| 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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