| import OAI.Combinatorics.ExpectationThreshold.SelectorCountingEncodingBound |
| import OAI.Combinatorics.ExpectationThreshold.SelectorCommonWitness |
| import OAI.Combinatorics.ExpectationThreshold.SelectorProfile |
| import OAI.Combinatorics.ExpectationThreshold.SelectorEntropy |
| import OAI.Combinatorics.ExpectationThreshold.ProbabilityLikelihood |
| import OAI.Combinatorics.ExpectationThreshold.ColorArithmetic |
| import Mathlib.Tactic.Positivity |
|
|
| namespace OAI |
|
|
| /-! |
| # The actual contribution of one full selector profile |
|
|
| The common heavy sets are obtained separately for each occurring profile. |
| The entropy estimate removes their sizes before profiles are summed, so no |
| uniform choice of heavy sets across distinct profiles is required. |
| -/ |
|
|
| noncomputable section |
| open scoped BigOperators |
|
|
| namespace TalagrandThreshold.Selector |
|
|
| variable {α : Type*} [Fintype α] [DecidableEq α] |
|
|
| / |
| claimed weighted contribution whenever its rows fit the actual heavy sets. -/ |
| theorem profile_group_cost_le (p : ℝ) (s : ℕ) (hp : 0 < p) |
| (hs : p * scaleSum s ≤ 1 / 2) |
| (G : Finset (Coloring α s)) (z : Coloring α s) (n : Profile s) |
| (R : Fin s → Finset α) |
| (hEarlier : ∀ a ∈ G, Earlier a z) |
| (hprofile : ∀ a ∈ G, profile a z = n) |
| (hrows : ∀ a ∈ G, ∀ i, movedRow a z i ⊆ R i) |
| (hR : ∀ i, (R i).card ≤ 2 ^ (i.val + 1) * profileSize n) : |
| (∑ a ∈ G, colorMass p s a * p ^ (changes a z).card) ≤ |
| colorMass p s z * |
| (∏ i : Fin s, (3 / (64 : ℝ) ^ (i.val + 1)) ^ (rowCount n i)) * |
| (∏ i : Fin s, ((wordCount (rowCount n i) (n i) : ℝ) * |
| ∏ h : Color s, colorProb p s h ^ (n i h))) := by |
| classical |
| let C : ℝ := ∏ i : Fin s, ((R i).card.choose (rowCount n i) : ℝ) |
| let W : ℝ := ∏ i : Fin s, (wordCount (rowCount n i) (n i) : ℝ) |
| let L : ℝ := ∏ i : Fin s, (p / colorProb p s i.castSucc) ^ (rowCount n i) |
| let Q : ℝ := ∏ i : Fin s, ∏ h : Color s, colorProb p s h ^ (n i h) |
| let E : ℝ := ∏ i : Fin s, (3 * 4 ^ (i.val + 1) : ℝ) ^ (rowCount n i) |
| let K : ℝ := ∏ i : Fin s, (3 / (64 : ℝ) ^ (i.val + 1)) ^ (rowCount n i) |
| have hW : 0 ≤ W := Finset.prod_nonneg fun i _ => Nat.cast_nonneg _ |
| have hL : 0 ≤ L := Finset.prod_nonneg fun i _ => |
| pow_nonneg (div_nonneg hp.le (colorProb_nonneg p s hp hs _)) _ |
| have hQ : 0 ≤ Q := Finset.prod_nonneg fun i _ => |
| Finset.prod_nonneg fun h _ => pow_nonneg (colorProb_nonneg p s hp hs h) _ |
| have hP : 0 ≤ colorMass p s z := colorMass_nonneg p s hp hs z |
| have hcard : (G.card : ℝ) ≤ C * W := by |
| have h := card_le_profile_encoding G z n R hEarlier hprofile hrows |
| have hcast : (G.card : ℝ) ≤ |
| ∏ i : Fin s, ((R i).card.choose (rowCount n i) : ℝ) * |
| (wordCount (rowCount n i) (n i) : ℝ) := by |
| exact_mod_cast h |
| simpa only [Finset.prod_mul_distrib] using hcast |
| have hentropy : C ≤ E := |
| selector_entropy_bound s (fun i => (R i).card) (rowCount n) hR |
| have hnormalize : E * L = K := by |
| dsimp [E, L, K] |
| rw [← Finset.prod_mul_distrib] |
| apply Finset.prod_congr rfl |
| intro i _ |
| exact profile_coefficient_pow_eq hp s i (rowCount n i) |
| have hterm (a : Coloring α s) (ha : a ∈ G) : |
| colorMass p s a * p ^ (changes a z).card = colorMass p s z * L * Q := by |
| have h := colorMass_profile_likelihood p s hp hs a z (hEarlier a ha) |
| rw [hprofile a ha] at h |
| have hsize : (changes a z).card = profileSize n := by |
| rw [← profileSize_profile, hprofile a ha] |
| rw [hsize, mul_comm] |
| simpa only [Finset.prod_mul_distrib, mul_assoc] using h |
| calc |
| (∑ a ∈ G, colorMass p s a * p ^ (changes a z).card) = |
| (G.card : ℝ) * (colorMass p s z * L * Q) := by |
| simp only [Finset.sum_congr rfl hterm, Finset.sum_const, nsmul_eq_mul] |
| _ ≤ (C * W) * (colorMass p s z * L * Q) := |
| mul_le_mul_of_nonneg_right hcard (mul_nonneg (mul_nonneg hP hL) hQ) |
| _ = C * (W * (colorMass p s z * L * Q)) := by ring |
| _ ≤ E * (W * (colorMass p s z * L * Q)) := |
| mul_le_mul_of_nonneg_right hentropy |
| (mul_nonneg hW (mul_nonneg (mul_nonneg hP hL) hQ)) |
| _ = colorMass p s z * (E * L) * (W * Q) := by ring |
| _ = colorMass p s z * K * (W * Q) := by rw [hnormalize] |
| _ = _ := by |
| dsimp [K, W, Q] |
| rw [← Finset.prod_mul_distrib] |
|
|
| / |
| counting premises are discharged by the concrete selector construction. -/ |
| theorem group_cost_le {F : Family α} (mu : MassFamily F) |
| (p : ℝ) (s : ℕ) (hp : 0 < p) (hs : p * scaleSum s ≤ 1 / 2) |
| (z : Coloring α s) (n : Profile s) : |
| (∑ a ∈ groupOriginals mu z n, colorMass p s a * p ^ (changes a z).card) ≤ |
| colorMass p s z * |
| (∏ i : Fin s, (3 / (64 : ℝ) ^ (i.val + 1)) ^ (rowCount n i)) * |
| (∏ i : Fin s, ((wordCount (rowCount n i) (n i) : ℝ) * |
| ∏ h : Color s, colorProb p s h ^ (n i h))) := by |
| classical |
| by_cases hne : (groupOriginals mu z n).Nonempty |
| · obtain ⟨R, hR, hrows⟩ := exists_common_heavy_sets mu z n hne |
| apply profile_group_cost_le p s hp hs (groupOriginals mu z n) z n R |
| · intro a ha |
| have h := (Finset.mem_filter.mp ha).2 |
| obtain ⟨H₀, hmin⟩ := h.2.1 |
| exact hmin.1.1.1 |
| · intro a ha |
| exact (Finset.mem_filter.mp ha).2.2.2 |
| · exact hrows |
| · exact hR |
| · have hempty : groupOriginals mu z n = ∅ := Finset.not_nonempty_iff_eq_empty.mp hne |
| rw [hempty, Finset.sum_empty] |
| exact mul_nonneg |
| (mul_nonneg (colorMass_nonneg p s hp hs z) |
| (Finset.prod_nonneg fun i _ => pow_nonneg (by positivity) _)) |
| (Finset.prod_nonneg fun i _ => mul_nonneg (Nat.cast_nonneg _) |
| (Finset.prod_nonneg fun h _ => pow_nonneg (colorProb_nonneg p s hp hs h) _)) |
|
|
| end TalagrandThreshold.Selector |
|
|
| end |
|
|
| end OAI |
|
|
