| import Mathlib |
| import OAI.Combinatorics.MatchingCount.Algorithm.RationalEstimator |
|
|
| namespace OAI |
|
|
| namespace MatchingFPRAS.FiniteTrial |
| open Finset Enlargement Transport Switching |
|
|
| variable {G : GraphInput} {j K S : ℕ} [Nontrivial (Fin G.n)] |
| attribute [local instance] NumberedHierarchy.labelOrder sym2Order bagEdgeOrder bagEdgeFintype |
| attribute [local instance] UnitRefresh.Computable.allowedDec prodOrder sumOrder |
| noncomputable local instance (w : Fin G.n → Fin G.n → ℚ) (K : ℕ) : |
| DecidableEq (SourceLabel (fun x y=>(w x y:ℝ)) K) := Classical.decEq _ |
| noncomputable local instance (w : Fin G.n → Fin G.n → ℚ) (K : ℕ) : |
| Fintype (BagEdge (sourceBags (fun x y=>(w x y:ℝ)) K)) := Fintype.ofFinite _ |
| def ScaleState.Balanced (hn : 2≤G.n) (s : ScaleState G j) : Prop := s.toReal.Balanced hn |
| omit [Nontrivial (Fin G.n)] in |
| lemma initial_balanced (hn : 2≤G.n) (he : Even G.n) : (initialScaleState G hn).Balanced hn := |
| Trial.initial_balanced G hn he |
| noncomputable def stageData (p : Trial.Parameters G) (hj : j≤p.K) (s : ScaleState G j) |
| (hs : s.Balanced p.hn) : StageData G.n p.K p.K where |
| hn:=p.hn |
| w a b:=(s.activity a b:ℝ) |
| next a b:=(s.nextActivity a b:ℝ) |
| positive a b:=by exact_mod_cast s.activity_pos a b |
| next_positive a b:=by exact_mod_cast s.nextActivity_pos a b |
| partition_positive:=by rw [s.activity_cast]; exact s.toReal.partition_pos p.even |
| lower:=by rw [s.activity_cast]; exact hs.1 |
| upper:=by rw [s.activity_cast]; exact hs.2 |
| cool_lower a b _:=by |
| have h:=(s.toReal.cooling a b).1 |
| rw [←s.activity_cast,←s.nextActivity_cast] at h |
| exact h |
| cool_upper a b _:=by |
| have h:=(s.toReal.cooling a b).2 |
| rw [←s.activity_cast,←s.nextActivity_cast] at h |
| exact h |
| cap:=by rw [s.activity_cast]; exact s.toReal.cap hj |
| stages:=p.K_pos |
| epsilon:=p.epsilon |
| epsilon_positive:=p.epsilon_real_pos |
| epsilon_lt_one:=p.epsilon_real_lt_one |
|
|
| theorem successful_update (p : Trial.Parameters G) (hj : j≤p.K) (s : ScaleState G j) |
| (hs : s.Balanced p.hn) (x : Fin S → SourceMarked (fun a b=>(s.activity a b:ℝ)) p.K) |
| (hx : StageGood (fun a b=>(s.activity a b:ℝ)) (fun a b=>(s.nextActivity a b:ℝ)) p.K p.sigma x) : |
| (s.update p.hn (RationalSource.sourceMarkedEquiv s.activity p.K ∘ x)).Balanced p.hn ∧ |
| |(s.ratioEstimate (RationalSource.sourceMarkedEquiv s.activity p.K ∘ x):ℝ)/ |
| (partition (Trial.raw G (j+1)) ∅/partition (Trial.raw G j) ∅)-1|≤(p.epsilon:ℝ)/(100*p.K) := by |
| let c:=stageData p hj s hs |
| let obs : EmpiricalData c:=empiricalData c x hx |
| have hh:=actual_successful_stage c obs |
| dsimp only at hh |
| let y:=RationalSource.sourceMarkedEquiv s.activity p.K ∘ x |
| have hm : (fun a b=>(s.matrix y a b:ℝ))=Enlargement.estimateMatrix (Enlargement.sourceD (V:=Fin G.n) p.K) obs.dbar obs.zbar := by |
| rw [s.matrix_cast,sourceD_eq] |
| rfl |
| have hv : (fun i=>(s.updateVector p.hn y i:ℝ))= |
| RealStage.scalingTrace c.hn (Enlargement.estimateMatrix (Enlargement.sourceD (V:=Fin G.n) p.K) obs.dbar obs.zbar) G.n := by |
| funext i |
| rw [ScaleState.updateVector,Counting.scalingTrace_cast,hm] |
| constructor |
| · unfold ScaleState.Balanced Trial.ScaleState.Balanced |
| rw [←ScaleState.activity_cast,ScaleState.update_activity] |
| have ha : (fun a b=>((s.nextActivity a b*s.updateVector p.hn y a*s.updateVector p.hn y b:ℚ):ℝ))= |
| (fun a b=>c.next a b*(RealStage.scalingTrace c.hn (Enlargement.estimateMatrix |
| (Enlargement.sourceD (V:=Fin G.n) p.K) obs.dbar obs.zbar) G.n a)* |
| (RealStage.scalingTrace c.hn (Enlargement.estimateMatrix |
| (Enlargement.sourceD (V:=Fin G.n) p.K) obs.dbar obs.zbar) G.n b)) := by |
| funext a z |
| simp only [Rat.cast_mul,congrFun hv] |
| rfl |
| rw [ha] |
| exact ⟨fun i=>(hh.2.1 i).1,fun i=>(hh.2.1 i).2⟩ |
| · rw [s.ratioEstimate_cast] |
| have hr : partition c.next ∅/partition c.w ∅= |
| partition (Trial.raw G (j+1)) ∅/partition (Trial.raw G j) ∅ := by |
| change partition (fun a b=>(s.nextActivity a b:ℝ)) ∅/partition (fun a b=>(s.activity a b:ℝ)) ∅=_ |
| rw [s.nextActivity_cast,s.activity_cast,s.toReal.ratio] |
| rw [←hr] |
| exact hh.2.2 |
|
|
| lemma ScaleState.ratioEstimate_nonneg (s : ScaleState G j) |
| (x : Fin S → SourceMarked (fun a b=>(s.activity a b:ℝ)) K) : |
| 0 ≤ s.ratioEstimate (RationalSource.sourceMarkedEquiv s.activity K ∘ x) := by |
| have h : (0:ℝ)≤(s.ratioEstimate (RationalSource.sourceMarkedEquiv s.activity K ∘ x):ℝ) := by |
| rw [s.ratioEstimate_cast] |
| unfold RealStage.ratioEstimate |
| split_ifs with h |
| · apply div_nonneg _ h.le |
| unfold Enlargement.sampleAverage |
| apply div_nonneg _ (Nat.cast_nonneg S) |
| apply sum_nonneg |
| intro i _ |
| exact (observableD_range_off (fun a b=>(s.activity a b:ℝ)) (fun a b=>(s.nextActivity a b:ℝ)) |
| (fun a b=>by exact_mod_cast s.activity_pos a b) |
| (fun a b=>by exact_mod_cast (s.nextActivity_pos a b).le) |
| (fun a b _=>by have h:=(s.toReal.cooling a b).2; rw [←s.activity_cast,←s.nextActivity_cast] at h; exact h) K (x i)).1 |
| · norm_num |
| exact_mod_cast h |
| end MatchingFPRAS.FiniteTrial |
| |
| end OAI |
| |
