| import OAI.Probability.MatroidProphet.SafeParity |
| import OAI.Probability.MatroidProphet.Residual.Statistics |
|
|
| namespace OAI |
|
|
| namespace MatroidProphet |
| open Finset Pivots |
| variable {α : Type*} [Fintype α] [DecidableEq α] |
| attribute [local instance] Classical.propDecidable |
|
|
| omit [DecidableEq α] in |
| lemma nominalBirth_eq_iff_departure (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (D C : ℕ → Set α) (h : ℕ) (d : α) (hd : d ∉ M.closure ∅) (b : ℤ) : |
| nominalBirth M hE κ D C h d = b ↔ |
| d ∈ nominalPath M hE κ D C h b ∧ d ∉ nominalPath M hE κ D C h (b-1) := by |
| rw [mem_nominal_iff_birth_le M hE κ D C h d hd b, |
| mem_nominal_iff_birth_le M hE κ D C h d hd (b-1)] |
| omega |
|
|
| omit [DecidableEq α] in |
| lemma isSafeSample_iff_window (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (D : ℕ → Set α) (G : ℕ → Finset α) (h : ℕ) (U : Set α) (I C T : Finset α) |
| (hIU : (I : Set α) ⊆ U) (d : α) (hdI : d ∈ I) (hd : d ∉ M.closure ∅) (parity : Bool) : |
| IsSafeSample M hE κ D G h d C T parity ↔ |
| ∃ b : ParityWindow (activation h) (boolParity parity), |
| d ∈ safeLayerSet M hE κ D (fun i => (C : Set α) ∩ (G i : Set α)) |
| (fun i => (T : Set α) ∩ (G i : Set α)) h U (I : Set α) (boolParity parity) b := by |
| constructor |
| · intro hs |
| obtain ⟨hlo, hhi, hp, hex⟩ := hs |
| let birth := nominalBirth M hE κ D (fun i => (C : Set α) ∩ (G i : Set α)) h d |
| have hbwin : birth ∈ Set.Icc (activation h-2) 1 := by |
| change activation h+2 ≤ birth at hlo |
| change birth ≤ 0 at hhi |
| constructor <;> omega |
| have hbpar : birth % 2 = ((boolParity parity).val : ℤ) := by |
| rw [boolParity_val] |
| exact hp |
| refine ⟨⟨⟨birth, hbwin⟩, hbpar⟩, ?_⟩ |
| have hdep := (nominalBirth_eq_iff_departure M hE κ D _ h d hd birth).mp rfl |
| exact ⟨⟨hdI, hIU hdI, hlo, hhi, hdep.1, hdep.2⟩, hex⟩ |
| · rintro ⟨b, hb⟩ |
| rcases hb with ⟨⟨_, _, hlo, hhi, hmem, hnot⟩, hex⟩ |
| have hbirth := (nominalBirth_eq_iff_departure M hE κ D _ h d hd b.val.val).mpr ⟨hmem, hnot⟩ |
| unfold IsSafeSample |
| rw [hbirth] |
| have hp := b.property.trans (boolParity_val parity) |
| exact ⟨hlo, hhi, hp, hex⟩ |
|
|
| theorem safeSampleCount_eq_layerStatistic |
| {α : Type u_1} [Fintype α] [DecidableEq α] (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (D : ℕ → Set α) (G : ℕ → Finset α) (h : ℕ) (U : Set α) (I C T : Finset α) |
| (hIU : (I : Set α) ⊆ U) (hIL : ∀ d ∈ I, d ∉ M.closure ∅) (parity : Bool) : |
| safeSampleCount M hE κ D G h I C T parity = |
| (safeLayerStatistic M hE κ D (fun i => (C : Set α) ∩ (G i : Set α)) |
| (fun i => (T : Set α) ∩ (G i : Set α)) h U (I : Set α) (boolParity parity) : ℝ) := by |
| let W := ParityWindow (activation h) (boolParity parity) |
| let L : W → Set α := safeLayerSet M hE κ D (fun i => (C : Set α) ∩ (G i : Set α)) |
| (fun i => (T : Set α) ∩ (G i : Set α)) h U (I : Set α) (boolParity parity) |
| have hcount (b : W) : (L b).ncard = (I.filter (fun d => d ∈ L b)).card := by |
| have hs : L b = ((I.filter (fun d => d ∈ L b) : Finset α) : Set α) := by |
| ext d |
| constructor |
| · intro hd |
| exact mem_filter.mpr ⟨hd.1.1, hd⟩ |
| · intro hd |
| exact (mem_filter.mp hd).2 |
| exact (congrArg Set.ncard hs).trans (Set.ncard_coe_finset _) |
| have hlabel (d : α) (hd : d ∈ I) : |
| (∑ b : W, if d ∈ L b then (1:ℝ) else 0) = if IsSafeSample M hE κ D G h d C T parity then 1 else 0 := by |
| rw [sum_unique_indicators] |
| · have he : (∃ b ∈ (univ : Finset W), d ∈ L b) ↔ IsSafeSample M hE κ D G h d C T parity := by |
| simp only [mem_univ, true_and] |
| exact (isSafeSample_iff_window M hE κ D G h U I C T hIU d hd (hIL d hd) parity).symm |
| simp only [he] |
| · intro b hb c hc hdb hdc |
| by_contra hbc |
| have hdisj := nominalLayerSet_disjoint M hE κ D (fun i => (C : Set α) ∩ (G i : Set α)) |
| h U (boolParity parity) hbc |
| exact Set.disjoint_left.mp hdisj hdb.1.2 hdc.1.2 |
| calc |
| safeSampleCount M hE κ D G h I C T parity = |
| ∑ d ∈ I, if IsSafeSample M hE κ D G h d C T parity then (1:ℝ) else 0 := by |
| rw [safeSampleCount_eq_card] |
| simp [safeSampleSet] |
| _ = ∑ d ∈ I, ∑ b : W, if d ∈ L b then (1:ℝ) else 0 := |
| sum_congr rfl (fun d hd => (hlabel d hd).symm) |
| _ = ∑ b : W, ∑ d ∈ I, if d ∈ L b then (1:ℝ) else 0 := sum_comm |
| _ = ∑ b : W, ((L b).ncard : ℝ) := by |
| apply sum_congr rfl |
| intro b hb |
| rw [hcount] |
| simp |
| _ = _ := by simp only [safeLayerStatistic, Nat.cast_sum]; rfl |
|
|
| theorem safeLayerStatistic_lower (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (hκ : 0 < κ) (D : ℕ → Set α) (G : ℕ → Finset α) |
| (hG : Pairwise (fun i j => Disjoint (G i) (G j))) (q : α → ℝ) |
| (hq0 : ∀ e, 0 ≤ q e) (hq1 : ∀ e, q e ≤ 1) |
| (h : ℕ) (U I : Set α) (hIU : I ⊆ U) (hI : M.Indep I) : |
| (((2:ℝ)^12)⁻¹) / 4 * I.ncard - (D h).ncard / (κ:ℝ) ≤ |
| bitsExpectation q univ (fun C => bitsExpectation q univ (fun T => |
| fairParityExpectation (fun parity => (safeLayerStatistic M hE κ D |
| (fun i => (C : Set α) ∩ (G i : Set α)) (fun i => (T : Set α) ∩ (G i : Set α)) |
| h U I (boolParity parity) : ℝ)))) := by |
| let J := I.toFinset |
| have hJI : (J : Set α) = I := by simp [J] |
| have hJcard : J.card = I.ncard := by rw [← Set.ncard_coe_finset, hJI] |
| have hJ : M.Indep (J : Set α) := hJI.symm ▸ hI |
| have hJU : (J : Set α) ⊆ U := hJI.symm ▸ hIU |
| have hl := safeSampleCount_lower M hE κ hκ D G hG q hq0 hq1 h J hJ |
| rw [hJcard] at hl |
| have heq : bitsExpectation q univ (fun C => bitsExpectation q univ (fun T => |
| fairParityExpectation (safeSampleCount M hE κ D G h J C T))) = |
| bitsExpectation q univ (fun C => bitsExpectation q univ (fun T => |
| fairParityExpectation (fun parity => (safeLayerStatistic M hE κ D |
| (fun i => (C : Set α) ∩ (G i : Set α)) (fun i => (T : Set α) ∩ (G i : Set α)) |
| h U I (boolParity parity) : ℝ)))) := by |
| apply bitsExpectation_congr |
| intro C hC |
| apply bitsExpectation_congr |
| intro T hT |
| congr 1 |
| funext parity |
| rw [safeSampleCount_eq_layerStatistic M hE κ D G h U J C T hJU |
| (fun d hd => indep_not_loop M J hJ hd) parity, hJI] |
| exact hl.trans_eq heq |
|
|
| end MatroidProphet |
|
|
| end OAI |
|
|
