| 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 |
|
|
