| import Mathlib |
| import OAI.Combinatorics.MatchingCount.Algorithm.OnlineGuardLaw |
| import OAI.Combinatorics.MatchingCount.Algorithm.OnlineMedian |
|
|
| namespace OAI |
|
|
| namespace MatchingFPRAS.OnlineAmplification |
| open PhysicalStream TreeTyped |
|
|
| abbrev TrialInput := List Bool×OnlineTrial.State |
|
|
| def trials (x : ℕ×TrialInput) : Experiment (List TreeFraction.Fraction) := |
| OnlineTree.traverseM OnlineGuard.trial (List.replicate x.1 x.2) |
| noncomputable def trialsRealizer : OnlineTree.Realizer trials := |
| OnlineTree.composition (OnlineTree.deterministic OnlineSample.repeatInputRealizer) |
| (OnlineTree.traverse OnlineGuard.trialRealizer) |
| def run (x : ℕ×TrialInput) : Experiment TreeFraction.Fraction := (trials x).map OnlineMedian.median |
| noncomputable def runRealizer : OnlineTree.Realizer run := |
| (OnlineTree.composition trialsRealizer (OnlineTree.deterministic OnlineMedian.medianRealizer)).congr (by |
| intro x h; simp only [run,Experiment.mean_map,Experiment.mean_bind,Experiment.mean]) |
|
|
| variable {G : GraphInput} [Nontrivial (Fin G.n)] |
| noncomputable def initial (p : Trial.Parameters G) : TrialInput := |
| (List.replicate p.K false,OnlineTrial.represent p (FiniteTrial.firstLayer p) PUnit.unit) |
| lemma trials_mean (p : Trial.Parameters G) (r : ℕ) (h : List TreeFraction.Fraction→ℚ) : |
| (trials (r,initial p)).mean h=qaverage (fun bs : Fin r→(Fin (FiniteTrial.guardedBits p)→Bool)=> |
| h (List.ofFn (fun i=>TreeFraction.repr (FiniteTrial.guardedBitOutput p (bs i))))) := by |
| rw [trials,←List.ofFn_const] |
| exact mean_traverse_ofFn OnlineGuard.trial r (fun _=>initial p) |
| (fun _ b=>TreeFraction.repr (FiniteTrial.guardedBitOutput p b)) (fun _=>OnlineGuard.trial_mean p) h |
| lemma run_mean (p : Trial.Parameters G) (δ : ℚ) (h : TreeFraction.Fraction→ℚ) : |
| (run (2*Amplification.halfRepetitions δ+1,initial p)).mean h= |
| qaverage (fun b=>h (TreeFraction.repr (FiniteTrial.guardedAmplifiedBitOutput p δ b))) := by |
| rw [run,Experiment.mean_map,trials_mean] |
| simp_rw [OnlineMedian.median_repr _ (fun _=>FiniteTrial.guardedBitOutput_nonneg p _)] |
| change _=qaverage ((fun bs : Fin (2*Amplification.halfRepetitions δ+1)→(Fin (FiniteTrial.guardedBits p)→Bool)=> |
| h (TreeFraction.repr (Amplification.median (fun i=>FiniteTrial.guardedBitOutput p (bs i))))) ∘ |
| Transport.matrixBitsEquiv (2*Amplification.halfRepetitions δ+1) (FiniteTrial.guardedBits p)) |
| exact (qaverage_equiv _ _).symm |
| end MatchingFPRAS.OnlineAmplification |
|
|
| end OAI |
|
|
