fin n) K) T c.epsilon) x) : EmpiricalData c where |
| abar := sampleAverage (observableA c.w K) x |
| dbar := sampleAverage (observableD c.w c.next K) x |
| zbar e := sampleAverage (observableProbe c.w c.next K e) x |
| errorA := by simpa only [sourceObservable,ite_true,sourceMean,meanA] using h (Sum.inl 0) |
| errorD := by simpa only [sourceObservable,show (1:Fin 2)≠0 by decide,ite_false,sourceMean,meanD] using h (Sum.inl 1) |
| errorProbe e := by simpa only [sourceObservable,sourceMean,meanProbe] using h (Sum.inr e) |
|
|
| lemma StageData.sigma_pos (c : StageData n K T) : 0 < RealStage.sigma (sourceD (V:=Fin n) K) T c.epsilon := by |
| unfold RealStage.sigma RealStage.alpha |
| have hD : 0 < sourceD (V:=Fin n) K := sourceD_one K |
| have ht : 0 < T := c.stages |
| have he := c.epsilon_positive |
| exact lt_min (by positivity) (by positivity) |
|
|
| noncomputable def stageSampleCount (n D T : ℕ) (ε : ℝ) : ℕ := |
| ⌈10*((RealStage.sigma D T ε)⁻¹)^2*((n:ℝ)+T+1)⌉₊ |
|
|
| lemma StageData.sampleCount_pos (c : StageData n K T) : |
| 0 < stageSampleCount n (sourceD (V:=Fin n) K) T c.epsilon := by |
| unfold stageSampleCount |
| apply Nat.ceil_pos.mpr |
| have hs := c.sigma_pos |
| positivity |
|
|
| lemma StageData.sampleCount_bound (c : StageData n K T) : |
| 10*((n:ℝ)+T+1) ≤ |
| (stageSampleCount n (sourceD (V:=Fin n) K) T c.epsilon:ℝ)* |
| (RealStage.sigma (sourceD (V:=Fin n) K) T c.epsilon)^2 := by |
| have hs := c.sigma_pos |
| have h := mul_le_mul_of_nonneg_right |
| (Nat.le_ceil (10*((RealStage.sigma (sourceD (V:=Fin n) K) T c.epsilon)⁻¹)^2*((n:ℝ)+T+1))) |
| (sq_nonneg (RealStage.sigma (sourceD (V:=Fin n) K) T c.epsilon)) |
| convert h using 1 |
| all_goals first | rfl | field_simp [ne_of_gt hs] |
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| theorem source_ideal_budget (w w₁ : V → V → ℝ) (hw : ∀ x y,0 < w x y) |
| (hw₁ : ∀ x y,0 ≤ w₁ x y) (hle : ∀ x y,x≠y → w₁ x y ≤ w x y) |
| (hz : 0 < partition w ∅) (K T S : ℕ) (hT : 0<T) (hS : 0<S) |
| (σ : ℝ) (hσ : 0≤σ) (hsize : 10*((Fintype.card V:ℝ)+T+1) ≤ (S:ℝ)*σ^2) : |
| ∑ x : Fin S → SourceMarked w K,(if ¬StageGood w w₁ K σ x |
| then ∏ i,sourceLaw w K (sourceD (V:=V) K) (x i) else 0) < 1/(100*T) := by |
| exact (source_ideal_concentration w w₁ hw hw₁ hle hz K S hS σ hσ).trans_lt |
| (MatchingFPRAS.Concentration.stage_hoeffding_budget (Fintype.card V) T S hT σ hsize) |
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| open MatchingFPRAS.Switching MatchingFPRAS.Subdivision MatchingFPRAS.RealGeometry |
| open MatchingFPRAS.Hierarchy MatchingFPRAS.Transport Finset |
| open MatchingFPRAS.Replica MatchingFPRAS.Replica.Ladder |
| open scoped Classical |
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| variable {V : Type*} [Fintype V] [LinearOrder V] [Nontrivial V] |
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| noncomputable local instance (w : V → V → ℝ) (K : ℕ) : DecidableEq (SourceLabel w K) := Classical.decEq _ |
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| noncomputable local instance (w : V → V → ℝ) (K : ℕ) : Fintype (BagEdge (sourceBags w K)) := Fintype.ofFinite _ |
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| noncomputable local instance (w : V → V → ℝ) (K : ℕ) : DecidableEq (SourcePM w K) := Classical.decEq _ |
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| theorem source_nearby_budget (w w₁ : V → V → ℝ) (hw : ∀ x y,0<w x y) |
| (hw₁ : ∀ x y,0≤w₁ x y) (hle : ∀ x y,x≠y → w₁ x y ≤ w x y) |
| (hz : 0<Switching.partition w ∅) (K T S : ℕ) (hT : 0<T) (hS : 0<S) |
| (σ : ℝ) (hσ : 0≤σ) (hsize : 10*((Fintype.card V:ℝ)+T+1) ≤ (S:ℝ)*σ^2) |
| (p : SourceMarked w K → ℝ) (hp : ∀ a,0≤p a) (hpsum : ∑ a,p a=1) |
| (htv : Variation.tv p (sourceLaw w K (sourceD (V:=V) K)) ≤ 3/(1000*T*S)) : |
| Variation.event (Variation.iid p S) (fun x => ¬StageGood w w₁ K σ x) < 13/(1000*T) := by |
| have hpos := sourceBagZ_pos w hw hz K (sourceD (V:=V) K) |
| have hq := sourceLaw_nonneg w hw K _ hpos.le |
| have hqs := sourceLaw_sum w K _ hpos.ne' |
| have hiid := Variation.iid_tv p (sourceLaw w K (sourceD (V:=V) K)) hp hpsum hq hqs S |
| have he := Variation.event_sub_le (Variation.iid p S) |
| (Variation.iid (sourceLaw w K (sourceD (V:=V) K)) S) |
| (by rw [Variation.iid_sum p hpsum,Variation.iid_sum _ hqs]) |
| (fun x => ¬StageGood w w₁ K σ x) |
| have hv := (le_abs_self _).trans (he.trans hiid) |
| have hh := source_ideal_budget w w₁ hw hw₁ hle hz K T S hT hS σ hσ hsize |
| have ht := mul_le_mul_of_nonneg_left htv (Nat.cast_nonneg S : (0:ℝ)≤S) |
| have hid : (1:ℝ)/(100*T)+(S:ℝ)*(3/(1000*T*S))=13/(1000*T) := by |
| have ht0 : (T:ℝ)≠0 := by exact_mod_cast hT.ne' |
| have hs0 : (S:ℝ)≠0 := by exact_mod_cast hS.ne' |
| field_simp |
| ring |
| have hh' : Variation.event (Variation.iid (sourceLaw w K (sourceD (V:=V) K)) S) |
| (fun x => ¬StageGood w w₁ K σ x) < 1/(100*T) := by |
| convert hh using 1 |
| unfold Variation.event Variation.iid |
| congr 1 |
| funext x |
| split_ifs <;> rfl |
| linarith |
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| theorem source_run_stage_budget (hn : 2 ≤ Fintype.card V) |
| (w w₁ : V → V → ℝ) (hw : ∀ x y,0<w x y) |
| (hw₁ : ∀ x y,0≤w₁ x y) (hle : ∀ x y,x≠y → w₁ x y ≤ w x y) |
| (hz : 0<Switching.partition w ∅) (K T S : ℕ) (hT : 0<T) (hS : 0<S) |
| (hK : ∀ x y,x≠y → edgeActivity w x y ≤ (4:ℝ)^K) |
| (σ : ℝ) (hσ : 0≤σ) (hsize : 10*((Fintype.card V:ℝ)+T+1) ≤ (S:ℝ)*σ^2) |
| (L : ℕ) (hL : max (sourceD (V:=V) K:ℝ) (3*4^K) ≤ (tierBase (SourceVertex V K))^L) |
| (hq : 0<ladderQ (Fintype.card (SourceVertex V K)) (sourceD (V:=V) K)) |
| (z : Slot L (ladderQ (Fintype.card (SourceVertex V K)) (sourceD (V:=V) K)) → SourcePM w K) : |
| let ξ : ℝ := 1/(1000*T*S) |
| let p := sourceRun w K hK L _ hq z |
| (ladderSteps (Fintype.card (SourceVertex V K)) (sourceD (V:=V) K) K L ξ) |
| Variation.event (Variation.iid p S) (fun x => ¬StageGood w w₁ K σ x) < 13/(1000*T) := by |
| dsimp only |
| apply source_nearby_budget w w₁ hw hw₁ hle hz K T S hT hS σ hσ hsize |
| · exact sourceRun_nonneg w hw K hK L _ hq z _ |
| · exact sourceRun_sum w hw K hK L _ hq z _ |
| · have ht : (1:ℝ)≤T := by exact_mod_cast hT |
| have hs : (1:ℝ)≤S := by exact_mod_cast hS |
| have hd : (1:ℝ)<1000*T*S := by nlinarith |
| exact (sourceRun_tv hn w hw K hK L hL hq z _ (by positivity) |
| ((div_lt_one (by positivity)).mpr hd)).trans (by gcongr; norm_num) |
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| end MatchingFPRAS.Enlargement |
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| end OAI |
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