| 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 |
|
|
| / |
| 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 |
|
|
| / |
| 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 |
| |
