| import Mathlib |
| import OAI.Probability.LogConcave.Sampling.ChangingChain |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace LogConcaveSampling |
| open MeasureTheory ProbabilityTheory Function |
| open scoped Classical |
|
|
| variable {d : ℕ} |
|
|
| lemma zero_average_lipschitz {μ : Measure (Point d)} [IsProbabilityMeasure μ] |
| (hi : Integrable (fun x => ‖x‖) μ) {f : Point d → ℝ} |
| (hf : Measurable f) {C L : ℝ} (hb : ∀x,|f x|≤C) |
| (hl : ∀x y,|f x-f y|≤L*dist x y) : |
| |f 0-(∫x,f x ∂μ)|≤L*(∫x,‖x‖ ∂μ) := by |
| have he : f 0-(∫x,f x ∂μ)=∫x,(f 0-f x) ∂μ := by |
| rw [integral_sub (integrable_const _) (bounded_integrable hf hb)] |
| simp |
| rw [he,←Real.norm_eq_abs,←integral_const_mul] |
| apply (norm_integral_le_integral_norm _).trans |
| apply integral_mono ((integrable_const (f 0)).sub (bounded_integrable hf hb)).norm (hi.const_mul L) |
| intro x |
| simpa only [Pi.sub_apply,Real.norm_eq_abs,dist_zero_left] using hl 0 x |
|
|
| lemma proximal_iterate_lipschitz {V : Point d → ℝ} (hV : Admissible V) {h : ℝ} (hh : 0<h) |
| {f : Point d → ℝ} {L C : ℝ} (hf : Measurable f) (hC : 0≤C) |
| (hb : ∀x,|f x|≤C) (hL : 0≤L) (hl : ∀x y,|f x-f y|≤L*dist x y) (n : ℕ) : |
| ∀x y,|markovAction (proximalKernel V hV h) n f x-markovAction (proximalKernel V hV h) n f y| |
| ≤(L/(1+h)^n)*dist x y := by |
| let := proximalKernel_markov hV hh |
| induction n with |
| | zero => simpa only [markovAction,iterate_zero,id_eq,pow_zero,div_one] using hl |
| | succ n ih => |
| have he := proximal_action_lipschitz hV hh |
| (markovAction_measurable _ hf n) hC (markovAction_bounded _ hC hb n) (by positivity) ih |
| intro x y |
| simp only [markovAction,iterate_succ_apply'] |
| convert! he x y using 1 |
| simp only [pow_succ,div_div] |
|
|
| lemma proximal_iterate_smoothing {V : Point d → ℝ} (hV : Admissible V) {h : ℝ} (hh : 0<h) |
| {f : Point d → ℝ} (hf : Measurable f) (hb : ∀x,0≤f x ∧ f x≤1) (n : ℕ) : |
| ∀x y,|markovAction (proximalKernel V hV h) (n+1) f x- |
| markovAction (proximalKernel V hV h) (n+1) f y|≤ |
| (1/((1+h)^(n+1)*Real.sqrt h))*dist x y := by |
| let := proximalKernel_markov hV hh |
| have hbf (x) : |f x|≤1 := by rw [abs_of_nonneg (hb x).1]; exact (hb x).2 |
| have he := proximal_iterate_lipschitz hV hh (kernelAction_measurable _ hf) (by norm_num : (0:ℝ)≤1) |
| (kernelAction_bounded _ (by norm_num : (0:ℝ)≤1) hbf) |
| (by positivity : 0≤1/((1+h)*Real.sqrt h)) (proximal_action_smoothing hV hh hf hb) n |
| intro x y |
| simp only [markovAction,iterate_succ_apply] |
| convert! he x y using 1 |
| congr 1 |
| simp only [div_div,pow_succ] |
| congr 1 |
| ring |
|
|
| theorem proximal_chain_mixing {V : Point d → ℝ} (hV : Admissible V) {h : ℝ} (hh : 0<h) |
| (n : ℕ) : |
| TVAtMost (markovChain (proximalKernel V hV h) (Measure.dirac 0) (n+1)) (noisedLaw V h) |
| (((1+Real.sqrt h)*Real.sqrt d)/((1+h)^(n+1)*Real.sqrt h)) := by |
| let := proximalKernel_markov hV hh |
| let := noisedLaw_probability hV h |
| intro A hA |
| let f : Point d → ℝ := A.indicator (fun _ => 1) |
| have hf : Measurable f := measurable_const.indicator hA |
| have hb (x) : 0≤f x ∧ f x≤1 := by by_cases hx : x∈A <;> simp [f,hx] |
| have hbf (x) : |f x|≤1 := by rw [abs_of_nonneg (hb x).1]; exact (hb x).2 |
| have hact := markovChain_integral (proximalKernel V hV h) (Measure.dirac 0) hf (C:=1) (by norm_num) hbf (n+1) |
| have hint := markovChain_integral (proximalKernel V hV h) (noisedLaw V h) hf (C:=1) (by norm_num) hbf (n+1) |
| rw [markovChain_invariant _ _ (proximalKernel_invariant hV hh)] at hint |
| have he := zero_average_lipschitz (noisedLaw_first_moment hV h).1 |
| (markovAction_measurable _ hf (n+1)) (markovAction_bounded _ (by norm_num : (0:ℝ)≤1) hbf (n+1)) |
| (proximal_iterate_smoothing hV hh hf hb n) |
| simp only [integral_dirac] at hact |
| rw [←hint,←hact] at he |
| have hif (μ : Measure (Point d)) : (∫x,f x ∂μ)=μ.real A := by |
| simp only [f,integral_indicator hA,setIntegral_const,smul_eq_mul,mul_one] |
| simp only [hif] at he |
| apply he.trans |
| calc |
| _ ≤ (1/((1+h)^(n+1)*Real.sqrt h))*((1+Real.sqrt h)*Real.sqrt d) := |
| mul_le_mul_of_nonneg_left (noisedLaw_first_moment hV h).2 (by positivity) |
| _ = _ := by ring |
|
|
| end LogConcaveSampling |
|
|
| end |
|
|
| end |
|
|
| section |
|
|
| noncomputable section |
| namespace LogConcaveSampling |
| open MeasureTheory ProbabilityTheory Function |
| open scoped Classical |
|
|
| variable {d : ℕ} |
|
|
| lemma proximal_chain_observable {V : Point d → ℝ} (hV : Admissible V) {h : ℝ} (hh : 0<h) |
| {f : Point d → ℝ} {L C : ℝ} (hf : Measurable f) (hC : 0≤C) |
| (hb : ∀x,|f x|≤C) (hL : 0≤L) (hl : ∀x y,|f x-f y|≤L*dist x y) (n : ℕ) : |
| |(∫x,f x ∂markovChain (proximalKernel V hV h) (Measure.dirac 0) n)- |
| (∫x,f x ∂noisedLaw V h)|≤L*((1+Real.sqrt h)*Real.sqrt d) := by |
| let := proximalKernel_markov hV hh |
| let := noisedLaw_probability hV h |
| have hact := markovChain_integral (proximalKernel V hV h) (Measure.dirac 0) hf hC hb n |
| have hint := markovChain_integral (proximalKernel V hV h) (noisedLaw V h) hf hC hb n |
| rw [markovChain_invariant _ _ (proximalKernel_invariant hV hh)] at hint |
| have he := zero_average_lipschitz (noisedLaw_first_moment hV h).1 |
| (markovAction_measurable _ hf n) (markovAction_bounded _ hC hb n) |
| (proximal_iterate_lipschitz hV hh hf hC hb hL hl n) |
| simp only [integral_dirac] at hact |
| rw [←hint,←hact] at he |
| apply he.trans |
| have hpow : (1:ℝ)≤(1+h)^n := one_le_pow₀ (by linarith) |
| have hdv : L/(1+h)^n≤L := (div_le_self hL hpow) |
| exact mul_le_mul hdv (noisedLaw_first_moment hV h).2 (integral_nonneg (fun _ => norm_nonneg _)) hL |
|
|
| lemma TVAtMost.cap_one {E : Type*} [MeasurableSpace E] |
| {μ ν : Measure E} [IsProbabilityMeasure μ] [IsProbabilityMeasure ν] |
| {e : ℝ} (he : TVAtMost μ ν e) : TVAtMost μ ν (min e 1) := by |
| intro A hA |
| refine le_min (he A hA) ?_ |
| exact abs_sub_le_iff.mpr ⟨by linarith [measureReal_le_one (μ:=μ) (s:=A),measureReal_nonneg (μ:=ν) (s:=A)], |
| by linarith [measureReal_le_one (μ:=ν) (s:=A),measureReal_nonneg (μ:=μ) (s:=A)]⟩ |
|
|
| lemma proximal_local_cap_lipschitz {a b : ℝ} (hb : 0≤b) : |
| ∀x y : Point d,|min (a+b*‖x‖) 1-min (a+b*‖y‖) 1|≤b*dist x y := by |
| intro x y |
| have ht := abs_min_sub_min_le_max (a+b*‖x‖) (1:ℝ) (a+b*‖y‖) 1 |
| simp only [sub_self,abs_zero] at ht |
| apply ht.trans |
| have he : a+b*‖x‖-(a+b*‖y‖)=b*(‖x‖-‖y‖) := by ring |
| rw [he,abs_mul,abs_of_nonneg hb] |
| apply max_le |
| · simpa only [dist_eq_norm] using mul_le_mul_of_nonneg_left (abs_norm_sub_norm_le x y) hb |
| · positivity |
|
|
| theorem proximal_chain_perturbation {V : Point d → ℝ} (hV : Admissible V) {h : ℝ} (hh : 0<h) |
| (κ : Kernel (Point d) (Point d)) [IsMarkovKernel κ] {a b : ℝ} (ha : 0≤a) (hb : 0≤b) |
| (he : ∀x,TVAtMost (κ x) (proximalKernel V hV h x) (a+b*‖x‖)) (n : ℕ) : |
| TVAtMost (markovChain κ (Measure.dirac 0) n) |
| (markovChain (proximalKernel V hV h) (Measure.dirac 0) n) |
| (n*(a+2*b*((1+Real.sqrt h)*Real.sqrt d))) := by |
| let := proximalKernel_markov hV hh |
| let := noisedLaw_probability hV h |
| let e : Point d → ℝ := fun x => min (a+b*‖x‖) 1 |
| have hm : Measurable e := by dsimp [e]; fun_prop |
| have he0 (x) : 0≤e x := le_min (by positivity) (by norm_num) |
| have he1 (x) : |e x|≤1 := by rw [abs_of_nonneg (he0 x)]; exact min_le_right _ _ |
| apply markovChain_perturbation κ (proximalKernel V hV h) (Measure.dirac 0) |
| (fun x => (he x).cap_one) (fun n => bounded_integrable hm he1) |
| intro k |
| have hstat : (∫x,e x ∂noisedLaw V h)≤a+b*((1+Real.sqrt h)*Real.sqrt d) := by |
| calc |
| _ ≤ ∫x,(a+b*‖x‖) ∂noisedLaw V h := integral_mono (bounded_integrable hm he1) |
| ((integrable_const a).add ((noisedLaw_first_moment hV h).1.const_mul b)) |
| (fun x => min_le_left _ _) |
| _ = a+b*(∫x,‖x‖ ∂noisedLaw V h) := by |
| rw [integral_add (integrable_const a) ((noisedLaw_first_moment hV h).1.const_mul b),integral_const_mul] |
| simp |
| _ ≤ _ := add_le_add_right (mul_le_mul_of_nonneg_left (noisedLaw_first_moment hV h).2 hb) a |
| have hc := proximal_chain_observable hV hh hm (by norm_num : (0:ℝ)≤1) he1 hb |
| (proximal_local_cap_lipschitz hb) k |
| linarith [(le_abs_self ((∫x,e x ∂markovChain (proximalKernel V hV h) (Measure.dirac 0) k)- |
| (∫x,e x ∂noisedLaw V h)))] |
|
|
| end LogConcaveSampling |
|
|
| end |
|
|
| end |
|
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| end OAI |
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