| import OAI.Computability.UniqueGames.Foundations.SamplingLemmas |
| import OAI.Computability.UniqueGames.Games.Variation |
|
|
| namespace OAI |
|
|
| section |
|
|
| /-! |
| Direct diagonal-overlap comparison for the finite correlated sampler. |
| All quantities are finite weight arrays; no reference sampler or embedding |
| claim is assumed. The final theorem consumes the per-label lower bound that |
| is proved for the actual finite threshold sampler in CorrelatedSampling. |
| -/ |
|
|
| namespace UniqueGamesTheorem.Foundations.Repetition |
|
|
| open scoped BigOperators |
| open Information |
|
|
| private theorem min_half_formula (x y : ℝ) : |
| min x y = (x + y - |x - y|) / 2 := by |
| rcases le_total x y with h | h |
| · rw [min_eq_left h, abs_of_nonpos (sub_nonpos.mpr h)] |
| ring |
| · rw [min_eq_right h, abs_of_nonneg (sub_nonneg.mpr h)] |
| ring |
|
|
| theorem sum_min_one_sub_tv {Ω : Type*} [Fintype Ω] |
| (p q : Ω → ℝ) (hp : IsProbability p) (hq : IsProbability q) : |
| (∑ z, min (p z) (q z)) = 1 - totalVariation p q := by |
| classical |
| simp_rw [min_half_formula] |
| simp only [div_eq_mul_inv, ← Finset.sum_mul, Finset.sum_sub_distrib, |
| Finset.sum_add_distrib, hp.2, hq.2, totalVariation] |
| ring |
|
|
| private theorem discount_le_self {c t : ℝ} (hc : 0 ≤ c) (ht : 0 ≤ t) : |
| c / (1 + t) ≤ c := by |
| apply (div_le_iff₀ (by linarith : 0 < 1 + t)).2 |
| nlinarith [mul_nonneg hc ht] |
|
|
| private theorem discount_loss_le_mul {c t : ℝ} (hc : 0 ≤ c) (ht : 0 ≤ t) : |
| c - c / (1 + t) ≤ c * t := by |
| have hd : 0 < 1 + t := by linarith |
| have hid : c - c / (1 + t) = c * t / (1 + t) := by |
| field_simp [hd.ne'] |
| ring |
| rw [hid] |
| apply (div_le_iff₀ hd).2 |
| nlinarith [mul_nonneg (mul_nonneg hc ht) ht] |
|
|
| noncomputable def discountedCommonMass {X S : Type*} [Fintype X] [Fintype S] |
| (p a b : X × S → ℝ) (t : X → ℝ) : ℝ := |
| ∑ z, min (p z) (min (a z) (b z)) / (1 + t z.1) |
|
|
| / |
| theorem discounted_common_mass_bounds {X S : Type*} [Fintype X] [Fintype S] |
| (p a b : X × S → ℝ) (μ t : X → ℝ) |
| (hp : IsProbability p) (ha : IsProbability a) (hb : IsProbability b) |
| (ht : ∀ x, 0 ≤ t x) |
| (hrow : ∀ x, (∑ s, a (x, s)) = μ x) |
| (htv : (∑ x, μ x * t x) = totalVariation a b) : |
| 1 - 2 * totalVariation p a - 2 * totalVariation p b ≤ discountedCommonMass p a b t ∧ |
| discountedCommonMass p a b t ≤ 1 := by |
| classical |
| let C : X × S → ℝ := fun z => min (p z) (min (a z) (b z)) |
| have hC0 : ∀ z, 0 ≤ C z := by |
| intro z |
| exact le_min (hp.1 z) (le_min (ha.1 z) (hb.1 z)) |
| have hCp : ∀ z, C z ≤ p z := by |
| intro z |
| exact min_le_left _ _ |
| have hCa : ∀ z, C z ≤ a z := by |
| intro z |
| exact (min_le_right _ _).trans (min_le_left _ _) |
| have hmass : 1 - totalVariation p a - totalVariation p b ≤ ∑ z, C z := by |
| have hpoint : ∀ z, min (p z) (a z) + min (p z) (b z) - p z ≤ C z := by |
| intro z |
| dsimp [C] |
| refine le_min ?_ (le_min ?_ ?_) |
| · linarith [min_le_left (p z) (a z), min_le_left (p z) (b z)] |
| · linarith [min_le_right (p z) (a z), min_le_left (p z) (b z)] |
| · linarith [min_le_right (p z) (b z), min_le_left (p z) (a z)] |
| have hsum := Finset.sum_le_sum |
| (fun z (_ : z ∈ (Finset.univ : Finset (X × S))) => hpoint z) |
| simp only [Finset.sum_sub_distrib, Finset.sum_add_distrib] at hsum |
| rw [sum_min_one_sub_tv p a hp ha, sum_min_one_sub_tv p b hp hb, hp.2] at hsum |
| linarith |
| have hrowC : ∀ x, (∑ s, C (x, s)) ≤ μ x := by |
| intro x |
| calc |
| (∑ s, C (x, s)) ≤ ∑ s, a (x, s) := Finset.sum_le_sum (fun s _ => hCa (x, s)) |
| _ = μ x := hrow x |
| have hweighted : (∑ z, C z * t z.1) ≤ totalVariation a b := by |
| rw [← htv, Fintype.sum_prod_type] |
| apply Finset.sum_le_sum |
| intro x _ |
| change (∑ s, C (x, s) * t x) ≤ μ x * t x |
| rw [← Finset.sum_mul] |
| exact mul_le_mul_of_nonneg_right (hrowC x) (ht x) |
| have hloss : (∑ z, C z) - discountedCommonMass p a b t ≤ ∑ z, C z * t z.1 := by |
| change (∑ z, C z) - (∑ z, C z / (1 + t z.1)) ≤ ∑ z, C z * t z.1 |
| rw [← Finset.sum_sub_distrib] |
| exact Finset.sum_le_sum (fun z _ => discount_loss_le_mul (hC0 z) (ht z.1)) |
| have htriangle : totalVariation a b ≤ totalVariation p a + totalVariation p b := by |
| have h := totalVariation_triangle a p b |
| rw [totalVariation_symm a p] at h |
| exact h |
| have hupper : discountedCommonMass p a b t ≤ 1 := by |
| change (∑ z, C z / (1 + t z.1)) ≤ 1 |
| calc |
| _ ≤ ∑ z, p z := Finset.sum_le_sum |
| (fun z _ => (discount_le_self (hC0 z) (ht z.1)).trans (hCp z)) |
| _ = 1 := hp.2 |
| exact ⟨by linarith, hupper⟩ |
|
|
| noncomputable def diagonalWeights {X S : Type*} [Fintype X] [Fintype S] |
| (p : X × S → ℝ) : X × S × S → ℝ := by |
| classical |
| exact fun z => if z.2.1 = z.2.2 then p (z.1, z.2.1) else 0 |
|
|
| theorem diagonalWeights_isProbability {X S : Type*} [Fintype X] [Fintype S] |
| (p : X × S → ℝ) (hp : IsProbability p) : IsProbability (diagonalWeights p) := by |
| classical |
| constructor |
| · intro z |
| dsimp [diagonalWeights] |
| split |
| · exact hp.1 _ |
| · exact le_rfl |
| · simpa [diagonalWeights, Fintype.sum_prod_type] using hp.2 |
|
|
| theorem diagonal_overlap_identity {X S : Type*} [Fintype X] [Fintype S] |
| (p : X × S → ℝ) (sim : X × S × S → ℝ) |
| (hp : IsProbability p) (hsim : IsProbability sim) : |
| totalVariation sim (diagonalWeights p) = |
| 1 - ∑ z : X × S, min (p z) (sim (z.1, z.2, z.2)) := by |
| classical |
| have h := sum_min_one_sub_tv sim (diagonalWeights p) hsim (diagonalWeights_isProbability p hp) |
| have heq : (∑ z, min (sim z) (diagonalWeights p z)) = |
| ∑ z : X × S, min (p z) (sim (z.1, z.2, z.2)) := by |
| simp only [Fintype.sum_prod_type, diagonalWeights] |
| apply Finset.sum_congr rfl |
| intro x _ |
| apply Finset.sum_congr rfl |
| intro s _ |
| have hpoint : ∀ s', min (sim (x, s, s')) (if s = s' then p (x, s) else 0) = |
| if s = s' then min (p (x, s)) (sim (x, s, s')) else 0 := by |
| intro s' |
| by_cases he : s = s' |
| · simp [he, min_comm] |
| · simp [he, min_eq_right (hsim.1 (x, s, s'))] |
| simp only [hpoint] |
| simp |
| rw [heq] at h |
| linarith |
|
|
| / |
| target's question marginal differs from the simulator's question marginal. -/ |
| theorem diagonal_sampler_totalVariation_le {X S : Type*} [Fintype X] [Fintype S] |
| (p : X × S → ℝ) (μ : X → ℝ) (L R : X → S → ℝ) |
| (sim : X × S × S → ℝ) (hp : IsProbability p) (hμ : IsProbability μ) |
| (hL : ∀ x, IsProbability (L x)) (hR : ∀ x, IsProbability (R x)) |
| (hsim : IsProbability sim) {τ : ℝ} (hτ0 : 0 ≤ τ) (hτ1 : τ ≤ 1) |
| (hdiag : ∀ x s, μ x * min (L x s) (R x s) / |
| (1 + totalVariation (L x) (R x)) * (1 - τ) ≤ sim (x, s, s)) : |
| totalVariation sim (diagonalWeights p) ≤ |
| 2 * totalVariation p (fun z => μ z.1 * L z.1 z.2) + |
| 2 * totalVariation p (fun z => μ z.1 * R z.1 z.2) + τ := by |
| classical |
| let a : X × S → ℝ := fun z => μ z.1 * L z.1 z.2 |
| let b : X × S → ℝ := fun z => μ z.1 * R z.1 z.2 |
| let t : X → ℝ := fun x => totalVariation (L x) (R x) |
| let C : X × S → ℝ := fun z => min (p z) (min (a z) (b z)) |
| have ha : IsProbability a := kernelProduct_isProbability μ L hμ hL |
| have hb : IsProbability b := kernelProduct_isProbability μ R hμ hR |
| have ht : ∀ x, 0 ≤ t x := fun x => totalVariation_nonneg (L x) (R x) |
| have hrow : ∀ x, (∑ s, a (x, s)) = μ x := by |
| intro x |
| simp only [a, ← Finset.mul_sum, (hL x).2, mul_one] |
| have htv : (∑ x, μ x * t x) = totalVariation a b := |
| (totalVariation_joint_common_weights μ L R hμ.1).symm |
| obtain ⟨hBlower, hBupper⟩ := discounted_common_mass_bounds p a b μ t hp ha hb ht hrow htv |
| have hpoint : ∀ z : X × S, |
| (1 - τ) * (C z / (1 + t z.1)) ≤ min (p z) (sim (z.1, z.2, z.2)) := by |
| intro z |
| have hden : 0 < 1 + t z.1 := by linarith [ht z.1] |
| have hC0 : 0 ≤ C z := le_min (hp.1 z) (le_min (ha.1 z) (hb.1 z)) |
| have hCp : C z ≤ p z := min_le_left _ _ |
| have hdisc0 : 0 ≤ C z / (1 + t z.1) := div_nonneg hC0 hden.le |
| have hmin : min (a z) (b z) = μ z.1 * min (L z.1 z.2) (R z.1 z.2) := by |
| dsimp [a, b] |
| rcases le_total (L z.1 z.2) (R z.1 z.2) with h | h |
| · rw [min_eq_left h, min_eq_left (mul_le_mul_of_nonneg_left h (hμ.1 z.1))] |
| · rw [min_eq_right h, min_eq_right (mul_le_mul_of_nonneg_left h (hμ.1 z.1))] |
| have hCmin : C z ≤ μ z.1 * min (L z.1 z.2) (R z.1 z.2) := by |
| rw [← hmin] |
| exact min_le_right _ _ |
| apply le_min |
| · calc |
| _ ≤ 1 * (C z / (1 + t z.1)) := |
| mul_le_mul_of_nonneg_right (by linarith) hdisc0 |
| _ ≤ C z := by simpa using discount_le_self hC0 (ht z.1) |
| _ ≤ p z := hCp |
| · have hd := mul_le_mul_of_nonneg_left |
| (div_le_div_of_nonneg_right hCmin hden.le) (by linarith : 0 ≤ 1 - τ) |
| calc |
| _ ≤ (1 - τ) * (μ z.1 * min (L z.1 z.2) (R z.1 z.2) / (1 + t z.1)) := hd |
| _ ≤ sim (z.1, z.2, z.2) := by |
| simpa only [t, mul_comm] using hdiag z.1 z.2 |
| have hsum := Finset.sum_le_sum (fun z (_ : z ∈ (Finset.univ : Finset (X × S))) => hpoint z) |
| rw [← Finset.mul_sum] at hsum |
| change (1 - τ) * discountedCommonMass p a b t ≤ _ at hsum |
| rw [diagonal_overlap_identity p sim hp hsim] |
| have hτB := mul_le_mul_of_nonneg_left hBupper hτ0 |
| change 1 - _ ≤ 2 * totalVariation p a + 2 * totalVariation p b + τ |
| nlinarith |
|
|
| end UniqueGamesTheorem.Foundations.Repetition |
|
|
| end |
|
|
| end OAI |
|
|
