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cdec422 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 | import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.AmbientDegree
import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.ExtractionRates
namespace OAI
/-! Joint tensor extraction, compatibility and entropy estimates. -/
noncomputable section
namespace MatrixMultiplication.JointAmbientRateLimit
open MatrixMultiplication.Foundation JointPopulation JointAmbientDegree Filter
open scoped BigOperators Topology
attribute [local instance] Classical.propDecidable
theorem tendsto_log_succ_div_nat_of_ratio {n : ℕ → ℕ} {mass : ℝ}
(hsize : Tendsto (fun N => (n N : ℝ) / (N : ℝ)) atTop (𝓝 mass)) :
Tendsto (fun N => Real.log (n N + 1 : ℕ) / (N : ℝ)) atTop (𝓝 0) := by
let C : ℝ := |mass| + 2
have hC : 0 < C := by dsimp [C]; positivity
have hmC : mass < C := by dsimp [C]; linarith [le_abs_self mass]
have hlog : Tendsto (fun N : ℕ => Real.log (N : ℝ) / (N : ℝ)) atTop (𝓝 0) := by
simpa only [Function.comp_def, id_eq] using
Real.isLittleO_log_id_atTop.tendsto_div_nhds_zero.comp tendsto_natCast_atTop_atTop
have hc : Tendsto (fun N : ℕ => Real.log (C + 1) / (N : ℝ)) atTop (𝓝 0) :=
tendsto_const_nhds.div_atTop tendsto_natCast_atTop_atTop
have hu : Tendsto (fun N : ℕ =>
(Real.log (C + 1) + Real.log (N : ℝ)) / (N : ℝ)) atTop (𝓝 0) := by
simpa only [add_div, add_zero] using hc.add hlog
apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds hu
· exact Eventually.of_forall fun N =>
div_nonneg (Real.log_nonneg (by exact_mod_cast Nat.succ_le_succ (Nat.zero_le (n N))))
(Nat.cast_nonneg _)
· filter_upwards [(tendsto_order.1 hsize).2 C hmC,
eventually_ge_atTop (1 : ℕ)] with N hn hN
have hNr : (1 : ℝ) ≤ N := by exact_mod_cast hN
have hNp : (0 : ℝ) < N := lt_of_lt_of_le zero_lt_one hNr
have hsizeN := (div_lt_iff₀ hNp).mp hn
have hbound : (n N + 1 : ℕ) ≤ (C + 1) * (N : ℝ) := by
push_cast
nlinarith
have hl := Real.log_le_log (by positivity : (0 : ℝ) < (n N + 1 : ℕ)) hbound
rw [Real.log_mul (by positivity : C + 1 ≠ 0) hNp.ne'] at hl
exact div_le_div_of_nonneg_right hl hNp.le
variable {H : Type*} [Fintype H] [DecidableEq H]
omit [Fintype H] [DecidableEq H] in
theorem tendsto_classExponent_div_nat
(counts : ℕ → H → Shape → ℕ) (s : Fin 3) (h : H)
(cap : ℕ → ℝ) (mass capRate marginalRate : ℝ)
(hsize : Tendsto (fun N => (∑ a, counts N h a : ℕ) / (N : ℝ))
atTop (𝓝 mass))
(hcap : Tendsto (fun N => ((∑ a, counts N h a : ℕ) / (N : ℝ)) * cap N)
atTop (𝓝 capRate))
(hmarginal : Tendsto (fun N =>
((∑ a, counts N h a : ℕ) / (N : ℝ)) *
finiteEntropy (fun a => (wordPopulation
(sideWord (counts N) s (triple (counts N) (canonicalTarget (counts N))) h) a : ℝ) /
Fintype.card (Positions (counts N) h))) atTop (𝓝 marginalRate)) :
Tendsto (fun N =>
classExponent (counts N) s (triple (counts N) (canonicalTarget (counts N))) h (cap N) /
(N : ℝ)) atTop (𝓝 (capRate - marginalRate)) := by
let A : ℝ := (Fintype.card Shape : ℝ) + (Fintype.card (Fin 17) : ℝ) + 2
have hlog := tendsto_log_succ_div_nat_of_ratio hsize
have hinv : Tendsto (fun N : ℕ => (1 : ℝ) / N) atTop (𝓝 0) :=
tendsto_const_nhds.div_atTop tendsto_natCast_atTop_atTop
have htotal := (hcap.sub hmarginal).add
(((hinv.add hlog).const_mul A).add (hlog.const_mul (Fintype.card Shape : ℝ)))
simp only [add_zero, mul_zero] at htotal
convert htotal using 1
ext N
dsimp only [classExponent, A, Positions]
simp only [Fintype.card_fin, div_eq_mul_inv]
ring
omit [DecidableEq H] in
theorem tendsto_sum_classExponent_div_nat
(counts : ℕ → H → Shape → ℕ) (s : Fin 3)
(cap : ℕ → H → ℝ) (mass capRate marginalRate : H → ℝ)
(hsize : ∀ h, Tendsto (fun N => (∑ a, counts N h a : ℕ) / (N : ℝ))
atTop (𝓝 (mass h)))
(hcap : ∀ h, Tendsto (fun N => ((∑ a, counts N h a : ℕ) / (N : ℝ)) * cap N h)
atTop (𝓝 (capRate h)))
(hmarginal : ∀ h, Tendsto (fun N =>
((∑ a, counts N h a : ℕ) / (N : ℝ)) *
finiteEntropy (fun a => (wordPopulation
(sideWord (counts N) s (triple (counts N) (canonicalTarget (counts N))) h) a : ℝ) /
Fintype.card (Positions (counts N) h))) atTop (𝓝 (marginalRate h))) :
Tendsto (fun N => (∑ h,
classExponent (counts N) s (triple (counts N) (canonicalTarget (counts N))) h (cap N h)) /
(N : ℝ)) atTop (𝓝 (∑ h, (capRate h - marginalRate h))) := by
have hs := tendsto_finsetSum Finset.univ (fun h _ =>
tendsto_classExponent_div_nat counts s h (fun N => cap N h)
(mass h) (capRate h) (marginalRate h) (hsize h) (hcap h) (hmarginal h))
simpa only [Finset.sum_div] using hs
theorem eventually_all_target_degree_upper
(counts : ℕ → H → Shape → ℕ) (sum : H → ℕ)
(cap : ℕ → H → ℝ) (sideRate : Fin 3 → ℝ) (D : ℝ) (hD : 0 ≤ D)
(hentropy : ∀ᶠ N in atTop, ∀ h w, localAllowed (counts N) sum h w →
finiteEntropy (fun a => (wordPopulation w a : ℝ) /
Fintype.card (Positions (counts N) h)) ≤ cap N h)
(hrate : ∀ s, Tendsto (fun N => (∑ h,
classExponent (counts N) s (triple (counts N) (canonicalTarget (counts N))) h (cap N h)) /
(N : ℝ)) atTop (𝓝 (sideRate s)))
(hside : ∀ s, sideRate s ≤ D) (η : ℝ) (hη : 0 < η) :
∀ᶠ N in atTop, ∀ e : Target (counts N),
Real.log ((neighbors (counts N) sum (triple (counts N) e)).card : ℝ) / (N : ℝ) ≤
D + η := by
have hs : ∀ᶠ N in atTop, ∀ s : Fin 3,
(∑ h, classExponent (counts N) s
(triple (counts N) (canonicalTarget (counts N))) h (cap N h)) / (N : ℝ) ≤
D + η / 2 := by
rw [Filter.eventually_all]
intro s
filter_upwards [(tendsto_order.1 (hrate s)).2 (sideRate s + η / 2) (by linarith)]
with N hN
linarith [hside s]
have hc : Tendsto (fun N : ℕ => Real.log 3 / (N : ℝ)) atTop (𝓝 0) :=
tendsto_const_nhds.div_atTop tendsto_natCast_atTop_atTop
filter_upwards [hentropy, hs, (tendsto_order.1 hc).2 (η / 2) (by linarith),
eventually_ge_atTop (1 : ℕ)] with N he hs hc hN
intro e
by_cases hz : (neighbors (counts N) sum (triple (counts N) e)).card = 0
· simp only [hz, Nat.cast_zero, Real.log_zero, zero_div]
linarith
have hNp : (0 : ℝ) < N := by exact_mod_cast hN
have hupper (s : Fin 3) :
(∑ h, classExponent (counts N) s (triple (counts N) e) h (cap N h)) ≤
(N : ℝ) * (D + η / 2) := by
simp_rw [classExponent_target_eq]
simpa only [mul_comm] using (div_le_iff₀ hNp).mp (hs s)
have hb := div_le_div_of_nonneg_right
(log_neighbors_card_le (counts N) sum (triple (counts N) e) (cap N) he
((N : ℝ) * (D + η / 2)) hupper (Nat.pos_of_ne_zero hz)) hNp.le
rw [add_div, mul_div_cancel_left₀ _ hNp.ne'] at hb
linarith
end MatrixMultiplication.JointAmbientRateLimit
end
end OAI
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