File size: 3,591 Bytes
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 | import Mathlib
import OAI.Probability.LogConcave.JetEstimates.ClosedWireLintegralBound
import OAI.Probability.LogConcave.JetEstimates.Reindex
import OAI.Probability.LogConcave.LowerBounds.HardEigenOutput
import OAI.Probability.LogConcave.OraclePrograms.NoisyCenterSquaredError
import OAI.Probability.LogConcave.Sampling.MatrixColor
import OAI.Probability.LogConcave.OraclePrograms.DampingCoefficient
import OAI.Probability.LogConcave.Sampling.HarmonicMeanTaylorRms
import OAI.Probability.LogConcave.Dynamics.StationaryIntegral
import OAI.Probability.LogConcave.Numerics.TaylorRemainderRms
import OAI.Probability.LogConcave.OraclePrograms.ShiftInvariant
import OAI.Probability.LogConcave.OraclePrograms.MeanTreeBase
import OAI.Probability.LogConcave.OraclePrograms.PairDistBounds
import OAI.Probability.LogConcave.OraclePrograms.SampleCorrect
import OAI.Probability.LogConcave.Sampling.MixingEnvelope
namespace OAI
/-! Subpolynomial upper bounds and logarithmic lower bounds for exact first-order sampling, with optimal dimension exponent zero. -/
section
noncomputable section
namespace LogConcaveSampling
open Filter MeasureTheory
open scoped Topology ENNReal
lemma CanSample.queryComplexity_le {d q : ℕ} (hq : CanSample d q) :
(queryComplexity d).toENNReal≤ENNReal.ofReal (q:ℝ) := by
have hle : queryComplexity d≤(q:ℕ∞) := sInf_le ⟨q,hq,rfl⟩
simpa only [ENat.toENNReal_coe,ENNReal.ofReal_natCast] using ENat.toENNReal_mono hle
theorem source_subpolynomial_upper :
∀ ε : ℝ,0<ε → ∃ C : ℝ,∀ d : ℕ,2≤d →
(queryComplexity d).toENNReal≤ENNReal.ofReal (C*(d:ℝ)^ε) := by
intro ε hε
obtain ⟨d₀,hd₀⟩ := eventually_atTop.mp (eventually_subpolynomial_sampler hε)
let C : ℝ := (10*2^d₀:ℕ)+1
have hC : 1≤C := by dsimp [C]; have := (Nat.cast_nonneg (10*2^d₀) : (0:ℝ)≤_); linarith
refine ⟨C,?_⟩
intro d hd
have hd1 : (1:ℝ)≤d := by exact_mod_cast (by omega : 1≤d)
have hp : 1≤(d:ℝ)^ε := Real.one_le_rpow hd1 hε.le
by_cases hlarge : d₀≤d
· obtain ⟨q,hq,hqbound⟩ := hd₀ d hlarge
exact hq.queryComplexity_le.trans (ENNReal.ofReal_le_ofReal
(hqbound.trans (by nlinarith)))
· have hsmall : d≤d₀ := by omega
have hpow : (10*2^d:ℕ)≤10*2^d₀ := by gcongr
have hq : ((10*2^d:ℕ):ℝ)≤C := by dsimp [C]; exact_mod_cast (by omega : 10*2^d≤10*2^d₀+1)
exact (finite_dimension_sampler d).queryComplexity_le.trans
(ENNReal.ofReal_le_ofReal (hq.trans (by nlinarith)))
theorem source_exponent_zero : ExponentBound 0 := by
refine ⟨le_rfl,?_⟩
intro ε hε
obtain ⟨d₀,hd₀⟩ := eventually_atTop.mp (eventually_subpolynomial_sampler hε)
refine ⟨d₀,?_⟩
intro d hd
obtain ⟨q,hq,hqbound⟩ := hd₀ d hd
simpa only [zero_add] using hq.queryComplexity_le.trans (ENNReal.ofReal_le_ofReal hqbound)
theorem source_gammaStar_zero : gammaStar=0 := by
apply le_antisymm
· exact csInf_le ⟨0,fun _ h => h.1⟩ source_exponent_zero
· exact le_csInf ⟨0,source_exponent_zero⟩ (fun _ h => h.1)
theorem exact_source_main :
(∀ ε : ℝ, 0 < ε → ∃ C : ℝ, ∀ d : ℕ, 2 ≤ d →
(queryComplexity d).toENNReal ≤ ENNReal.ofReal (C * (d : ℝ) ^ ε)) ∧
(∃ c : ℝ, 0 < c ∧ ∃ d₀ : ℕ, 2 ≤ d₀ ∧ ∀ d : ℕ, d₀ ≤ d →
ENNReal.ofReal (c * Real.log (d : ℝ)) ≤ (queryComplexity d).toENNReal) ∧
gammaStar = 0 := by
exact ⟨source_subpolynomial_upper,LowerBound.source_logarithmic_lower,source_gammaStar_zero⟩
end LogConcaveSampling
end
end
end OAI
|