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d5b7741 | 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 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 | import OAI.Combinatorics.ContingencyTables.Sampling.ExactIntegerOnline
import OAI.Combinatorics.ContingencyTables.Sampling.OriginalTableBox
import OAI.Combinatorics.ContingencyTables.Sampling.ExactIntegerPhysical
namespace OAI
/-! # Returning a matrix from the exact integer-weighted trial -/
namespace ContingencyTables.ExactMatrixTrial
open MatchingFPRAS MatchingFPRAS.TreeTyped MatchingFPRAS.PhysicalStream
open DensePrograms ProfilePrograms
/-- Enumerated candidates together with a feasible fallback. -/
abbrev Lookup := List (Option MatrixCode)×MatrixCode
abbrev Input := Lookup×List ℕ
def lookup (z : Lookup×ℕ) : MatrixCode := (z.1.1.getD z.2 none).getD z.1.2
def lookupRealizer : Realizer lookup :=
composition (f := fun z : Lookup×ℕ => (z.1.2,z.1.1.getD z.2 none))
(g := fun z : MatrixCode×Option MatrixCode => z.2.getD z.1)
(pair (composition first second)
(composition (pair (composition first first) second) (getDFast none)))
FirstSuccessProgram.defaultRealizer
lemma polynomial_lookup : PolynomialTime lookupRealizer :=
polynomial_composition (polynomial_pair (polynomial_composition polynomial_first polynomial_second)
(polynomial_composition (polynomial_pair (polynomial_composition polynomial_first polynomial_first)
polynomial_second) (polynomial_getDFast none))) FirstSuccessProgram.polynomial_default
def decode (z : Lookup×Option ℕ) : Option MatrixCode :=
z.2.map (fun i => lookup (z.1,i))
def decodeRealizer : Realizer decode :=
(choose (composition second FirstSuccessProgram.isSomeRealizer)
(composition (composition
(pair first (composition (pair (constant (0:ℕ)) second) FirstSuccessProgram.defaultRealizer))
lookupRealizer) some) (constant none)).congr (by
intro z
rcases z with ⟨cache,o⟩
cases o <;> rfl)
lemma polynomial_decode : PolynomialTime decodeRealizer := by
unfold decodeRealizer
apply polynomial_congr
exact polynomial_choose (polynomial_composition polynomial_second FirstSuccessProgram.polynomial_isSome)
(polynomial_composition (polynomial_composition (polynomial_pair polynomial_first
(polynomial_composition (polynomial_pair (polynomial_constant 0) polynomial_second)
FirstSuccessProgram.polynomial_default)) polynomial_lookup) polynomial_some)
(polynomial_constant none)
def trial (z : Input) : Experiment (Option MatrixCode) :=
(ExactIntegerOnline.trial z.2).map (fun i => decode (z.1,i))
noncomputable def trialRealizer : OnlineTree.Realizer trial :=
(OnlineTree.composition
(OnlineTree.pair (OnlineTree.deterministic first)
(OnlineTree.composition (OnlineTree.deterministic second) ExactIntegerOnline.trialRealizer))
(OnlineTree.deterministic decodeRealizer)).congr (by
intro z g
simp only [Experiment.mean_bind,Experiment.mean,Experiment.mean_map,trial])
lemma polynomial_trial : OnlineTree.PolynomialTime trialRealizer := by
unfold trialRealizer
apply OnlineTree.work_congr
exact OnlineTree.polynomial_composition
(OnlineTree.polynomial_pair (OnlineTree.polynomial_deterministic polynomial_first)
(OnlineTree.polynomial_composition (OnlineTree.polynomial_deterministic polynomial_second)
ExactIntegerOnline.polynomial_trial))
(OnlineTree.polynomial_deterministic polynomial_decode)
noncomputable section
open ExactIntegerChoice ResidualMixture
open scoped BigOperators Classical
lemma trial_mean {K : ℕ} (candidates : Lookup) (w : Fin K → ℕ)
(hw : 0<totalWeight w) (g : Option MatrixCode → ℚ) :
(trial (candidates,List.ofFn w)).mean g=
∑ y : Option (Fin K),
(trialLaw w hw (Nat.le_pow_clog (by decide : 1<2) (totalWeight w))).mass y*
g (y.map (fun i => lookup (candidates,i.val))) := by
rw [trial,Experiment.mean_map,ExactIntegerOnline.trial_mean w hw]
apply Finset.sum_congr rfl
intro y _
cases y <;> rfl
lemma trial_failure {K : ℕ} (candidates : Lookup) (w : Fin K → ℕ)
(hw : 0<totalWeight w) :
(trial (candidates,List.ofFn w)).mean (fun o => if o=none then 1 else 0)=
failure w (Q := 2^(Nat.clog 2 (totalWeight w))) := by
rw [trial_mean _ w hw,Fintype.sum_option]
simp only [Option.map_none,ite_true,Option.map_some,reduceCtorEq,ite_false,mul_zero,
Finset.sum_const_zero,add_zero,mul_one,trialLaw_none]
lemma prefix_mean {K : ℕ} (candidates : Lookup) (w : Fin K → ℕ)
(hw : 0<totalWeight w) (N : ℕ) (g : Option MatrixCode → ℚ) :
(OnlineIntegerRetry.prefixExperiment (f := trial) (candidates,List.ofFn w) N).mean g=
∑ y : Option (Fin K),
(retryIntegerLaw w hw (Nat.le_pow_clog (by decide : 1<2) (totalWeight w)) N).mass y*
g (y.map (fun i => lookup (candidates,i.val))) :=
ExactIntegerOnline.prefix_mean trial (candidates,List.ofFn w) _ _
(trial_mean candidates w hw) N g
lemma prefix_failure {K : ℕ} (candidates : Lookup) (w : Fin K → ℕ)
(hw : 0<totalWeight w) (N : ℕ) :
(OnlineIntegerRetry.prefixExperiment (f := trial) (candidates,List.ofFn w) N).mean
(fun o => if o=none then 1 else 0)≤(1/2:ℚ)^N := by
rw [prefix_mean candidates w hw,Fintype.sum_option]
simp only [Option.map_none,ite_true,Option.map_some,reduceCtorEq,ite_false,mul_zero,
Finset.sum_const_zero,add_zero,mul_one]
exact retryIntegerLaw_failure_bound w hw _
(clog_power_lt_twice (totalWeight w) hw) N
/-- Exact weighted matrix selection has a summable physical halting tail. -/
theorem expected_time {K : ℕ} (candidates : Lookup) (w : Fin K → ℕ)
(hw : 0<totalWeight w) (s : TreeData.Data) (H : ℕ)
(hH : StackCost.StackBound
(OnlineTree.initial (OnlineIntegerRetry.program trialRealizer)
(TreeTyped.encode (candidates,List.ofFn w)) s).stk H) :
Summable (PhysicalRetryTail.tail (OnlineRetryRuntime.machineTable trialRealizer)
(OnlineRetryRuntime.initialCfg trialRealizer (candidates,List.ofFn w) s)) ∧
(∑' t,PhysicalRetryTail.tail (OnlineRetryRuntime.machineTable trialRealizer)
(OnlineRetryRuntime.initialCfg trialRealizer (candidates,List.ofFn w) s) t)≤
2*(OnlineRetryRuntime.attemptBound trialRealizer (candidates,List.ofFn w) H:ℝ) :=
OnlineRetryRuntime.expected_time trialRealizer (candidates,List.ofFn w) s H hH
(prefix_failure candidates w hw)
end
end ContingencyTables.ExactMatrixTrial
end OAI
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