| import Mathlib |
| import OAI.Combinatorics.MatchingCount.Complexity.OnlineHistoryCost |
|
|
| namespace OAI |
|
|
| namespace MatchingFPRAS.OnlineTrial |
| open TreeTyped PhysicalStream Polynomial OnlineGraph OnlineSample |
|
|
| noncomputable def initialWeightPolynomial : Polynomial ℕ := 2*X+3+statePolynomial |
| lemma initial_weight {G : GraphInput} [Nontrivial (Fin G.n)] (f : Edges→Edges) (p : Trial.Parameters G) : |
| weight (initial f p) ≤ initialWeightPolynomial.eval (sizeNorm p (graph f (ExecutableAdaptive.graphRep G))) := by |
| have hs:=history_size f p p.K (initial f p).2 (history_initial f p) |
| have hw:=weight_le_stateSize (initial f p).2 |
| have hk:=(sizeNorm_fields p (graph f (ExecutableAdaptive.graphRep G))).2.1 |
| rw [weight_prod] |
| have hl : weight (initial f p).1=2*p.K+1 := by simp only [initial,weight_replicate,show weight false=1 from rfl];omega |
| rw [hl] |
| simp only [initialWeightPolynomial,Polynomial.eval_add,Polynomial.eval_mul,Polynomial.eval_X,Polynomial.eval_ofNat] |
| omega |
| end MatchingFPRAS.OnlineTrial |
| |
| namespace MatchingFPRAS.OnlineGuard |
| open TreeTyped PhysicalStream Polynomial OnlineGraph OnlineTrial |
| |
| lemma trial_bound : ∃ q : Polynomial ℕ,∀ (G : GraphInput) [Nontrivial (Fin G.n)] |
| (f : Edges→Edges) (_hf : ∀ es e,e∈f es↔e∈es) (p : Trial.Parameters G), |
| trialRealizer.cost (OnlineTrial.initial f p)+(trial (OnlineTrial.initial f p)).worst weight≤ |
| q.eval (sizeNorm p (graph f (ExecutableAdaptive.graphRep G))) := by |
| obtain ⟨q,hq⟩:=sweep_bound |
| obtain ⟨r,hr⟩:=work_checked |
| refine ⟨q+2*(r.comp statePolynomial),?_⟩ |
| intro G inst f hf p |
| let n:=sizeNorm p (graph f (ExecutableAdaptive.graphRep G)) |
| have hs:=hq G f hf p |
| have hb : (sweep (OnlineTrial.initial f p)).All (fun s=>checkedRealizer.cost s+(checked s).worst weight≤r.eval (statePolynomial.eval n)) := |
| (sweep_history f hf p).mono (fun s hs=>(hr s).trans (polynomial_eval_monotone r (history_size f p 0 s hs))) |
| have h1:=Experiment.worst_le _ _ _ (hb.mono (fun s hs=>Nat.le_trans (Nat.le_add_right _ _) hs)) |
| have h2:=Experiment.worst_le _ _ _ (hb.mono (fun s hs=>Nat.le_trans (Nat.le_add_left _ _) hs)) |
| change sweepRealizer.cost _+(sweep _).worst checkedRealizer.cost+((sweep _).bind checked).worst weight≤_ |
| rw [Experiment.worst_bind] |
| simp only [Polynomial.eval_add,Polynomial.eval_mul,Polynomial.eval_ofNat,Polynomial.eval_comp] |
| dsimp only [n] at * |
| omega |
| end MatchingFPRAS.OnlineGuard |
|
|
| namespace MatchingFPRAS.OnlineAmplification |
| open TreeTyped PhysicalStream Polynomial OnlineGraph OnlineTrial OnlineSample |
|
|
| lemma run_bound : ∃ q : Polynomial ℕ,∀ (G : GraphInput) [Nontrivial (Fin G.n)] |
| (f : Edges→Edges) (_hf : ∀ es e,e∈f es↔e∈es) (p : Trial.Parameters G) (r : ℕ), |
| runRealizer.cost (r,OnlineTrial.initial f p)+(run (r,OnlineTrial.initial f p)).worst weight≤ |
| q.eval (r+sizeNorm p (graph f (ExecutableAdaptive.graphRep G))) := by |
| obtain ⟨q,hq⟩:=OnlineGuard.trial_bound |
| obtain ⟨s,hs⟩:=OnlineTree.traverse_bound OnlineGuard.trialRealizer |
| obtain ⟨u,hu⟩:=work_repeatInput (A:=TrialInput) |
| obtain ⟨v,hv⟩:=OnlineMedian.polynomial_median |
| let I := initialWeightPolynomial |
| let R : Polynomial ℕ := 5*X+2+I |
| let B : Polynomial ℕ := X*(I+1)+1+q |
| let T := u.comp R+s.comp B |
| refine ⟨T+2*(v.comp T),?_⟩ |
| intro G inst f hf p r |
| let n:=sizeNorm p (graph f (ExecutableAdaptive.graphRep G)) |
| let z:=r+n |
| let x : TrialInput:=OnlineTrial.initial f p |
| have hn:n≤z := by dsimp [z];omega |
| have hr:r≤z := by dsimp [z];omega |
| have hx : weight x≤I.eval z := (initial_weight f p).trans (polynomial_eval_monotone I hn) |
| have hinput : r+weight (r,x) ≤R.eval z := by |
| have hw:=weight_nat_numeric r |
| simp only [weight_prod,R,Polynomial.eval_add,Polynomial.eval_mul,Polynomial.eval_X,Polynomial.eval_ofNat] |
| omega |
| have hrep := (hu (r,x)).trans (polynomial_eval_monotone u hinput) |
| have hwrep : weight (List.replicate r x)≤B.eval z := by |
| rw [weight_replicate] |
| simp only [B,Polynomial.eval_add,Polynomial.eval_mul,Polynomial.eval_X,Polynomial.eval_one] |
| nlinarith |
| have ht : ∀ a∈List.replicate r x,OnlineGuard.trialRealizer.cost a+(OnlineGuard.trial a).worst weight≤B.eval z := by |
| intro a ha |
| have he: a=x := (List.mem_replicate.mp ha).2 |
| rw [he] |
| exact ((hq G f hf p).trans (polynomial_eval_monotone q hn)).trans (by |
| simp only [B,Polynomial.eval_add,Polynomial.eval_mul,Polynomial.eval_X,Polynomial.eval_one];omega) |
| have htrav:=hs (List.replicate r x) (B.eval z) hwrep ht |
| have htrial : trialsRealizer.cost (r,x)+(trials (r,x)).worst weight≤T.eval z := by |
| change repeatInputRealizer.cost (r,x)+(OnlineTree.traverse OnlineGuard.trialRealizer).cost (List.replicate r x)+ |
| (OnlineTree.traverseM OnlineGuard.trial (List.replicate r x)).worst weight≤_ |
| simp only [T,Polynomial.eval_add,Polynomial.eval_comp] |
| change repeatInputRealizer.cost (r,x)+weight (List.replicate r x)≤u.eval (R.eval z) at hrep |
| omega |
| have htail : (trials (r,x)).worst (fun as=>OnlineMedian.medianRealizer.cost as+weight (OnlineMedian.median as))≤v.eval (T.eval z) := by |
| apply Experiment.worst_le _ _ _ |
| exact (Experiment.all_worst (trials (r,x)) weight).mono (fun as hw=>(hv as).trans |
| (polynomial_eval_monotone v (hw.trans (by omega)))) |
| have hcost : (trials (r,x)).worst OnlineMedian.medianRealizer.cost≤v.eval (T.eval z) := |
| (Experiment.worst_mono _ _ _ ((Experiment.all_true _).mono (fun _ _=>Nat.le_add_right _ _))).trans htail |
| have hout : (trials (r,x)).worst (weight ∘ OnlineMedian.median)≤v.eval (T.eval z) := |
| (Experiment.worst_mono _ _ _ ((Experiment.all_true _).mono (fun _ _=>Nat.le_add_left _ _))).trans htail |
|
|
| have hboth : (trials (r,x)).worst OnlineMedian.medianRealizer.cost+ |
| (trials (r,x)).worst (weight ∘ OnlineMedian.median)≤2*v.eval (T.eval z) := by omega |
|
|
| clear hcost hout htail |
| change trialsRealizer.cost (r,x)+(trials (r,x)).worst OnlineMedian.medianRealizer.cost+ |
| (run (r,x)).worst weight≤_ |
| rw [run,Experiment.worst_map] |
| simp only [Polynomial.eval_add,Polynomial.eval_mul,Polynomial.eval_ofNat,Polynomial.eval_comp] |
| change _ ≤ T.eval z+2*v.eval (T.eval z) |
| omega |
| end MatchingFPRAS.OnlineAmplification |
|
|
| end OAI |
|
|
