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e320e82 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 | import OAI.Combinatorics.ExpectationThreshold.SelectorEntropyDyadic
import Mathlib.Data.Nat.Choose.Bounds
import Mathlib.Analysis.Complex.Exponential
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Ring
namespace OAI
/-!
# The selector's finite entropy estimate
After scaling row `i` by `4^(i+1)`, its binomial coefficient is bounded by
one term of the exponential series at `T * 2^(-(i+1))`. The sum of these
arguments is at most `T`, so their product is at most `exp T ≤ 3^T`.
This proves the row-count estimate directly, including rows of count zero.
-/
namespace TalagrandThreshold
open scoped BigOperators
/-- A finite series and its explicit remainder already give the needed slack. -/
private theorem selector_exp_one_le_three : Real.exp 1 ≤ 3 := by
have h := Real.exp_bound' (x := (1 : ℝ)) (n := 3)
(by norm_num) (by norm_num) (by norm_num)
exact h.trans (by norm_num [Finset.sum_range_succ])
/-- A scaled binomial coefficient is bounded by an exponential-series term. -/
theorem selector_choose_le_scaled_exp (m t : ℕ) (d y : ℝ)
(hd : 0 ≤ d) (hy : 0 ≤ y) (hm : (m : ℝ) ≤ d * y) :
(m.choose t : ℝ) ≤ d ^ t * Real.exp y := by
calc
(m.choose t : ℝ) ≤ (m : ℝ) ^ t / (t.factorial : ℝ) :=
Nat.choose_le_pow_div t m
_ ≤ (d * y) ^ t / (t.factorial : ℝ) :=
div_le_div_of_nonneg_right
(pow_le_pow_left₀ (Nat.cast_nonneg m) hm t) (Nat.cast_nonneg _)
_ = d ^ t * (y ^ t / (t.factorial : ℝ)) := by rw [mul_pow]; ring
_ ≤ d ^ t * Real.exp y :=
mul_le_mul_of_nonneg_left (Real.pow_div_factorial_le_exp y hy t)
(pow_nonneg hd t)
/-- The actual finite entropy bound for the selector's row sizes and counts.
The statement also permits total count zero. -/
theorem selector_entropy_bound (s : ℕ) (m t : Fin s → ℕ)
(hm : ∀ i, m i ≤ 2 ^ (i.val + 1) * (∑ j, t j)) :
(∏ i, ((m i).choose (t i) : ℝ)) ≤
∏ i, (3 * 4 ^ (i.val + 1) : ℝ) ^ (t i) := by
let T : ℕ := ∑ i, t i
have hT : (0 : ℝ) ≤ T := Nat.cast_nonneg T
have hrow (i : Fin s) :
((m i).choose (t i) : ℝ) ≤
((4 : ℝ) ^ (i.val + 1)) ^ (t i) *
Real.exp ((T : ℝ) * (1 / 2 : ℝ) ^ (i.val + 1)) := by
have hbound : (m i : ℝ) ≤ (2 : ℝ) ^ (i.val + 1) * (T : ℝ) := by
have hnat : m i ≤ 2 ^ (i.val + 1) * T := hm i
exact_mod_cast hnat
have hscale : (4 : ℝ) ^ (i.val + 1) *
((T : ℝ) * (1 / 2 : ℝ) ^ (i.val + 1)) =
(2 : ℝ) ^ (i.val + 1) * (T : ℝ) := by
calc
(4 : ℝ) ^ (i.val + 1) *
((T : ℝ) * (1 / 2 : ℝ) ^ (i.val + 1)) =
((4 : ℝ) * (1 / 2 : ℝ)) ^ (i.val + 1) * (T : ℝ) := by
rw [mul_pow]
ring
_ = (2 : ℝ) ^ (i.val + 1) * (T : ℝ) := by norm_num
apply selector_choose_le_scaled_exp
(m i) (t i) ((4 : ℝ) ^ (i.val + 1))
((T : ℝ) * (1 / 2 : ℝ) ^ (i.val + 1))
· exact pow_nonneg (by norm_num) _
· exact mul_nonneg hT (pow_nonneg (by norm_num) _)
· rw [hscale]
exact hbound
have hsum : (∑ i : Fin s, (T : ℝ) * (1 / 2 : ℝ) ^ (i.val + 1)) ≤
(T : ℝ) := by
rw [← Finset.mul_sum]
exact mul_le_of_le_one_right hT (selector_dyadic_sum_le_one s)
have hexp : Real.exp (T : ℝ) ≤ (3 : ℝ) ^ T := by
calc
Real.exp (T : ℝ) = Real.exp 1 ^ T := by
simpa only [mul_one] using Real.exp_nat_mul 1 T
_ ≤ (3 : ℝ) ^ T :=
pow_le_pow_left₀ (Real.exp_pos 1).le selector_exp_one_le_three T
have hprod : (0 : ℝ) ≤ ∏ i : Fin s, ((4 : ℝ) ^ (i.val + 1)) ^ (t i) :=
Finset.prod_nonneg (fun i _ => pow_nonneg (pow_nonneg (by norm_num) _) _)
have hfactor : (∏ i, (3 * 4 ^ (i.val + 1) : ℝ) ^ (t i)) =
(∏ i, ((4 : ℝ) ^ (i.val + 1)) ^ (t i)) * (3 : ℝ) ^ T := by
simp only [mul_pow, Finset.prod_mul_distrib, Finset.prod_pow_eq_pow_sum]
exact mul_comm _ _
calc
(∏ i, ((m i).choose (t i) : ℝ)) ≤
∏ i, (((4 : ℝ) ^ (i.val + 1)) ^ (t i) *
Real.exp ((T : ℝ) * (1 / 2 : ℝ) ^ (i.val + 1))) :=
Finset.prod_le_prod₀ (fun i _ => Nat.cast_nonneg _) (fun i _ => hrow i)
_ = (∏ i, ((4 : ℝ) ^ (i.val + 1)) ^ (t i)) *
Real.exp (∑ i : Fin s, (T : ℝ) * (1 / 2 : ℝ) ^ (i.val + 1)) := by
rw [Finset.prod_mul_distrib, ← Real.exp_sum]
_ ≤ (∏ i, ((4 : ℝ) ^ (i.val + 1)) ^ (t i)) * Real.exp (T : ℝ) :=
mul_le_mul_of_nonneg_left (Real.exp_le_exp_of_le hsum) hprod
_ ≤ (∏ i, ((4 : ℝ) ^ (i.val + 1)) ^ (t i)) * (3 : ℝ) ^ T :=
mul_le_mul_of_nonneg_left hexp hprod
_ = ∏ i, (3 * 4 ^ (i.val + 1) : ℝ) ^ (t i) := hfactor.symm
end TalagrandThreshold
end OAI
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