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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 99 100 101 102 103 104 105 106 107 108 109 110 111 | import Mathlib
import OAI.Probability.LogConcave.OraclePrograms.Fresh
import OAI.Probability.LogConcave.OraclePrograms.IndependentSeq
import OAI.Probability.LogConcave.Sampling.AverageLipschitz
namespace OAI
section
noncomputable section
namespace LogConcaveSampling.OracleCompiler
open MeasureTheory ProbabilityTheory Function
open scoped Classical NNReal
variable {d : ℕ}
def topProgram (S : ReservedProgram d) (h : ℝ) (N J : ℕ) :=
(Program.fresh (fun _ => proximalProgram S h) N).independentSeq
(Program.fresh (fun j => halvingProgram S (descentScale h j)) J)
def topSeedLaw (S : ReservedProgram d) (N J : ℕ) :=
((Measure.dirac (0:Point d)).prod (Measure.pi (fun _ : Fin N => gaussianTape d S.full.slots))).prod
(Measure.pi (fun _ : Fin J => gaussianTape d S.full.slots))
instance topSeedLaw_probability (S : ReservedProgram d) (N J : ℕ) :
IsProbabilityMeasure (topSeedLaw S N J) := by unfold topSeedLaw; infer_instance
lemma topProgram_law (S : ReservedProgram d) (h : ℝ) (N J : ℕ)
{V : Point d → ℝ} (hV : Admissible V) :
(topSeedLaw S N J).map ((topProgram S h N J).run V)=
changingChain (fun j => approximateHalvingKernel S (descentScale h j) V hV)
(markovChain (approximateProximalKernel S h V hV) (Measure.dirac 0) N) J := by
let (j : ℕ) : IsMarkovKernel (approximateHalvingKernel S (descentScale h j) V hV) :=
approximateHalvingKernel_markov _ _ _
let := approximateProximalKernel_markov S h hV
have hVm := hV.continuous_firstOrderReply.measurable
rw [topProgram,topSeedLaw,Program.independentSeq_law _ _ hVm,
Program.fresh_law (fun _ => proximalProgram S h) hVm]
rw [stateLaw_eq_changingChain _ _ _ (fun _ => (proximalProgram S h).measurable_run V hVm)
(fun _ => approximateProximalKernel S h V hV) (fun _ => approximateProximalKernel_apply S h hV),
changingChain_constant]
rw [Program.fresh_law _ hVm]
exact stateLaw_eq_changingChain _ _ _
(fun j => (halvingProgram S (descentScale h j)).measurable_run V hVm)
_ (fun j => approximateHalvingKernel_program S (descentScale h j) hV) J
lemma affine_correct_from_uniform (S : ℝ → ℝ → ReservedProgram d) {q K e ηmin : ℝ}
(hc : ∀{F : Point d → ℝ} {lam : ℝ≥0},Primitive F lam → 0<lam → (lam:ℝ)≤1 →
SampleCorrect F lam q K e ηmin S)
{V : Point d → ℝ} (hV : Admissible V) {h : ℝ} (hh : 0<h) (hh1 : 2*h≤1) (y : Point d) :
SampleCorrect (affinePotential V y (Real.sqrt h)) (2*h) q K e ηmin S := by
have hp : (0:ℝ)<2*(Real.sqrt h)^2 := by rw [Real.sq_sqrt hh.le]; positivity
have hl : 2*(Real.sqrt h)^2≤1 := by rwa [Real.sq_sqrt hh.le]
have he := hc (hV.affinePrimitive y (Real.sqrt h)) (show (0:ℝ≥0)<⟨2*(Real.sqrt h)^2,by positivity⟩ from hp) hl
change SampleCorrect _ (2*(Real.sqrt h)^2) q K e ηmin S at he
simpa only [Real.sq_sqrt hh.le] using he
theorem topProgram_accuracy (S : ℝ → ℝ → ReservedProgram d) {q K e ηmin h : ℝ}
(hh : 0<h) (hq : 2*h≤q) (hq1 : q<1) (he : 0≤e) (hη : ηmin≤1)
(hres : (S 1 1).reserve=1/2)
(hc : ∀{F : Point d → ℝ} {lam : ℝ≥0},Primitive F lam → 0<lam → (lam:ℝ)≤1 →
SampleCorrect F lam q K e ηmin S)
(n J : ℕ) {V : Point d → ℝ} (hV : Admissible V) :
TVAtMost ((topSeedLaw (S 1 1) (n+1) J).map ((topProgram (S 1 1) h (n+1) J).run V))
(gibbs V)
(((n+1)*18+J*10)*e*Real.sqrt d+
((1+Real.sqrt h)*Real.sqrt d)/((1+h)^(n+1)*Real.sqrt h)+
2*(2*Real.sqrt (descentScale h J)+descentScale h J)*d) := by
let T := S 1 1
have hh1 : h≤1 := by linarith
have hs : Real.sqrt h≤1 := by simpa using Real.sqrt_le_sqrt hh1
have hloc := approximateProximalKernel_tv S hV hh hq hq1 he hη hres
(affine_correct_from_uniform S hc hV hh (by linarith))
let := approximateProximalKernel_markov T h hV
let := proximalKernel_markov hV hh
let := noisedLaw_probability hV h
let := noisedLaw_probability hV (descentScale h J)
let (j : ℕ) : IsMarkovKernel (approximateHalvingKernel T (descentScale h j) V hV) :=
approximateHalvingKernel_markov _ _ _
have hp := proximal_chain_perturbation hV hh (approximateProximalKernel T h V hV)
(a:=2*e*Real.sqrt d) (b:=4*e*Real.sqrt h) (by positivity) (by positivity)
(fun x => by convert! hloc x using 1; ring) (n+1)
have hp' : TVAtMost (markovChain (approximateProximalKernel T h V hV) (Measure.dirac 0) (n+1))
(markovChain (proximalKernel V hV h) (Measure.dirac 0) (n+1))
((n+1)*(18*e*Real.sqrt d)) := by
apply hp.mono
push_cast
apply mul_le_mul_of_nonneg_left _ (by positivity)
have hss : Real.sqrt h*(1+Real.sqrt h)≤2 := by nlinarith [Real.sqrt_nonneg h]
nlinarith [mul_le_mul_of_nonneg_right hss (by positivity : 0≤e*Real.sqrt d)]
have hm := hp'.trans (proximal_chain_mixing hV hh n)
have hd := halving_chain_error hV hh hh1 he
(fun j => approximateHalvingKernel T (descentScale h j) V hV)
(fun j x => approximateHalvingKernel_tv S hV (descentScale_pos hh j)
(by linarith [descentScale_le hh.le j]) hq1 he hη hres
(affine_correct_from_uniform S hc hV (descentScale_pos hh j)
(by linarith [descentScale_le hh.le j])) x) J
have ht := ((changingChain_tv _ hm J).trans hd).trans
(noisedLaw_tv hV (descentScale_pos hh J))
rw [topProgram_law _ _ _ _ hV]
apply ht.mono
ring_nf
rfl
end LogConcaveSampling.OracleCompiler
end
end
end OAI
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