| import Mathlib |
| import OAI.Probability.SKGap.Entropy.EntropyBound |
|
|
| namespace OAI |
|
|
| namespace SKGap.GaussianStep |
| open scoped BigOperators |
| noncomputable section |
| variable {n : ℕ} |
|
|
| lemma cond_sub_const (p : Prior n) (hp : ∀x,0<p x) (f : Spin n→ℝ) (c : ℝ) |
| (i : Fin n) (x : Spin n) : cond p i (fun x => f x-c) x=cond p i f x-c := by |
| unfold cond |
| field_simp [(add_pos (hp x) (hp (flip i x))).ne'] |
| ring |
|
|
| lemma energy_sub_const (p : Prior n) (hp : ∀x,0<p x) (f : Spin n→ℝ) (c : ℝ) : |
| energy p (fun x => f x-c)=energy p f := by |
| unfold energy |
| simp_rw [cond_sub_const p hp f c,sub_sub_sub_cancel_right] |
|
|
| lemma var_center (p : Prior n) (f : Spin n→ℝ) : |
| var p (fun x => f x-avg p f)=var p f := by |
| unfold var |
| rw [avg_center] |
| simp only [sub_zero] |
|
|
| end |
| end SKGap.GaussianStep |
|
|
| namespace SKGap.GaussianHistory |
| open MeasureTheory ProbabilityTheory Real Set Filter |
| open scoped BigOperators ENNReal NNReal |
| open GaussianStep |
| noncomputable section |
| variable {n : ℕ} |
|
|
| lemma localization_scalar {den N E r c e s v w : ℝ} (hd : 0<den) (hN : 0<N) |
| (H : den*r*N≤den*(c*E+e*N)+N*(s*(v+w*E/N)+1)) : |
| (r-(e+(s*v+1)/den))*N≤(c+s*w/den)*E := by |
| have he : den*((r-(e+(s*v+1)/den))*N-(c+s*w/den)*E)= |
| den*r*N-(den*(c*E+e*N)+N*(s*(v+w*E/N)+1)) := by |
| field_simp |
| ring |
| have hh : den*((r-(e+(s*v+1)/den))*N-(c+s*w/den)*E)≤0 := by |
| rw [he] |
| linarith |
| have hz : (r-(e+(s*v+1)/den))*N-(c+s*w/den)*E ≤ 0 := by |
| by_contra hz |
| have hm := mul_pos hd (lt_of_not_ge hz) |
| linarith |
| linarith |
|
|
| def localizationLoss (n : ℕ) (t W A a b T : ℝ) : ℝ := |
| 16*(n:ℝ)^2*T*t+(exp (4*T/a)*(T*(W*(n:ℝ)+2*A*sqrt (n:ℝ)+4/a)+64*(n:ℝ)^2*T*t)+1)/(b*(n:ℝ)) |
|
|
| def localizationCost (n : ℕ) (A a b D L T : ℝ) : ℝ := |
| D*T+L+exp (4*T/a)*T*A*sqrt (n:ℝ)/(b*(n:ℝ)) |
|
|
| lemma centered_localization_bound (p : Prior n) (hp : ∀x,0<p x) (hn : 0<n) |
| (f : Spin n→ℝ) (hf : 0<avg p (fun x => (f x)^2)) (hc : avg p f=0) |
| {t : ℝ} (ht : 0≤t) (htn : 4*(n:ℝ)*t≤1) |
| (K : ℕ) (B G : ∀k,Set (History n k)) |
| (hB : ∀k,MeasurableSet (B k)) (hG : ∀k,MeasurableSet (G k)) |
| (E : Set (History n K)) (hE : MeasurableSet E) |
| (hprefix : ∀k<K,∀h:History n (k+1),h∈G (k+1)→h.1∈G k) |
| (hGE : (G K)ᶜ⊆E) |
| (D W A a b L T : ℝ) (hD : 0≤D) (hW : 0≤W) (hA : 0≤A) |
| (ha : 0<a) (hb : 0<b) (hL : 0≤L) (hDt : D*t≤1/2) (hT : (K:ℝ)*t=T) |
| (hsmall : ∀k<K,eventProbability p t (B k)≤exp (-a*(n:ℝ))) |
| (hEsmall : eventProbability p t E≤exp (-b*(n:ℝ))) |
| (hmean : ∀k<K,∀h:History n k,h∉B k → driftSquare p (squarePrior p f hf) t h ≤ |
| W*(n:ℝ)+A*sqrt (n:ℝ)*(2+energy (posterior p t h) f/avg (posterior p t h) (fun x => (f x)^2))) |
| (hcov : ∀k<K,∀h:History n k,h∈G k→ |
| (∑i,(∑x,posterior p t h x*(f x-avg (posterior p t h) f)*spinValue (x i))^2)≤ |
| D*(var (posterior p t h) f+energy (posterior p t h) f)) |
| (hterminal : ∀h:History n K,h∉E→var (posterior p t h) f≤L*energy (posterior p t h) f) : |
| (exp (-2*D*T)-localizationLoss n t W A a b T)*var p f≤ |
| localizationCost n A a b D L T*energy p f := by |
| have hh:=localization_bound p hp f hf ht htn K B G hB hG E hE hprefix hGE |
| D W A a b L T hD hW hA ha hb hL hDt hT hsmall hEsmall hmean hcov hterminal |
| have hv : var p f=avg p (fun x => (f x)^2) := by rw [var_eq,hc]; ring |
| rw [hv] at hh ⊢ |
| let v:=T*(W*(n:ℝ)+2*A*sqrt (n:ℝ)+4/a)+64*(n:ℝ)^2*T*t |
| let w:=T*A*sqrt (n:ℝ) |
| have hbody : T*(W*(n:ℝ)+A*sqrt (n:ℝ)*(2+energy p f/avg p (fun x => (f x)^2))+4/a)+ |
| 64*(n:ℝ)^2*T*t = v+w*energy p f/avg p (fun x => (f x)^2) := by dsimp [v,w]; ring |
| rw [hbody] at hh |
| simpa only [localizationLoss,localizationCost,v,w,mul_assoc] using |
| localization_scalar (mul_pos hb (by exact_mod_cast hn)) hf hh |
|
|
| theorem poincare_of_history_estimates (p : Prior n) (hp : ∀x,0<p x) (hn : 0<n) |
| {t : ℝ} (ht : 0≤t) (htn : 4*(n:ℝ)*t≤1) |
| (K : ℕ) (B G : ∀k,Set (History n k)) |
| (hB : ∀k,MeasurableSet (B k)) (hG : ∀k,MeasurableSet (G k)) |
| (E : Set (History n K)) (hE : MeasurableSet E) |
| (hprefix : ∀k<K,∀h:History n (k+1),h∈G (k+1)→h.1∈G k) |
| (hGE : (G K)ᶜ⊆E) |
| (D W A a b L T C : ℝ) (hD : 0≤D) (hW : 0≤W) (hA : 0≤A) |
| (ha : 0<a) (hb : 0<b) (hL : 0≤L) (hC : 0≤C) |
| (hDt : D*t≤1/2) (hT : (K:ℝ)*t=T) |
| (hsmall : ∀k<K,eventProbability p t (B k)≤exp (-a*(n:ℝ))) |
| (hEsmall : eventProbability p t E≤exp (-b*(n:ℝ))) |
| (hmean : ∀(f : Spin n→ℝ) (hf : 0<avg p (fun x => (f x)^2)),∀k<K,∀h:History n k,h∉B k → |
| driftSquare p (squarePrior p f hf) t h ≤ |
| W*(n:ℝ)+A*sqrt (n:ℝ)*(2+energy (posterior p t h) f/avg (posterior p t h) (fun x => (f x)^2))) |
| (hcov : ∀(f : Spin n→ℝ),∀k<K,∀h:History n k,h∈G k→ |
| (∑i,(∑x,posterior p t h x*(f x-avg (posterior p t h) f)*spinValue (x i))^2)≤ |
| D*(var (posterior p t h) f+energy (posterior p t h) f)) |
| (hterminal : ∀(f : Spin n→ℝ),∀h:History n K,h∉E→var (posterior p t h) f≤L*energy (posterior p t h) f) |
| (hLoss : localizationLoss n t W A a b T≤exp (-2*D*T)/2) |
| (hCost : localizationCost n A a b D L T≤C) : |
| ∀f : Spin n→ℝ,var p f≤(2*C/exp (-2*D*T))*energy p f := by |
| intro f |
| by_cases hv : 0<var p f |
| · let F:=fun x => f x-avg p f |
| have hF : 0<avg p (fun x => (F x)^2) := hv |
| have hh:=centered_localization_bound p hp hn F hF (avg_center p f) ht htn K B G hB hG E hE hprefix hGE |
| D W A a b L T hD hW hA ha hb hL hDt hT hsmall hEsmall (hmean F hF) (hcov F) (hterminal F) |
| dsimp [F] at hh |
| rw [var_center,energy_sub_const p hp] at hh |
| have hl:=mul_le_mul_of_nonneg_right hLoss hv.le |
| have hc:=mul_le_mul_of_nonneg_right hCost (energy_nonneg p f) |
| rw [div_mul_eq_mul_div] |
| apply (le_div_iff₀ (exp_pos (-2*D*T))).mpr |
| nlinarith |
| · have hz : var p f=0 := le_antisymm (le_of_not_gt hv) (var_nonneg p f) |
| rw [hz] |
| exact mul_nonneg (div_nonneg (by positivity) (exp_pos _).le) (energy_nonneg p f) |
|
|
| end |
| end SKGap.GaussianHistory |
|
|
| namespace SKGap.GaussianHistory |
| open MeasureTheory ProbabilityTheory Real Set Filter |
| open scoped BigOperators ENNReal NNReal |
| open GaussianStep |
| noncomputable section |
| variable {n : ℕ} |
|
|
| lemma eventProbability_lift (p : Prior n) {t : ℝ} (ht : 0≤t) {k : ℕ} |
| (E : Set (History n k)) (hE : MeasurableSet E) : |
| eventProbability p t (Prod.fst ⁻¹' E : Set (History n (k+1)))=eventProbability p t E := by |
| rw [eventProbability, ← integral_indicator (hE.preimage measurable_fst), |
| eventProbability, ← integral_indicator hE] |
| have hm : Measurable (E.indicator (fun _=> (1:ℝ))) := measurable_const.indicator hE |
| have he (h : History n k) : density p t h*E.indicator (fun _=>(1:ℝ)) h=E.indicator (density p t) h := by |
| by_cases hmem : h∈E <;> simp [hmem] |
| have hi : Integrable (fun h=>density p t h*E.indicator (fun _=> (1:ℝ)) h) (reference n k) := by |
| simpa only [he] using (integrable_density p t k).indicator hE |
| have H:=integral_lift p ht _ hm hi |
| have he' (h : History n (k+1)) : density p t h*E.indicator (fun _=>(1:ℝ)) h.1= |
| (Prod.fst ⁻¹' E).indicator (density p t (k:=k+1)) h := by |
| by_cases hmem : h.1∈E <;> simp [hmem] |
| simpa only [he,he'] using H |
|
|
| lemma eventProbability_union_le (p : Prior n) {t : ℝ} (_ht : 0≤t) {k : ℕ} |
| (E F : Set (History n k)) (hE : MeasurableSet E) (hF : MeasurableSet F) : |
| eventProbability p t (E∪F)≤eventProbability p t E+eventProbability p t F := by |
| have H := measureReal_union_le (μ := law p t k) E F |
| have he (A : Set (History n k)) (hA : MeasurableSet A) : |
| (law p t k).real A=eventProbability p t A := by |
| rw [Measure.real, ← eventProbability_eq p t A hA, ENNReal.toReal_ofReal (eventProbability_nonneg p t A)] |
| simpa only [he _ (hE.union hF),he E hE,he F hF] using H |
|
|
| def goodPrefix (B : ∀k, Set (History n k)) : ∀k,Set (History n k) |
| | 0 => (B 0)ᶜ |
| | k+1 => (Prod.fst ⁻¹' goodPrefix B k) ∩ (B (k+1))ᶜ |
|
|
| lemma measurable_goodPrefix (B : ∀k, Set (History n k)) (hB : ∀k,MeasurableSet (B k)) : |
| ∀k,MeasurableSet (goodPrefix B k) |
| | 0 => (hB 0).compl |
| | k+1 => ((measurable_goodPrefix B hB k).preimage measurable_fst).inter (hB (k+1)).compl |
|
|
| lemma goodPrefix_not_bad (B : ∀k, Set (History n k)) : |
| ∀k,∀h:History n k,h∈goodPrefix B k→h∉B k |
| | 0, _, h => h |
| | _+1, _, h => h.2 |
|
|
| lemma goodPrefix_previous (B : ∀k, Set (History n k)) (k : ℕ) |
| (h : History n (k+1)) (hh : h∈goodPrefix B (k+1)) : h.1∈goodPrefix B k := hh.1 |
|
|
| lemma goodPrefix_failure_bound (p : Prior n) {t : ℝ} (ht : 0≤t) |
| (B : ∀k,Set (History n k)) (hB : ∀k,MeasurableSet (B k)) : |
| ∀k,eventProbability p t (goodPrefix B k)ᶜ ≤ ∑i∈Finset.range (k+1),eventProbability p t (B i) |
| | 0 => by simp [goodPrefix] |
| | k+1 => by |
| have hE : (goodPrefix B (k+1))ᶜ = |
| (Prod.fst ⁻¹' (goodPrefix B k)ᶜ) ∪ B (k+1) := by |
| ext h |
| simp only [goodPrefix,mem_compl_iff,mem_preimage,mem_inter_iff,mem_union] |
| tauto |
| rw [hE,Finset.sum_range_succ] |
| have H : eventProbability p t (Prod.fst ⁻¹' (goodPrefix B k)ᶜ : Set (History n (k+1))) ≤ |
| ∑i∈Finset.range (k+1),eventProbability p t (B i) := by |
| rw [eventProbability_lift p ht _ (measurable_goodPrefix B hB k).compl] |
| exact goodPrefix_failure_bound p ht B hB k |
| exact (eventProbability_union_le p ht _ _ |
| ((measurable_goodPrefix B hB k).compl.preimage measurable_fst) (hB (k+1))).trans |
| (by linarith) |
|
|
| lemma eventProbability_mono (p : Prior n) {t : ℝ} (_ht : 0≤t) {k : ℕ} |
| {E F : Set (History n k)} (h : E⊆F) : eventProbability p t E≤eventProbability p t F := by |
| exact setIntegral_mono_set (integrable_density p t k).integrableOn |
| (ae_of_all _ (fun h=> (density_pos p t h).le)) (Filter.Eventually.of_forall h) |
|
|
| lemma bad_term_le_sum (p : Prior n) (t : ℝ) (B : ∀k,Set (History n k)) {k K : ℕ} |
| (hk : k≤K) : eventProbability p t (B k)≤∑i∈Finset.range (K+1),eventProbability p t (B i) := by |
| exact Finset.single_le_sum (fun i _=>eventProbability_nonneg p t (B i)) (by simpa using hk) |
|
|
| end |
| end SKGap.GaussianHistory |
|
|
| end OAI |
|
|
