| import OAI.Probability.MatroidProphet.Main |
| import OAI.Probability.MatroidSecretary.Transport.TransferSpecification |
|
|
| namespace OAI |
|
|
| /-! |
| # Source-facing sample-to-rank transfer |
|
|
| Source: `sections/transfer.tex`, `lem:transfer` / `eq:rank-transfer`. |
| The comparison set is the actual priority-greedy survivor group; its independence |
| and containment are proved, not supplied as extra assumptions. `mainMean` fixes |
| only H and averages the density, guard, test, and parity coins. `trueZ` uses zero |
| on unlisted outcomes, so the expectation is not conditioned on listing. |
| -/ |
|
|
| namespace MatroidProphet.SourceProof |
|
|
| open Finset MainAlgorithm |
|
|
| / |
| fields of the initial record are overwritten by independent product samples. -/ |
| lemma mainMean_filter_congr {n : ℕ} (d d' : MainMasks n) |
| (hH : d.H = d'.H) (f : MainMasks n → ℝ) : mainMean d f = mainMean d' f := by |
| rcases d with ⟨H, D, C, T, p⟩ |
| rcases d' with ⟨H', D', C', T', p'⟩ |
| change H = H' at hH |
| subst H' |
| rfl |
|
|
| / |
| lemma trueZ_unlisted {n : ℕ} (M : Matroid (Fin n)) (hE : M.E = Set.univ) |
| (d : MainMasks n) (s : Fin n → Option ℤ) (i : ℤ) |
| (hi : i ∉ groups M d s) : trueZ M hE d s i = 0 := by |
| classical |
| simp only [trueZ, ite_eq_right hi] |
|
|
| / |
| no positivity or listing assumption is imposed on the individual outcomes. -/ |
| lemma trueZ_zero_of_no_discovery {n : ℕ} (M : Matroid (Fin n)) |
| (hE : M.E = Set.univ) (d : MainMasks n) (s : Fin n → Option ℤ) (i : ℤ) |
| (hmiss : d.D ∩ trueGroup M d s i = ∅) : trueZ M hE d s i = 0 := by |
| apply trueZ_unlisted |
| intro hi |
| obtain ⟨e, heD, heU⟩ := (mem_listed_iff M d s i).mp ((mem_groups M d s i).mp hi) |
| have he : e ∈ d.D ∩ trueGroup M d s i := mem_inter.mpr ⟨heD, heU⟩ |
| rw [hmiss] at he |
| exact notMem_empty e he |
|
|
| / |
| also covers the `none` keys representing zero/nonpositive weights. -/ |
| theorem transfer_keys {n : ℕ} (M : Matroid (Fin n)) (hE : M.E = Set.univ) |
| (s : Fin n → Option ℤ) (H : Finset (Fin n)) (i : ℤ) : |
| let d : MainMasks n := ⟨H, ∅, ∅, ∅, false⟩ |
| let U := trueGroup M d s i |
| let Y := (positiveGreedy M s \ H).filter (fun e => s e = some i) |
| densityThreshold ≤ (U.card : ℝ) → |
| ((((2 : ℝ)^16)⁻¹ * Y.card - ((2 : ℝ)^23)⁻¹ * U.card) / (2 : ℝ)^100) ≤ |
| mainMean d (fun d' => (trueZ M hE d' s i : ℝ)) := by |
| classical |
| dsimp only |
| intro hn |
| let d : MainMasks n := ⟨H, ∅, ∅, ∅, false⟩ |
| let Y := (positiveGreedy M s \ H).filter (fun e => s e = some i) |
| have hY : (Y : Set (Fin n)) ⊆ (trueGroup M d s i : Set (Fin n)) := by |
| intro e he |
| rw [trueGroup_eq_candidate_filter] |
| rcases mem_filter.mp he with ⟨heG, hei⟩ |
| exact mem_filter.mpr ⟨positiveGreedy_survivors_subset M s H heG, hei⟩ |
| have hI : M.Indep (Y : Set (Fin n)) := |
| (positiveGreedy_survivors_indep M hE s H).subset |
| (fun _ he => (mem_filter.mp he).1) |
| have h := mainMean_trueZ_lower M hE d s i (Y : Set (Fin n)) hY hI hn |
| simpa only [Set.ncard_coe_finset, retainedFraction, densityThreshold, d, Y] using h |
|
|
| / |
| with literal constants gamma = 2^-16 and kappa = 2^100. -/ |
| theorem transfer {n : ℕ} (M : Matroid (Fin n)) (hE : M.E = Set.univ) |
| (w : Weights n) (_hw : ∀ e, 0 ≤ w e) (H : Finset (Fin n)) (i : ℤ) : |
| let s : Fin n → Option ℤ := fun e => roundedLevel weightBase (w e) |
| let d : MainMasks n := ⟨H, ∅, ∅, ∅, false⟩ |
| let U := trueGroup M d s i |
| let Y := (positiveGreedy M s \ H).filter (fun e => s e = some i) |
| densityThreshold ≤ (U.card : ℝ) → |
| ((((2 : ℝ)^16)⁻¹ * Y.card - ((2 : ℝ)^23)⁻¹ * U.card) / (2 : ℝ)^100) ≤ |
| mainMean d (fun d' => (trueZ M hE d' s i : ℝ)) := |
| transfer_keys M hE (fun e => roundedLevel weightBase (w e)) H i |
|
|
| / |
| theorem transfer_claim : TransferClaim := by |
| intro n M hE w hw H i |
| exact transfer M hE w hw H i |
|
|
| end MatroidProphet.SourceProof |
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| end OAI |
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