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e5a97c6 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 | 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
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