| import Mathlib |
| import OAI.Probability.SKGap.Localization.DriftSquare |
| import OAI.Probability.SKGap.Localization.Envelope |
| import OAI.Probability.SKGap.Localization.AffineUpper |
|
|
| namespace OAI |
|
|
| namespace SKGap.GaussianHistory |
| open MeasureTheory ProbabilityTheory Real Set Filter |
| open scoped BigOperators ENNReal NNReal |
| open GaussianStep DiscreteRecurrence |
| noncomputable section |
| variable {n : ℕ} |
|
|
| lemma entropy_bound (p : Prior n) (hp : ∀x,0<p x) (f : Spin n→ℝ) |
| (hf : 0<avg p (fun x => (f x)^2)) {t : ℝ} (ht : 0≤t) (htn : 4*(n:ℝ)*t≤1) |
| (K : ℕ) (B : ∀k,Set (History n k)) (hB : ∀k,MeasurableSet (B k)) |
| (W A a T : ℝ) (hW : 0≤W) (hA : 0≤A) (ha : 0<a) (hT : (K:ℝ)*t=T) |
| (hsmall : ∀k<K,eventProbability p t (B k)≤exp (-a*(n:ℝ))) |
| (hgood : ∀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))) : |
| entropy p (squarePrior p f hf) t K≤ |
| exp (4*T/a)*(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) := by |
| have hh:=affine_upper (entropy p (squarePrior p f hf) t) |
| (show 0≤4*t/a by positivity) |
| (show 0≤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^2 by have := energy_nonneg p f; positivity) |
| (entropy_zero p (squarePrior p f hf) t).le K |
| (fun k hk => by |
| simpa only [add_assoc] using entropy_good_step p hp f hf ht htn k (B k) (hB k) W A a hW hA ha |
| (hsmall k hk) (hgood k hk)) |
| have he : (K:ℝ)*(4*t/a)=4*T/a := by rw [←hT]; ring |
| rw [he] at hh |
| convert hh using 1 |
| rw [←hT] |
| ring |
|
|
| lemma envelope_bound (p : Prior n) (hp : ∀x,0<p x) (f : Spin n→ℝ) |
| {t : ℝ} (ht : 0≤t) (htn : 4*(n:ℝ)*t≤1) (K : ℕ) |
| (G : ∀k,Set (History n k)) (hG : ∀k,MeasurableSet (G k)) |
| (hprefix : ∀k<K,∀h:History n (k+1),h∈G (k+1)→h.1∈G k) |
| (D T : ℝ) (hD : 0≤D) (hDt : D*t≤1/2) (hT : (K:ℝ)*t=T) |
| (hgood : ∀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)) : |
| exp (-2*D*T)*var p f≤ envelopeIntegral p t (G K) f+D*T*energy p f+ |
| 16*(n:ℝ)^2*T*t*avg p (fun x => (f x)^2) := by |
| have hh:=affine_lower (fun k => envelopeIntegral p t (G k) f) |
| (show 0≤1-D*t by linarith) (show 1-D*t≤1 by nlinarith) |
| (show 0≤D*t*energy p f+16*(n:ℝ)^2*t^2*avg p (fun x => (f x)^2) by |
| have := energy_nonneg p f |
| have := avg_nonneg p (fun x => (f x)^2) (fun _ => sq_nonneg _) |
| positivity) K (fun k hk => by |
| have hs:=envelope_step_bound p hp (G k) (hG k) (G (k+1)) (hG (k+1)) |
| (hprefix k hk) f ht htn D hD (hgood k hk) |
| linarith) |
| have hz:=mul_le_mul_of_nonneg_left (variance_le_envelope_zero p t (G 0) f) |
| (pow_nonneg (show 0≤1-D*t by linarith) K) |
| have he:=mul_le_mul_of_nonneg_right (pow_one_sub_lower (mul_nonneg hD ht) hDt K) (var_nonneg p f) |
| have heq : -2*(K:ℝ)*(D*t)= -2*D*T := by rw [←hT]; ring |
| rw [heq] at he |
| rw [←hT] |
| rw [←hT] at he |
| nlinarith |
|
|
| theorem localization_bound (p : Prior n) (hp : ∀x,0<p x) (f : Spin n→ℝ) |
| (hf : 0<avg p (fun x => (f x)^2)) {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) : |
| b*(n:ℝ)*exp (-2*D*T)*var p f ≤ |
| b*(n:ℝ)*((D*T+L)*energy p f+16*(n:ℝ)^2*T*t*avg p (fun x => (f x)^2))+ |
| avg p (fun x => (f x)^2)*(exp (4*T/a)*(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)+1) := by |
| have hv:=envelope_bound p hp f ht htn K G hG hprefix D T hD hDt hT hcov |
| have hterm:=envelope_terminal_bound p hp f hf ht (G K) E (hG K) hE hGE L hL hterminal |
| have hentropy:=entropy_bound p hp f hf ht htn K B hB W A a T hW hA ha hT hsmall hmean |
| have he:=entropy_event_bound p (squarePrior p f hf) hp ht K E hE (b*(n:ℝ)) |
| (by simpa only [neg_mul] using hEsmall) |
| have heN:=mul_le_mul_of_nonneg_left he hf.le |
| have hvb:=mul_le_mul_of_nonneg_left hv (show 0≤b*(n:ℝ) by positivity) |
| have htb:=mul_le_mul_of_nonneg_left hterm (show 0≤b*(n:ℝ) by positivity) |
| have hHN:=mul_le_mul_of_nonneg_left hentropy hf.le |
| nlinarith |
|
|
| end |
| end SKGap.GaussianHistory |
|
|
| end OAI |
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