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dd14c26 | 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 | import Mathlib
import OAI.Probability.LogConcave.Sampling.LogMeshCountRate
namespace OAI
section
noncomputable section
namespace LogConcaveSampling
open Filter Quadrature MeanTree
open scoped Topology Classical
def sourceOccurrenceBudget (a b : ℝ) (n m N nc Nc d : ℕ) : ℕ :=
meanWork (Fintype.card (ProbabilityNode (sourceCorrelation a d) ((d:ℝ)^(-b)) (n+1)))
(n+2) m N nc Nc 0 0 0+
(Fintype.card (ProbabilityNode (sampleCorrelation ((d:ℝ)^(-a))) ((d:ℝ)^(-b)) (n+1))+1)^N
def occurrenceDegree (N Nc : ℕ) : ℕ := max (Nc*(2*N+2)) N
lemma sourceOccurrenceBudget_rate {a b : ℝ} (ha : 0<a) (hb : 0<b)
(n m N nc Nc : ℕ) :
LogPowerRate (fun d => (sourceOccurrenceBudget a b n m N nc Nc d:ℝ))
(-((occurrenceDegree N Nc:ℕ):ℝ)*b) := by
have he := sourceCorrelation_eventually ha
have hp := probabilityNode_card_rate hb.le (n+1)
(he.mono (fun _ h => h.1)) (he.mono (fun _ h => h.2.1))
(logMeshLength_rate a (he.mono (fun _ h => h.1))
(he.mono (fun _ h => h.2.1)) (he.mono (fun _ h => h.2.2.le)))
have hR : ∀ᶠ d : ℕ in atTop,0<(d:ℝ)^(-a) ∧ (d:ℝ)^(-a)≤1 := by
filter_upwards [eventually_ge_atTop (1:ℕ)] with d hd
have hd1 : (1:ℝ)≤d := by exact_mod_cast hd
exact ⟨Real.rpow_pos_of_pos (by linarith) _,Real.rpow_le_one_of_one_le_of_nonpos hd1 (by linarith)⟩
have hs := hR.mono (fun d hd => sampleCorrelation_properties hd.1 hd.2)
have hsp := probabilityNode_card_rate hb.le (n+1)
(hs.mono (fun _ h => by linarith [h.1])) (hs.mono (fun _ h => h.2.1))
(logMeshLength_sample_rate (hR.mono (fun _ h => ⟨le_rfl,h.2⟩)))
have hsample := (hsp.add ((LogPowerRate.const 1).mono (by linarith))).pow N
have hmean := meanWork_rate hb.le hp (n+2) m N nc Nc
have hB₁ : ((Nc*(2*N+2):ℕ):ℝ)≤(occurrenceDegree N Nc:ℝ) := by
exact_mod_cast le_max_left (Nc*(2*N+2)) N
have hB₂ : (N:ℝ)≤(occurrenceDegree N Nc:ℝ) := by exact_mod_cast le_max_right (Nc*(2*N+2)) N
have hm := hmean.mono (show -(occurrenceDegree N Nc:ℝ)*b≤-((Nc*(2*N+2):ℕ):ℝ)*b by nlinarith)
have hp' := hsample.mono (show -(occurrenceDegree N Nc:ℝ)*b≤(N:ℝ)*(-b) by nlinarith)
simpa only [sourceOccurrenceBudget,Nat.cast_add,Nat.cast_pow,Nat.cast_one] using hm.add hp'
namespace OracleCompiler
lemma literalSample_cost_le_source {a b : ℝ} {d : ℕ} (hd : 0<d)
(n m N nc Nc : ℕ) {η : ℝ} (hηR : (d:ℝ)^(-a)≤η) (hη1 : η≤1)
(r : ℝ) (e : ProbabilityNode (sampleCorrelation η) ((d:ℝ)^(-b)) (n+1)) :
cost (literalSample (d:=d) r (sampleCorrelation η) ((d:ℝ)^(-b)) (n+1) N e)≤
sourceOccurrenceBudget a b n m N nc Nc d := by
have hd0 : (0:ℝ)<d := by exact_mod_cast hd
have hR := Real.rpow_pos_of_pos hd0 (-a)
have hη := lt_of_lt_of_le hR hηR
have hR1 : (d:ℝ)^(-a)≤1 := hηR.trans hη1
have hmono := logMeshCount_mono (sampleCorrelation_antitone hR.le hηR)
(sampleCorrelation_properties hR hR1).2.1 (Real.rpow_pos_of_pos hd0 (-b))
have hcard : Fintype.card (ProbabilityNode (sampleCorrelation η) ((d:ℝ)^(-b)) (n+1))≤
Fintype.card (ProbabilityNode (sampleCorrelation ((d:ℝ)^(-a))) ((d:ℝ)^(-b)) (n+1)) := by
simp only [probabilityNode_card]
exact Nat.mul_le_mul_right _ hmono
rw [cost_literalSample]
have he := flowWork_bound (Fintype.card (ProbabilityNode (sampleCorrelation η) ((d:ℝ)^(-b)) (n+1))) N 0
simp only [zero_add,one_mul] at he
have hp := Nat.pow_le_pow_left (Nat.add_le_add_right hcard 1) N
unfold sourceOccurrenceBudget
omega
theorem exists_cost_mesh_exponent {t κ : ℝ} (ht : 0<t) (hκ : 0<κ)
(N Nc rank : ℕ) : ∃w : ℝ,0<w ∧ w<t ∧
∀n m nc,∀ᶠ d : ℕ in atTop,
((2*(40*sourceOccurrenceBudget (κ*t) (κ*w) n m N nc Nc d+40)^rank:ℕ):ℝ)≤
(d:ℝ)^(κ*t) := by
let H : ℝ := (rank:ℝ)*(occurrenceDegree N Nc:ℝ)+1
have hH : 0<H := by dsimp [H]; positivity
let w := t/(2*H)
have hw : 0<w := div_pos ht (mul_pos (by norm_num) hH)
have hHt : 1≤H := by
dsimp [H]
have : 0≤(rank:ℝ)*(occurrenceDegree N Nc:ℝ) := by positivity
linarith
have hwid : w*(2*H)=t := div_mul_cancel₀ t (ne_of_gt (mul_pos (by norm_num) hH))
have hwt : w<t := by nlinarith
refine ⟨w,hw,hwt,?_⟩
intro n m nc
have hc := sourceOccurrenceBudget_rate (mul_pos hκ ht) (mul_pos hκ hw) n m N nc Nc
have hr := routineBudget_rate (show 0≤(occurrenceDegree N Nc:ℝ)*(κ*w) by positivity) (by simpa only [neg_mul] using hc) rank
have hexp : -(κ*t)< -(rank:ℝ)*((occurrenceDegree N Nc:ℝ)*(κ*w)) := by
have he : (rank:ℝ)*(occurrenceDegree N Nc:ℝ)*w<t := by dsimp [H] at hwid hH hHt; nlinarith
nlinarith
have he := hr.eventually_small hexp
filter_upwards [he] with d hd
exact (le_abs_self _).trans (by simpa only [neg_neg] using hd)
end OracleCompiler
end LogConcaveSampling
end
end
end OAI
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